Theorem ssdifim 4292

Description: Implication of a class difference with a subclass. (Contributed by AV, 3-Jan-2022.)

Assertion

Ref	Expression
ssdifim	⊢ ((𝐴 ⊆ 𝑉 ∧ 𝐵 = (𝑉 ∖ 𝐴)) → 𝐴 = (𝑉 ∖ 𝐵))

Proof of Theorem ssdifim

Step	Hyp	Ref	Expression
1		dfss4 4288	. . 3 ⊢ (𝐴 ⊆ 𝑉 ↔ (𝑉 ∖ (𝑉 ∖ 𝐴)) = 𝐴)
2		eqcom 2747	. . 3 ⊢ ((𝑉 ∖ (𝑉 ∖ 𝐴)) = 𝐴 ↔ 𝐴 = (𝑉 ∖ (𝑉 ∖ 𝐴)))
3	1, 2	sylbb 219	. 2 ⊢ (𝐴 ⊆ 𝑉 → 𝐴 = (𝑉 ∖ (𝑉 ∖ 𝐴)))
4		difeq2 4143	. . 3 ⊢ (𝐵 = (𝑉 ∖ 𝐴) → (𝑉 ∖ 𝐵) = (𝑉 ∖ (𝑉 ∖ 𝐴)))
5	4	eqcomd 2746	. 2 ⊢ (𝐵 = (𝑉 ∖ 𝐴) → (𝑉 ∖ (𝑉 ∖ 𝐴)) = (𝑉 ∖ 𝐵))
6	3, 5	sylan9eq 2800	1 ⊢ ((𝐴 ⊆ 𝑉 ∧ 𝐵 = (𝑉 ∖ 𝐴)) → 𝐴 = (𝑉 ∖ 𝐵))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∖ cdif 3973   ⊆ wss 3976

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1089  df-tru 1540  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-rab 3444  df-v 3490  df-dif 3979  df-in 3983  df-ss 3993

This theorem is referenced by:  ssdifsym  4293  frgrwopregbsn  30349