Theorem prssad 33107

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 486	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ∈ V) → 𝐴 ∈ 𝑉)
3		simpr 490	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ∈ V) → 𝐵 ∈ V)
4		prssad.2	. . . . 5 ⊢ (𝜑 → {𝐴, 𝐵} ⊆ 𝐶)
5	4	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ∈ V) → {𝐴, 𝐵} ⊆ 𝐶)
6		prssg 4780	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ V) → ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) ↔ {𝐴, 𝐵} ⊆ 𝐶))
7	6	biimpar 483	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ V) ∧ {𝐴, 𝐵} ⊆ 𝐶) → (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶))
8	2, 3, 5, 7	syl21anc 851	. . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ V) → (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶))
9	8	simpld 500	. 2 ⊢ ((𝜑 ∧ 𝐵 ∈ V) → 𝐴 ∈ 𝐶)
10		prprc2 4727	. . . . 5 ⊢ (¬ 𝐵 ∈ V → {𝐴, 𝐵} = {𝐴})
11	10	adantl 487	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐵 ∈ V) → {𝐴, 𝐵} = {𝐴})
12	4	adantr 486	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐵 ∈ V) → {𝐴, 𝐵} ⊆ 𝐶)
13	11, 12	eqsstrrd 3966	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐵 ∈ V) → {𝐴} ⊆ 𝐶)
14		snssg 4744	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝐶 ↔ {𝐴} ⊆ 𝐶))
15	14	biimpar 483	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ {𝐴} ⊆ 𝐶) → 𝐴 ∈ 𝐶)
16	1, 13, 15	syl2an2r 698	. 2 ⊢ ((𝜑 ∧ ¬ 𝐵 ∈ V) → 𝐴 ∈ 𝐶)
17	9, 16	pm2.61dan 825	1 ⊢ (𝜑 → 𝐴 ∈ 𝐶)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  Vcvv 3451   ⊆ wss 3899  {csn 4584  {cpr 4586

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-v 3453  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4280  df-sn 4585  df-pr 4587

This theorem is used by:  tpssad  33117  constrllcllem  34366  constrlccllem  34367