Theorem tpnz 4746

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 4735	. 2 ⊢ 𝐴 ∈ {𝐴, 𝐵, 𝐶}
3	2	ne0ii 4298	1 ⊢ {𝐴, 𝐵, 𝐶} ≠ ∅

Syntax hints:   ∈ wcel 2143   ≠ wne 2958  Vcvv 3455  ∅c0 4287  {ctp 4594

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-v 3457  df-dif 3909  df-un 3911  df-nul 4288  df-sn 4591  df-pr 4593  df-tp 4595

This theorem is referenced by: (None)