Theorem tpsscd 33184

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 4706	. . . 4 ⊢ {𝐴, 𝐵, 𝐶} = {𝐵, 𝐶, 𝐴}
3		tprot 4706	. . . 4 ⊢ {𝐵, 𝐶, 𝐴} = {𝐶, 𝐴, 𝐵}
4	2, 3	eqtri 2783	. . 3 ⊢ {𝐴, 𝐵, 𝐶} = {𝐶, 𝐴, 𝐵}
5		tpsscd.2	. . 3 ⊢ (𝜑 → {𝐴, 𝐵, 𝐶} ⊆ 𝐷)
6	4, 5	eqsstrrid 3947	. 2 ⊢ (𝜑 → {𝐶, 𝐴, 𝐵} ⊆ 𝐷)
7	1, 6	tpssad 33182	1 ⊢ (𝜑 → 𝐶 ∈ 𝐷)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2144   ⊆ wss 3895  {ctp 4588

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-v 3452  df-ss 3896  df-dif 4047  df-un 4151  df-nul 4279  df-sn 4585  df-pr 4587  df-tp 4589

This theorem is used by:  constrlccllem  34431  constrcccllem  34432