Theorem tpnz 4457

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 4448	. 2 ⊢ 𝐴 ∈ {𝐴, 𝐵, 𝐶}
3	2	ne0ii 4067	1 ⊢ {𝐴, 𝐵, 𝐶} ≠ ∅

Syntax hints:   ∈ wcel 2140   ≠ wne 2933  Vcvv 3341  ∅c0 4059  {ctp 4326

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ne 2934  df-v 3343  df-dif 3719  df-un 3721  df-nul 4060  df-sn 4323  df-pr 4325  df-tp 4327

This theorem is referenced by: (None)