Genesis 2:23 And Adam said, This is now bone of my bones, and flesh of my flesh: she shall be called Woman, because she was taken out of Man. Notice there is no blood before the fall.
Luke 24:39 Behold my hands and my feet, that it is I myself: handle me, and see; for a spirit hath not flesh and bones, as ye see me have. Flesh and bones but no blood in the resurrected body of the Savior.
1 Corinthians 15
50 Now this I say, brethren, that flesh and blood cannot inherit the kingdom of God; neither doth corruption inherit incorruption.
51 Behold, I shew you a mystery; We shall not all sleep, but we shall all be changed,
52 In a moment, in the twinkling of an eye, at the last trump: for the trumpet shall sound, and the dead shall be raised incorruptible, and we shall be changed.
53 For this corruptible must put on incorruption, and this mortal must put on immortality.
At the resurrection, our bodies will be changed. Blood is what's corrupt, mortal. Blood cannot inherit the kingdom of God. Why did the Lord shed His blood on the cross? It's a picture of ridding the body of sin, what makes us corrupt. Under the law, they bled the lamb before the sacrifice. It was a picture of getting rid of sin.