Theorem opeluu 4721

Description: Each member of an ordered pair belongs to the union of the union of a class to which the ordered pair belongs. Lemma 3D of [Enderton] p. 41. (Contributed by NM, 31-Mar-1995.) (Revised by Mario Carneiro, 27-Feb-2016.)

Hypotheses

Ref	Expression
opeluu.1
opeluu.2

Assertion

Ref	Expression
opeluu

Proof of Theorem opeluu

Step	Hyp	Ref	Expression
1		opeluu.1	. . . 4
2	1	prid1 4138	. . 3
3		opeluu.2	. . . . 5
4	1, 3	opi2 4720	. . . 4
5		elunii 4254	. . . 4
6	4, 5	mpan 670	. . 3
7		elunii 4254	. . 3
8	2, 6, 7	sylancr 663	. 2
9	3	prid2 4139	. . 3
10		elunii 4254	. . 3
11	9, 6, 10	sylancr 663	. 2
12	8, 11	jca 532	1

Syntax hints:  ->wi 4  /\wa 369  e.wcel 1818  cvv 3109  {cpr 4031  <.cop 4035  U.cuni 4249

This theorem is referenced by:  asymref  5388  asymref2  5389  wrdexb  12558

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-v 3111  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-uni 4250