Theorem tpnzd 4805

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 4794	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴, 𝐵, 𝐶})
3		ne0i 4364	. 2 ⊢ (𝐴 ∈ {𝐴, 𝐵, 𝐶} → {𝐴, 𝐵, 𝐶} ≠ ∅)
4	1, 2, 3	3syl 18	1 ⊢ (𝜑 → {𝐴, 𝐵, 𝐶} ≠ ∅)

Syntax hints:   → wi 4   ∈ wcel 2108   ≠ wne 2946  ∅c0 4352  {ctp 4652

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-v 3490  df-dif 3979  df-un 3981  df-nul 4353  df-sn 4649  df-pr 4651  df-tp 4653

This theorem is referenced by:  raltpd  4806  fr3nr  7807  limsupequzlem  45643  etransclem48  46203