Theorem tpsscd 32689

Description: If an ordered triple is a subset of a class, the third element of the triple is an element of that class. (Contributed by Thierry Arnoux, 2-Nov-2025.)

Hypotheses

Ref	Expression
tpsscd.1	⊢ (𝜑 → 𝐶 ∈ 𝑉)
tpsscd.2	⊢ (𝜑 → {𝐴, 𝐵, 𝐶} ⊆ 𝐷)

Assertion

Ref	Expression
tpsscd	⊢ (𝜑 → 𝐶 ∈ 𝐷)

Proof of Theorem tpsscd

Step	Hyp	Ref	Expression
1		tpsscd.1	. 2 ⊢ (𝜑 → 𝐶 ∈ 𝑉)
2		tprot 4707	. . . 4 ⊢ {𝐴, 𝐵, 𝐶} = {𝐵, 𝐶, 𝐴}
3		tprot 4707	. . . 4 ⊢ {𝐵, 𝐶, 𝐴} = {𝐶, 𝐴, 𝐵}
4	2, 3	eqtri 2784	. . 3 ⊢ {𝐴, 𝐵, 𝐶} = {𝐶, 𝐴, 𝐵}
5		tpsscd.2	. . 3 ⊢ (𝜑 → {𝐴, 𝐵, 𝐶} ⊆ 𝐷)
6	4, 5	eqsstrrid 3975	. 2 ⊢ (𝜑 → {𝐶, 𝐴, 𝐵} ⊆ 𝐷)
7	1, 6	tpssad 32687	1 ⊢ (𝜑 → 𝐶 ∈ 𝐷)

Syntax hints:   → wi 4   ∈ wcel 2141   ⊆ wss 3904  {ctp 4585

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-v 3455  df-dif 3907  df-un 3909  df-ss 3921  df-nul 4286  df-sn 4582  df-pr 4584  df-tp 4586

This theorem is referenced by:  constrlccllem  34011  constrcccllem  34012