Theorem tpnzd 4760

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 4749	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴, 𝐵, 𝐶})
3		ne0i 4321	. 2 ⊢ (𝐴 ∈ {𝐴, 𝐵, 𝐶} → {𝐴, 𝐵, 𝐶} ≠ ∅)
4	1, 2, 3	3syl 18	1 ⊢ (𝜑 → {𝐴, 𝐵, 𝐶} ≠ ∅)

Syntax hints:   → wi 4   ∈ wcel 2107   ≠ wne 2931  ∅c0 4313  {ctp 4610

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-ne 2932  df-v 3465  df-dif 3934  df-un 3936  df-nul 4314  df-sn 4607  df-pr 4609  df-tp 4611

This theorem is referenced by:  raltpd  4761  fr3nr  7774  limsupequzlem  45694  etransclem48  46254