Theorem tpnz 4707

Description: A triplet 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 4697	. 2 ⊢ 𝐴 ∈ {𝐴, 𝐵, 𝐶}
3	2	ne0ii 4302	1 ⊢ {𝐴, 𝐵, 𝐶} ≠ ∅

Syntax hints:   ∈ wcel 2110   ≠ wne 3016  Vcvv 3494  ∅c0 4290  {ctp 4564

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-v 3496  df-dif 3938  df-un 3940  df-nul 4291  df-sn 4561  df-pr 4563  df-tp 4565

This theorem is referenced by: (None)