Theorem vvin 3568

Description: Two classes are both the universal class if and only if their intersection is the universal class. Dual of un00 3566. (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 3427	. . 3 ⊢ ((𝐴 = V ∧ 𝐵 = V) → (𝐴 ∩ 𝐵) = (V ∩ V))
2		inv1 3559	. . 3 ⊢ (V ∩ V) = V
3	1, 2	eqtrdi 2287	. 2 ⊢ ((𝐴 = V ∧ 𝐵 = V) → (𝐴 ∩ 𝐵) = V)
4		inss1 3451	. . . . 5 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
5		sseq1 3271	. . . . 5 ⊢ ((𝐴 ∩ 𝐵) = V → ((𝐴 ∩ 𝐵) ⊆ 𝐴 ↔ V ⊆ 𝐴))
6	4, 5	mpbii 148	. . . 4 ⊢ ((𝐴 ∩ 𝐵) = V → V ⊆ 𝐴)
7		vss 3567	. . . 4 ⊢ (V ⊆ 𝐴 ↔ 𝐴 = V)
8	6, 7	sylib 122	. . 3 ⊢ ((𝐴 ∩ 𝐵) = V → 𝐴 = V)
9		inss2 3452	. . . . 5 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵
10		sseq1 3271	. . . . 5 ⊢ ((𝐴 ∩ 𝐵) = V → ((𝐴 ∩ 𝐵) ⊆ 𝐵 ↔ V ⊆ 𝐵))
11	9, 10	mpbii 148	. . . 4 ⊢ ((𝐴 ∩ 𝐵) = V → V ⊆ 𝐵)
12		vss 3567	. . . 4 ⊢ (V ⊆ 𝐵 ↔ 𝐵 = V)
13	11, 12	sylib 122	. . 3 ⊢ ((𝐴 ∩ 𝐵) = V → 𝐵 = V)
14	8, 13	jca 306	. 2 ⊢ ((𝐴 ∩ 𝐵) = V → (𝐴 = V ∧ 𝐵 = V))
15	3, 14	impbii 126	1 ⊢ ((𝐴 = V ∧ 𝐵 = V) ↔ (𝐴 ∩ 𝐵) = V)

Syntax hints:   ∧ wa 104   ↔ wb 105   = wceq 1402  Vcvv 2821   ∩ cin 3219   ⊆ wss 3220

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-in 3226  df-ss 3233

This theorem is referenced by: (None)