Theorem tpnzd 4739

Description: An unordered triple containing a set is not empty. (Contributed by Thierry Arnoux, 8-Apr-2019.)

Hypothesis

Ref	Expression
tpnzd.1	⊢ (𝜑 → 𝐴 ∈ 𝑉)

Assertion

Ref	Expression
tpnzd	⊢ (𝜑 → {𝐴, 𝐵, 𝐶} ≠ ∅)

Proof of Theorem tpnzd

Step	Hyp	Ref	Expression
1		tpnzd.1	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
2		tpid1g 4728	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴, 𝐵, 𝐶})
3		ne0i 4293	. 2 ⊢ (𝐴 ∈ {𝐴, 𝐵, 𝐶} → {𝐴, 𝐵, 𝐶} ≠ ∅)
4	1, 2, 3	3syl 18	1 ⊢ (𝜑 → {𝐴, 𝐵, 𝐶} ≠ ∅)

Syntax hints:   → wi 4   ∈ wcel 2142   ≠ wne 2957  ∅c0 4285  {ctp 4586

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1099  df-tru 1563  df-fal 1573  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-v 3456  df-dif 3907  df-un 3909  df-nul 4286  df-sn 4583  df-pr 4585  df-tp 4587

This theorem is referenced by:  raltpd  4740  fr3nr  7755  limsupequzlem  46296  etransclem48  46856