Theorem oprcl 3641

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 2635	. 2 |- ( C e. <. A , B >. -> E. y y e. <. A , B >. )
2		df-op 3450	. . . . . . 7 |- <. A , B >. = { x | ( A e. _V /\ B e. _V /\ x e. { { A } , { A , B } } ) }
3	2	eleq2i 2154	. . . . . 6 |- ( y e. <. A , B >. <-> y e. { x | ( A e. _V /\ B e. _V /\ x e. { { A } , { A , B } } ) } )
4		df-clab 2075	. . . . . 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 182	. . . . 5 |- ( y e. <. A , B >. <-> [ y / x ] ( A e. _V /\ B e. _V /\ x e. { { A } , { A , B } } ) )
6		3simpa 940	. . . . . 6 |- ( ( A e. _V /\ B e. _V /\ x e. { { A } , { A , B } } ) -> ( A e. _V /\ B e. _V ) )
7	6	sbimi 1694	. . . . 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 119	. . . 4 |- ( y e. <. A , B >. -> [ y / x ] ( A e. _V /\ B e. _V ) )
9		nfv 1466	. . . . 5 |- F/ x ( A e. _V /\ B e. _V )
10	9	sbf 1707	. . . 4 |- ( [ y / x ] ( A e. _V /\ B e. _V ) <-> ( A e. _V /\ B e. _V ) )
11	8, 10	sylib 120	. . 3 |- ( y e. <. A , B >. -> ( A e. _V /\ B e. _V ) )
12	11	exlimiv 1534	. 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 102   /\ w3a 924   E.wex 1426   e. wcel 1438   [wsb 1692   {cab 2074   _Vcvv 2619   {csn 3441   {cpr 3442   <.cop 3444

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-5 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-3an 926  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-v 2621  df-op 3450

This theorem is referenced by:  opth1  4054  opth  4055  0nelop  4066