Theorem vvin 4358

Description: Two classes are both the universal class if and only if their intersection is the universal class. Dual of un00 4356. (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 4111	. . 3 ⊢ ((𝐴 = V ∧ 𝐵 = V) → (𝐴 ∩ 𝐵) = (V ∩ V))
2		inv1 4347	. . 3 ⊢ (V ∩ V) = V
3	1, 2	eqtrdi 2811	. 2 ⊢ ((𝐴 = V ∧ 𝐵 = V) → (𝐴 ∩ 𝐵) = V)
4		inss1 4132	. . . . 5 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
5		sseq1 3934	. . . . 5 ⊢ ((𝐴 ∩ 𝐵) = V → ((𝐴 ∩ 𝐵) ⊆ 𝐴 ↔ V ⊆ 𝐴))
6	4, 5	mpbii 236	. . . 4 ⊢ ((𝐴 ∩ 𝐵) = V → V ⊆ 𝐴)
7		vss 4357	. . . 4 ⊢ (V ⊆ 𝐴 ↔ 𝐴 = V)
8	6, 7	sylib 221	. . 3 ⊢ ((𝐴 ∩ 𝐵) = V → 𝐴 = V)
9		inss2 4133	. . . . 5 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵
10		sseq1 3934	. . . . 5 ⊢ ((𝐴 ∩ 𝐵) = V → ((𝐴 ∩ 𝐵) ⊆ 𝐵 ↔ V ⊆ 𝐵))
11	9, 10	mpbii 236	. . . 4 ⊢ ((𝐴 ∩ 𝐵) = V → V ⊆ 𝐵)
12		vss 4357	. . . 4 ⊢ (V ⊆ 𝐵 ↔ 𝐵 = V)
13	11, 12	sylib 221	. . 3 ⊢ ((𝐴 ∩ 𝐵) = V → 𝐵 = V)
14	8, 13	jca 521	. 2 ⊢ ((𝐴 ∩ 𝐵) = V → (𝐴 = V ∧ 𝐵 = V))
15	3, 14	impbii 212	1 ⊢ ((𝐴 = V ∧ 𝐵 = V) ↔ (𝐴 ∩ 𝐵) = V)

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1569  Vcvv 3450   ⊆ wss 3896   ∩ cin 4088

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-ss 3897  df-in 4090

This theorem is used by: (None)