Theorem vvin 4372

Description: Two classes are both the universal class if and only if their intersection is the universal class. Dual of un00 4370. (Contributed by BJ, 12-Jul-2026.)

Assertion

Ref	Expression
vvin	⊢ ((𝐴 = V ∧ 𝐵 = V) ↔ (𝐴 ∩ 𝐵) = V)

Proof of Theorem vvin

Step	Hyp	Ref	Expression
1		ineq12 4176	. . 3 ⊢ ((𝐴 = V ∧ 𝐵 = V) → (𝐴 ∩ 𝐵) = (V ∩ V))
2		inv1 4362	. . 3 ⊢ (V ∩ V) = V
3	1, 2	eqtrdi 2820	. 2 ⊢ ((𝐴 = V ∧ 𝐵 = V) → (𝐴 ∩ 𝐵) = V)
4		inss1 4197	. . . . 5 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
5		sseq1 3970	. . . . 5 ⊢ ((𝐴 ∩ 𝐵) = V → ((𝐴 ∩ 𝐵) ⊆ 𝐴 ↔ V ⊆ 𝐴))
6	4, 5	mpbii 236	. . . 4 ⊢ ((𝐴 ∩ 𝐵) = V → V ⊆ 𝐴)
7		vss 4371	. . . 4 ⊢ (V ⊆ 𝐴 ↔ 𝐴 = V)
8	6, 7	sylib 221	. . 3 ⊢ ((𝐴 ∩ 𝐵) = V → 𝐴 = V)
9		inss2 4198	. . . . 5 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵
10		sseq1 3970	. . . . 5 ⊢ ((𝐴 ∩ 𝐵) = V → ((𝐴 ∩ 𝐵) ⊆ 𝐵 ↔ V ⊆ 𝐵))
11	9, 10	mpbii 236	. . . 4 ⊢ ((𝐴 ∩ 𝐵) = V → V ⊆ 𝐵)
12		vss 4371	. . . 4 ⊢ (V ⊆ 𝐵 ↔ 𝐵 = V)
13	11, 12	sylib 221	. . 3 ⊢ ((𝐴 ∩ 𝐵) = V → 𝐵 = V)
14	8, 13	jca 520	. 2 ⊢ ((𝐴 ∩ 𝐵) = V → (𝐴 = V ∧ 𝐵 = V))
15	3, 14	impbii 212	1 ⊢ ((𝐴 = V ∧ 𝐵 = V) ↔ (𝐴 ∩ 𝐵) = V)

Syntax hints:   ↔ wb 209   ∧ wa 400   = wceq 1567  Vcvv 3463   ∩ cin 3912   ⊆ wss 3913

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1570  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-rab 3424  df-v 3465  df-in 3920  df-ss 3930

This theorem is referenced by: (None)