Theorem tpsscd 32627

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 4708	. . . 4 ⊢ {𝐴, 𝐵, 𝐶} = {𝐵, 𝐶, 𝐴}
3		tprot 4708	. . . 4 ⊢ {𝐵, 𝐶, 𝐴} = {𝐶, 𝐴, 𝐵}
4	2, 3	eqtri 2760	. . 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 32625	1 ⊢ (𝜑 → 𝐶 ∈ 𝐷)

Syntax hints:   → wi 4   ∈ wcel 2114   ⊆ wss 3903  {ctp 4586

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-v 3444  df-dif 3906  df-un 3908  df-ss 3920  df-nul 4288  df-sn 4583  df-pr 4585  df-tp 4587

This theorem is referenced by:  constrlccllem  33930  constrcccllem  33931