Theorem opprc 3854

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 3652	. 2 |- <. A , B >. = { x | ( A e. _V /\ B e. _V /\ x e. { { A } , { A , B } } ) }
2		3simpa 997	. . . . 5 |- ( ( A e. _V /\ B e. _V /\ x e. { { A } , { A , B } } ) -> ( A e. _V /\ B e. _V ) )
3	2	con3i 633	. . . 4 |- ( -. ( A e. _V /\ B e. _V ) -> -. ( A e. _V /\ B e. _V /\ x e. { { A } , { A , B } } ) )
4	3	alrimiv 1898	. . 3 |- ( -. ( A e. _V /\ B e. _V ) -> A. x -. ( A e. _V /\ B e. _V /\ x e. { { A } , { A , B } } ) )
5		abeq0 3499	. . 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 2252	1 |- ( -. ( A e. _V /\ B e. _V ) -> <. A , B >. = (/) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   /\ w3a 981   A.wal 1371   = wceq 1373   e. wcel 2178   {cab 2193   _Vcvv 2776   (/)c0 3468   {csn 3643   {cpr 3644   <.cop 3646

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778  df-dif 3176  df-nul 3469  df-op 3652

This theorem is referenced by:  opprc1  3855  opprc2  3856  ovprc  6003