Theorem prssad 32464

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 4785	. . . . 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 837	. . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ V) → (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶))
9	8	simpld 494	. 2 ⊢ ((𝜑 ∧ 𝐵 ∈ V) → 𝐴 ∈ 𝐶)
10		prprc2 4732	. . . . 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 3984	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐵 ∈ V) → {𝐴} ⊆ 𝐶)
14		snssg 4749	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝐶 ↔ {𝐴} ⊆ 𝐶))
15	14	biimpar 477	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ {𝐴} ⊆ 𝐶) → 𝐴 ∈ 𝐶)
16	1, 13, 15	syl2an2r 685	. 2 ⊢ ((𝜑 ∧ ¬ 𝐵 ∈ V) → 𝐴 ∈ 𝐶)
17	9, 16	pm2.61dan 812	1 ⊢ (𝜑 → 𝐴 ∈ 𝐶)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3450   ⊆ wss 3916  {csn 4591  {cpr 4593

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-v 3452  df-dif 3919  df-un 3921  df-ss 3933  df-nul 4299  df-sn 4592  df-pr 4594

This theorem is referenced by:  tpssad  32474  constrllcllem  33748  constrlccllem  33749