Theorem tpnz 4785

Description: An unordered triple containing a set is not empty. (Contributed by NM, 10-Apr-1994.)

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	tpid1 4774	. 2 ⊢ 𝐴 ∈ {𝐴, 𝐵, 𝐶}
3	2	ne0ii 4351	1 ⊢ {𝐴, 𝐵, 𝐶} ≠ ∅

Syntax hints:   ∈ wcel 2107   ≠ wne 2939  Vcvv 3479  ∅c0 4340  {ctp 4636

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-tru 1541  df-fal 1551  df-ex 1778  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-v 3481  df-dif 3967  df-un 3969  df-nul 4341  df-sn 4633  df-pr 4635  df-tp 4637

This theorem is referenced by: (None)