Theorem tpsscd 32636

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 4688	. . . 4 ⊢ {𝐴, 𝐵, 𝐶} = {𝐵, 𝐶, 𝐴}
3		tprot 4688	. . . 4 ⊢ {𝐵, 𝐶, 𝐴} = {𝐶, 𝐴, 𝐵}
4	2, 3	eqtri 2763	. . 3 ⊢ {𝐴, 𝐵, 𝐶} = {𝐶, 𝐴, 𝐵}
5		tpsscd.2	. . 3 ⊢ (𝜑 → {𝐴, 𝐵, 𝐶} ⊆ 𝐷)
6	4, 5	eqsstrrid 3961	. 2 ⊢ (𝜑 → {𝐶, 𝐴, 𝐵} ⊆ 𝐷)
7	1, 6	tpssad 32634	1 ⊢ (𝜑 → 𝐶 ∈ 𝐷)

Syntax hints:   → wi 4   ∈ wcel 2119   ⊆ wss 3890  {ctp 4566

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-ne 2936  df-v 3434  df-dif 3893  df-un 3895  df-ss 3907  df-nul 4269  df-sn 4563  df-pr 4565  df-tp 4567

This theorem is referenced by:  constrlccllem  33944  constrcccllem  33945