Theorem prssbd 32678

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

Hypotheses

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

Assertion

Ref	Expression
prssbd	⊢ (𝜑 → 𝐵 ∈ 𝐶)

Proof of Theorem prssbd

Step	Hyp	Ref	Expression
1		simpr 488	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ V) → 𝐴 ∈ V)
2		prssbd.1	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑉)
3	2	adantr 484	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ V) → 𝐵 ∈ 𝑉)
4		prssbd.2	. . . . 5 ⊢ (𝜑 → {𝐴, 𝐵} ⊆ 𝐶)
5	4	adantr 484	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ V) → {𝐴, 𝐵} ⊆ 𝐶)
6		prssg 4776	. . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ 𝑉) → ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) ↔ {𝐴, 𝐵} ⊆ 𝐶))
7	6	biimpar 481	. . . 4 ⊢ (((𝐴 ∈ V ∧ 𝐵 ∈ 𝑉) ∧ {𝐴, 𝐵} ⊆ 𝐶) → (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶))
8	1, 3, 5, 7	syl21anc 848	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ V) → (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶))
9	8	simprd 499	. 2 ⊢ ((𝜑 ∧ 𝐴 ∈ V) → 𝐵 ∈ 𝐶)
10		prprc1 4723	. . . . 5 ⊢ (¬ 𝐴 ∈ V → {𝐴, 𝐵} = {𝐵})
11	10	adantl 485	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ V) → {𝐴, 𝐵} = {𝐵})
12	4	adantr 484	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ V) → {𝐴, 𝐵} ⊆ 𝐶)
13	11, 12	eqsstrrd 3971	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ V) → {𝐵} ⊆ 𝐶)
14		snssg 4741	. . . 4 ⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ 𝐶 ↔ {𝐵} ⊆ 𝐶))
15	14	biimpar 481	. . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ {𝐵} ⊆ 𝐶) → 𝐵 ∈ 𝐶)
16	2, 13, 15	syl2an2r 695	. 2 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ V) → 𝐵 ∈ 𝐶)
17	9, 16	pm2.61dan 822	1 ⊢ (𝜑 → 𝐵 ∈ 𝐶)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   = wceq 1559   ∈ wcel 2141  Vcvv 3453   ⊆ wss 3904  {csn 4581  {cpr 4583

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-v 3455  df-dif 3907  df-un 3909  df-ss 3921  df-nul 4286  df-sn 4582  df-pr 4584

This theorem is referenced by:  constrllcllem  34010  constrlccllem  34011