Theorem opnzi 4370

Description: An ordered pair is nonempty if the arguments are sets (it is also inhabited; see opm 4369). (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 4369	. . 3 |- ( E. x x e. <. A , B >. <-> ( A e. _V /\ B e. _V ) )
4	1, 2, 3	mpbir2an 955	. 2 |- E. x x e. <. A , B >.
5		n0r 3535	. 2 |- ( E. x x e. <. A , B >. -> <. A , B >. =/= (/) )
6	4, 5	ax-mp 5	1 |- <. A , B >. =/= (/)

Syntax hints:   E.wex 1545   e. wcel 2209   =/= wne 2420   _Vcvv 2821   (/)c0 3520   <.cop 3708

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-v 2823  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714

This theorem is referenced by:  0nelxp  4797  0neqopab  6123