Theorem opnzi 4236

Description: An ordered pair is nonempty if the arguments are sets (it is also inhabited; see opm 4235). (Contributed by Mario Carneiro, 26-Apr-2015.)

Hypotheses

Ref	Expression
opth1.1	|- A e. _V
opth1.2	|- B e. _V

Assertion

Ref	Expression
opnzi	|- <. A , B >. =/= (/)

Proof of Theorem opnzi

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		opth1.1	. . 3 |- A e. _V
2		opth1.2	. . 3 |- B e. _V
3		opm 4235	. . 3 |- ( E. x x e. <. A , B >. <-> ( A e. _V /\ B e. _V ) )
4	1, 2, 3	mpbir2an 942	. 2 |- E. x x e. <. A , B >.
5		n0r 3437	. 2 |- ( E. x x e. <. A , B >. -> <. A , B >. =/= (/) )
6	4, 5	ax-mp 5	1 |- <. A , B >. =/= (/)

Syntax hints:   E.wex 1492   e. wcel 2148   =/= wne 2347   _Vcvv 2738   (/)c0 3423   <.cop 3596

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-pow 4175

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-v 2740  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602

This theorem is referenced by:  0nelxp  4655  0neqopab  5920