Theorem opeluu 4200

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 3498	. . 3 |- A e. { A , B }
3		opeluu.2	. . . . 5 |- B e. _V
4	1, 3	opi2 3988	. . . 4 |- { A , B } e. <. A , B >.
5		elunii 3606	. . . 4 |- ( ( { A , B } e. <. A , B >. /\ <. A , B >. e. C ) -> { A , B } e. U. C )
6	4, 5	mpan 414	. . 3 |- ( <. A , B >. e. C -> { A , B } e. U. C )
7		elunii 3606	. . 3 |- ( ( A e. { A , B } /\ { A , B } e. U. C ) -> A e. U. U. C )
8	2, 6, 7	sylancr 405	. 2 |- ( <. A , B >. e. C -> A e. U. U. C )
9	3	prid2 3499	. . 3 |- B e. { A , B }
10		elunii 3606	. . 3 |- ( ( B e. { A , B } /\ { A , B } e. U. C ) -> B e. U. U. C )
11	9, 6, 10	sylancr 405	. 2 |- ( <. A , B >. e. C -> B e. U. U. C )
12	8, 11	jca 300	1 |- ( <. A , B >. e. C -> ( A e. U. U. C /\ B e. U. U. C ) )

Syntax hints:   -> wi 4   /\ wa 102   e. wcel 1433   _Vcvv 2601   {cpr 3399   <.cop 3401   U.cuni 3601

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-un 2977  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602

This theorem is referenced by:  asymref  4730