Theorem opeluu 4498

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 3739	. . 3 |- A e. { A , B }
3		opeluu.2	. . . . 5 |- B e. _V
4	1, 3	opi2 4278	. . . 4 |- { A , B } e. <. A , B >.
5		elunii 3855	. . . 4 |- ( ( { A , B } e. <. A , B >. /\ <. A , B >. e. C ) -> { A , B } e. U. C )
6	4, 5	mpan 424	. . 3 |- ( <. A , B >. e. C -> { A , B } e. U. C )
7		elunii 3855	. . 3 |- ( ( A e. { A , B } /\ { A , B } e. U. C ) -> A e. U. U. C )
8	2, 6, 7	sylancr 414	. 2 |- ( <. A , B >. e. C -> A e. U. U. C )
9	3	prid2 3740	. . 3 |- B e. { A , B }
10		elunii 3855	. . 3 |- ( ( B e. { A , B } /\ { A , B } e. U. C ) -> B e. U. U. C )
11	9, 6, 10	sylancr 414	. 2 |- ( <. A , B >. e. C -> B e. U. U. C )
12	8, 11	jca 306	1 |- ( <. A , B >. e. C -> ( A e. U. U. C /\ B e. U. U. C ) )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2176   _Vcvv 2772   {cpr 3634   <.cop 3636   U.cuni 3850

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4163  ax-pr 4254

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-un 3170  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851

This theorem is referenced by:  asymref  5069  wrdexb  11008