Theorem tpsscd 32516

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 4702	. . . 4 ⊢ {𝐴, 𝐵, 𝐶} = {𝐵, 𝐶, 𝐴}
3		tprot 4702	. . . 4 ⊢ {𝐵, 𝐶, 𝐴} = {𝐶, 𝐴, 𝐵}
4	2, 3	eqtri 2754	. . 3 ⊢ {𝐴, 𝐵, 𝐶} = {𝐶, 𝐴, 𝐵}
5		tpsscd.2	. . 3 ⊢ (𝜑 → {𝐴, 𝐵, 𝐶} ⊆ 𝐷)
6	4, 5	eqsstrrid 3974	. 2 ⊢ (𝜑 → {𝐶, 𝐴, 𝐵} ⊆ 𝐷)
7	1, 6	tpssad 32514	1 ⊢ (𝜑 → 𝐶 ∈ 𝐷)

Syntax hints:   → wi 4   ∈ wcel 2111   ⊆ wss 3902  {ctp 4580

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-v 3438  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4284  df-sn 4577  df-pr 4579  df-tp 4581

This theorem is referenced by:  constrlccllem  33761  constrcccllem  33762