Theorem prnz 2459

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 2450	. 2 |- A e. {A, B}
3		ne0i 2286	. 2 |- (A e. {A, B} -> {A, B} =/= (/))
4	2, 3	ax-mp 7	1 |- {A, B} =/= (/)

Syntax hints:   e. wcel 958   =/= wne 1585  Vcvv 1811  (/)c0 2280  {cpr 2410

This theorem is referenced by:  opprc1b 2796  fr2nr 2925  fiint 4559  fiintOLD 4560  shincl 9331  chincl 9383

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-un 2050  df-nul 2281  df-sn 2412  df-pr 2413

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