Theorem oprcl 3729

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 2702	. 2 |- ( C e. <. A , B >. -> E. y y e. <. A , B >. )
2		df-op 3536	. . . . . . 7 |- <. A , B >. = { x | ( A e. _V /\ B e. _V /\ x e. { { A } , { A , B } } ) }
3	2	eleq2i 2206	. . . . . 6 |- ( y e. <. A , B >. <-> y e. { x | ( A e. _V /\ B e. _V /\ x e. { { A } , { A , B } } ) } )
4		df-clab 2126	. . . . . 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 183	. . . . 5 |- ( y e. <. A , B >. <-> [ y / x ] ( A e. _V /\ B e. _V /\ x e. { { A } , { A , B } } ) )
6		3simpa 978	. . . . . 6 |- ( ( A e. _V /\ B e. _V /\ x e. { { A } , { A , B } } ) -> ( A e. _V /\ B e. _V ) )
7	6	sbimi 1737	. . . . 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 120	. . . 4 |- ( y e. <. A , B >. -> [ y / x ] ( A e. _V /\ B e. _V ) )
9		nfv 1508	. . . . 5 |- F/ x ( A e. _V /\ B e. _V )
10	9	sbf 1750	. . . 4 |- ( [ y / x ] ( A e. _V /\ B e. _V ) <-> ( A e. _V /\ B e. _V ) )
11	8, 10	sylib 121	. . 3 |- ( y e. <. A , B >. -> ( A e. _V /\ B e. _V ) )
12	11	exlimiv 1577	. 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 103   /\ w3a 962   E.wex 1468   e. wcel 1480   [wsb 1735   {cab 2125   _Vcvv 2686   {csn 3527   {cpr 3528   <.cop 3530

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-3an 964  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-v 2688  df-op 3536

This theorem is referenced by:  opth1  4158  opth  4159  0nelop  4170