Theorem prnz 3490

Description: A pair containing a set is not empty. (Contributed by NM, 9-Apr-1994.)

Hypothesis

Ref	Expression
prnz.1	|- A e. _V

Assertion

Ref	Expression
prnz	|- { A , B } =/= (/)

Proof of Theorem prnz

Step	Hyp	Ref	Expression
1		prnz.1	. . 3 |- A e. _V
2	1	prid1 3476	. 2 |- A e. { A , B }
3		ne0i 3230	. 2 |- ( A e. { A , B } -> { A , B } =/= (/) )
4	2, 3	ax-mp 7	1 |- { A , B } =/= (/)

Syntax hints:   e. wcel 1393   =/= wne 2204   _Vcvv 2557   (/)c0 3224   {cpr 3376

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-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  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-nul 3225  df-sn 3381  df-pr 3382

This theorem is referenced by:  prnzg  3492