Theorem oprcl 3886

Description: If an ordered pair has an element, then its arguments are sets. (Contributed by Mario Carneiro, 26-Apr-2015.)

Assertion

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

Proof of Theorem oprcl

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex2 2819	. 2 |- ( C e. <. A , B >. -> E. y y e. <. A , B >. )
2		df-op 3678	. . . . . . 7 |- <. A , B >. = { x | ( A e. _V /\ B e. _V /\ x e. { { A } , { A , B } } ) }
3	2	eleq2i 2298	. . . . . 6 |- ( y e. <. A , B >. <-> y e. { x | ( A e. _V /\ B e. _V /\ x e. { { A } , { A , B } } ) } )
4		df-clab 2218	. . . . . 6 |- ( y e. { x | ( A e. _V /\ B e. _V /\ x e. { { A } , { A , B } } ) } <-> [ y / x ] ( A e. _V /\ B e. _V /\ x e. { { A } , { A , B } } ) )
5	3, 4	bitri 184	. . . . 5 |- ( y e. <. A , B >. <-> [ y / x ] ( A e. _V /\ B e. _V /\ x e. { { A } , { A , B } } ) )
6		3simpa 1020	. . . . . 6 |- ( ( A e. _V /\ B e. _V /\ x e. { { A } , { A , B } } ) -> ( A e. _V /\ B e. _V ) )
7	6	sbimi 1812	. . . . 5 |- ( [ y / x ] ( A e. _V /\ B e. _V /\ x e. { { A } , { A , B } } ) -> [ y / x ] ( A e. _V /\ B e. _V ) )
8	5, 7	sylbi 121	. . . 4 |- ( y e. <. A , B >. -> [ y / x ] ( A e. _V /\ B e. _V ) )
9		nfv 1576	. . . . 5 |- F/ x ( A e. _V /\ B e. _V )
10	9	sbf 1825	. . . 4 |- ( [ y / x ] ( A e. _V /\ B e. _V ) <-> ( A e. _V /\ B e. _V ) )
11	8, 10	sylib 122	. . 3 |- ( y e. <. A , B >. -> ( A e. _V /\ B e. _V ) )
12	11	exlimiv 1646	. 2 |- ( E. y y e. <. A , B >. -> ( A e. _V /\ B e. _V ) )
13	1, 12	syl 14	1 |- ( C e. <. A , B >. -> ( A e. _V /\ B e. _V ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1004   E.wex 1540   [wsb 1810   e. wcel 2202   {cab 2217   _Vcvv 2802   {csn 3669   {cpr 3670   <.cop 3672

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-5 1495  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-v 2804  df-op 3678

This theorem is referenced by:  opth1  4328  opth  4329  0nelop  4340