Theorem tpssbd 32867

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

Hypotheses

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

Assertion

Ref	Expression
tpssbd	⊢ (𝜑 → 𝐵 ∈ 𝐷)

Proof of Theorem tpssbd

Step	Hyp	Ref	Expression
1		tpssbd.1	. 2 ⊢ (𝜑 → 𝐵 ∈ 𝑉)
2		tprot 4716	. . 3 ⊢ {𝐴, 𝐵, 𝐶} = {𝐵, 𝐶, 𝐴}
3		tpssbd.2	. . 3 ⊢ (𝜑 → {𝐴, 𝐵, 𝐶} ⊆ 𝐷)
4	2, 3	eqsstrrid 3977	. 2 ⊢ (𝜑 → {𝐵, 𝐶, 𝐴} ⊆ 𝐷)
5	1, 4	tpssad 32866	1 ⊢ (𝜑 → 𝐵 ∈ 𝐷)

Syntax hints:   → wi 4   ∈ wcel 2143   ⊆ wss 3906  {ctp 4594

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-v 3457  df-dif 3909  df-un 3911  df-ss 3923  df-nul 4288  df-sn 4591  df-pr 4593  df-tp 4595

This theorem is referenced by:  constrlccllem  34124  constrcccllem  34125