Theorem prssad 32614

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

Hypotheses

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

Assertion

Ref	Expression
prssad	⊢ (𝜑 → 𝐴 ∈ 𝐶)

Proof of Theorem prssad

Step	Hyp	Ref	Expression
1		prssad.1	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
2	1	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ∈ V) → 𝐴 ∈ 𝑉)
3		simpr 484	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ∈ V) → 𝐵 ∈ V)
4		prssad.2	. . . . 5 ⊢ (𝜑 → {𝐴, 𝐵} ⊆ 𝐶)
5	4	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ∈ V) → {𝐴, 𝐵} ⊆ 𝐶)
6		prssg 4763	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ V) → ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) ↔ {𝐴, 𝐵} ⊆ 𝐶))
7	6	biimpar 477	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ V) ∧ {𝐴, 𝐵} ⊆ 𝐶) → (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶))
8	2, 3, 5, 7	syl21anc 838	. . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ V) → (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶))
9	8	simpld 494	. 2 ⊢ ((𝜑 ∧ 𝐵 ∈ V) → 𝐴 ∈ 𝐶)
10		prprc2 4711	. . . . 5 ⊢ (¬ 𝐵 ∈ V → {𝐴, 𝐵} = {𝐴})
11	10	adantl 481	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐵 ∈ V) → {𝐴, 𝐵} = {𝐴})
12	4	adantr 480	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐵 ∈ V) → {𝐴, 𝐵} ⊆ 𝐶)
13	11, 12	eqsstrrd 3958	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐵 ∈ V) → {𝐴} ⊆ 𝐶)
14		snssg 4728	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝐶 ↔ {𝐴} ⊆ 𝐶))
15	14	biimpar 477	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ {𝐴} ⊆ 𝐶) → 𝐴 ∈ 𝐶)
16	1, 13, 15	syl2an2r 686	. 2 ⊢ ((𝜑 ∧ ¬ 𝐵 ∈ V) → 𝐴 ∈ 𝐶)
17	9, 16	pm2.61dan 813	1 ⊢ (𝜑 → 𝐴 ∈ 𝐶)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  Vcvv 3430   ⊆ wss 3890  {csn 4568  {cpr 4570

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-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-v 3432  df-dif 3893  df-un 3895  df-ss 3907  df-nul 4275  df-sn 4569  df-pr 4571

This theorem is referenced by:  tpssad  32624  constrllcllem  33912  constrlccllem  33913