Theorem tpnz 1848

Description: A triplet containing a set is not empty.

Hypothesis

Ref	Expression
tpnz.1	⊢ A ∈ V

Assertion

Ref	Expression
tpnz	⊢ ¬ {A, B, C} = ∅

Proof of Theorem tpnz

Step	Hyp	Ref	Expression
1		tpnz.1	. . 3 ⊢ A ∈ V
2	1	tpi1 1843	. 2 ⊢ A ∈ {A, B, C}
3		n0i 1712	. 2 ⊢ (A ∈ {A, B, C} → ¬ {A, B, C} = ∅)
4	2, 3	ax-mp 6	1 ⊢ ¬ {A, B, C} = ∅

Syntax hints:  ¬ wn 1   = wceq 1091   ∈ wcel 1092  Vcvv 1348  ∅c0 1707  {ctp 1813

This theorem is referenced by:  fr3nr 2178

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-in 1491  df-ss 1492  df-nul 1708  df-sn 1811  df-pr 1812  df-tp 1814

metamath.org