Theorem prnz 2463

Description: A pair containing a set is not empty.

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	pri1 2454	. 2 |- A e. {A, B}
3		ne0i 2289	. 2 |- (A e. {A, B} -> {A, B} =/= (/))
4	2, 3	ax-mp 7	1 |- {A, B} =/= (/)

Syntax hints:   e. wcel 960   =/= wne 1588  Vcvv 1814  (/)c0 2283  {cpr 2414

This theorem is referenced by:  opprc1b 2802  fr2nr 2931  fiint 4572  fiintOLD 4573  shincl 9326  chincl 9378

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-v 1815  df-dif 2052  df-un 2053  df-nul 2284  df-sn 2416  df-pr 2417

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