Theorem tpnzd 4718

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		tpid1g 4708	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴, 𝐵, 𝐶})
3		ne0i 4303	. 2 ⊢ (𝐴 ∈ {𝐴, 𝐵, 𝐶} → {𝐴, 𝐵, 𝐶} ≠ ∅)
4	1, 2, 3	3syl 18	1 ⊢ (𝜑 → {𝐴, 𝐵, 𝐶} ≠ ∅)

Syntax hints:   → wi 4   ∈ wcel 2113   ≠ wne 3019  ∅c0 4294  {ctp 4574

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-v 3499  df-dif 3942  df-un 3944  df-nul 4295  df-sn 4571  df-pr 4573  df-tp 4575

This theorem is referenced by:  raltpd  4719  fr3nr  7497  limsupequzlem  42009  etransclem48  42574