Theorem opnzi 3972

Description: An ordered pair is nonempty if the arguments are sets (it is also inhabited; see opm 3971). (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 3971	. . 3 |- ( E. x x e. <. A , B >. <-> ( A e. _V /\ B e. _V ) )
4	1, 2, 3	mpbir2an 849	. 2 |- E. x x e. <. A , B >.
5		n0r 3234	. 2 |- ( E. x x e. <. A , B >. -> <. A , B >. =/= (/) )
6	4, 5	ax-mp 7	1 |- <. A , B >. =/= (/)

Syntax hints:   E.wex 1381   e. wcel 1393   =/= wne 2204   _Vcvv 2557   (/)c0 3224   <.cop 3378

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384

This theorem is referenced by:  0nelxp  4372  0neqopab  5550