Theorem oprcl 3573

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 2570	. 2 |- ( C e. <. A , B >. -> E. y y e. <. A , B >. )
2		df-op 3384	. . . . . . 7 |- <. A , B >. = { x | ( A e. _V /\ B e. _V /\ x e. { { A } , { A , B } } ) }
3	2	eleq2i 2104	. . . . . 6 |- ( y e. <. A , B >. <-> y e. { x | ( A e. _V /\ B e. _V /\ x e. { { A } , { A , B } } ) } )
4		df-clab 2027	. . . . . 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 173	. . . . 5 |- ( y e. <. A , B >. <-> [ y / x ] ( A e. _V /\ B e. _V /\ x e. { { A } , { A , B } } ) )
6		3simpa 901	. . . . . 6 |- ( ( A e. _V /\ B e. _V /\ x e. { { A } , { A , B } } ) -> ( A e. _V /\ B e. _V ) )
7	6	sbimi 1647	. . . . 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 114	. . . 4 |- ( y e. <. A , B >. -> [ y / x ] ( A e. _V /\ B e. _V ) )
9		nfv 1421	. . . . 5 |- F/ x ( A e. _V /\ B e. _V )
10	9	sbf 1660	. . . 4 |- ( [ y / x ] ( A e. _V /\ B e. _V ) <-> ( A e. _V /\ B e. _V ) )
11	8, 10	sylib 127	. . 3 |- ( y e. <. A , B >. -> ( A e. _V /\ B e. _V ) )
12	11	exlimiv 1489	. 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 97   /\ w3a 885   E.wex 1381   e. wcel 1393   [wsb 1645   {cab 2026   _Vcvv 2557   {csn 3375   {cpr 3376   <.cop 3378

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-3an 887  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-v 2559  df-op 3384

This theorem is referenced by:  opth1  3973  opth  3974  0nelop  3985