Theorem prnz 1847

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 1841	. 2 |- A e. {A, B}
3		n0i 1712	. 2 |- (A e. {A, B} -> -. {A, B} = (/))
4	2, 3	ax-mp 6	1 |- -. {A, B} = (/)

Syntax hints:  -. wn 1   = wceq 1091   e. wcel 1092  Vcvv 1348  (/)c0 1707  {cpr 1809

This theorem is referenced by:  opprc1b 1906  fr2nr 2177  fiint 3445  shincl 5332  chincl 5382

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-nul 1708  df-sn 1811  df-pr 1812

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