Theorem tpnz 2464

Description: A triplet containing a set is not empty.

Hypothesis

Ref	Expression
tpnz.1	|- A e. V

Assertion

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

Proof of Theorem tpnz

Step	Hyp	Ref	Expression
1		tpnz.1	. . 3 |- A e. V
2	1	tpi1 2459	. 2 |- A e. {A, B, C}
3		ne0i 2289	. 2 |- (A e. {A, B, C} -> {A, B, C} =/= (/))
4	2, 3	ax-mp 7	1 |- {A, B, C} =/= (/)

Syntax hints:   e. wcel 960   =/= wne 1588  Vcvv 1814  (/)c0 2283  {ctp 2418

This theorem is referenced by:  fr3nr 2932

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-in 2054  df-ss 2056  df-nul 2284  df-sn 2416  df-pr 2417  df-tp 2419

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