Theorem opeluu 4939

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	|- A e. _V
opeluu.2	|- B e. _V

Assertion

Ref	Expression
opeluu	|- ( <. A , B >. e. C -> ( A e. U. U. C /\ B e. U. U. C ) )

Proof of Theorem opeluu

Step	Hyp	Ref	Expression
1		opeluu.1	. . . 4 |- A e. _V
2	1	prid1 4297	. . 3 |- A e. { A , B }
3		opeluu.2	. . . . 5 |- B e. _V
4	1, 3	opi2 4938	. . . 4 |- { A , B } e. <. A , B >.
5		elunii 4441	. . . 4 |- ( ( { A , B } e. <. A , B >. /\ <. A , B >. e. C ) -> { A , B } e. U. C )
6	4, 5	mpan 706	. . 3 |- ( <. A , B >. e. C -> { A , B } e. U. C )
7		elunii 4441	. . 3 |- ( ( A e. { A , B } /\ { A , B } e. U. C ) -> A e. U. U. C )
8	2, 6, 7	sylancr 695	. 2 |- ( <. A , B >. e. C -> A e. U. U. C )
9	3	prid2 4298	. . 3 |- B e. { A , B }
10		elunii 4441	. . 3 |- ( ( B e. { A , B } /\ { A , B } e. U. C ) -> B e. U. U. C )
11	9, 6, 10	sylancr 695	. 2 |- ( <. A , B >. e. C -> B e. U. U. C )
12	8, 11	jca 554	1 |- ( <. A , B >. e. C -> ( A e. U. U. C /\ B e. U. U. C ) )

Syntax hints:   -> wi 4   /\ wa 384   e. wcel 1990   _Vcvv 3200   {cpr 4179   <.cop 4183   U.cuni 4436

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437

This theorem is referenced by:  asymref  5512  asymref2  5513  wrdexb  13316