Theorem opprc 3883

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 3678	. 2 |- <. A , B >. = { x | ( A e. _V /\ B e. _V /\ x e. { { A } , { A , B } } ) }
2		3simpa 1020	. . . . 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 1922	. . 3 |- ( -. ( A e. _V /\ B e. _V ) -> A. x -. ( A e. _V /\ B e. _V /\ x e. { { A } , { A , B } } ) )
5		abeq0 3525	. . 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 2276	1 |- ( -. ( A e. _V /\ B e. _V ) -> <. A , B >. = (/) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   /\ w3a 1004   A.wal 1395   = wceq 1397   e. wcel 2202   {cab 2217   _Vcvv 2802   (/)c0 3494   {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-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-dif 3202  df-nul 3495  df-op 3678

This theorem is referenced by:  opprc1  3884  opprc2  3885  ovprc  6053