Theorem opprc 3904

Description: Expansion of an ordered pair when either member is a proper class. (Contributed by Mario Carneiro, 26-Apr-2015.)

Assertion

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

Proof of Theorem opprc

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-op 3698	. 2 |- <. A , B >. = { x | ( A e. _V /\ B e. _V /\ x e. { { A } , { A , B } } ) }
2		3simpa 1021	. . . . 5 |- ( ( A e. _V /\ B e. _V /\ x e. { { A } , { A , B } } ) -> ( A e. _V /\ B e. _V ) )
3	2	con3i 637	. . . 4 |- ( -. ( A e. _V /\ B e. _V ) -> -. ( A e. _V /\ B e. _V /\ x e. { { A } , { A , B } } ) )
4	3	alrimiv 1923	. . 3 |- ( -. ( A e. _V /\ B e. _V ) -> A. x -. ( A e. _V /\ B e. _V /\ x e. { { A } , { A , B } } ) )
5		abeq0 3539	. . 3 |- ( { x | ( A e. _V /\ B e. _V /\ x e. { { A } , { A , B } } ) } = (/) <-> A. x -. ( A e. _V /\ B e. _V /\ x e. { { A } , { A , B } } ) )
6	4, 5	sylibr 134	. 2 |- ( -. ( A e. _V /\ B e. _V ) -> { x | ( A e. _V /\ B e. _V /\ x e. { { A } , { A , B } } ) } = (/) )
7	1, 6	eqtrid 2277	1 |- ( -. ( A e. _V /\ B e. _V ) -> <. A , B >. = (/) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   /\ w3a 1005   A.wal 1396   = wceq 1398   e. wcel 2203   {cab 2218   _Vcvv 2813   (/)c0 3508   {csn 3689   {cpr 3690   <.cop 3692

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-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2815  df-dif 3213  df-nul 3509  df-op 3698

This theorem is referenced by:  opprc1  3905  opprc2  3906  ovprc  6086