Theorem tpnz 4783

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 4772	. 2 ⊢ 𝐴 ∈ {𝐴, 𝐵, 𝐶}
3	2	ne0ii 4337	1 ⊢ {𝐴, 𝐵, 𝐶} ≠ ∅

Syntax hints:   ∈ wcel 2105   ≠ wne 2939  Vcvv 3473  ∅c0 4322  {ctp 4632

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ne 2940  df-v 3475  df-dif 3951  df-un 3953  df-nul 4323  df-sn 4629  df-pr 4631  df-tp 4633

This theorem is referenced by: (None)