Theorem tpnzd 4531

Description: A triplet 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		tpid3g 4524	. . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐵, 𝐶, 𝐴})
3		tprot 4501	. . 3 ⊢ {𝐴, 𝐵, 𝐶} = {𝐵, 𝐶, 𝐴}
4	2, 3	syl6eleqr 2916	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴, 𝐵, 𝐶})
5		ne0i 4149	. 2 ⊢ (𝐴 ∈ {𝐴, 𝐵, 𝐶} → {𝐴, 𝐵, 𝐶} ≠ ∅)
6	1, 4, 5	3syl 18	1 ⊢ (𝜑 → {𝐴, 𝐵, 𝐶} ≠ ∅)

Syntax hints:   → wi 4   ∈ wcel 2166   ≠ wne 2998  ∅c0 4143  {ctp 4400

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-ext 2802

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3or 1114  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-clab 2811  df-cleq 2817  df-clel 2820  df-nfc 2957  df-ne 2999  df-v 3415  df-dif 3800  df-un 3802  df-nul 4144  df-sn 4397  df-pr 4399  df-tp 4401

This theorem is referenced by:  raltpd  4532  fr3nr  7239  limsupequzlem  40748  etransclem48  41292