Theorem ssdifim 4218

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 4214	. . 3 ⊢ (𝐴 ⊆ 𝑉 ↔ (𝑉 ∖ (𝑉 ∖ 𝐴)) = 𝐴)
2		eqcom 2738	. . 3 ⊢ ((𝑉 ∖ (𝑉 ∖ 𝐴)) = 𝐴 ↔ 𝐴 = (𝑉 ∖ (𝑉 ∖ 𝐴)))
3	1, 2	sylbb 219	. 2 ⊢ (𝐴 ⊆ 𝑉 → 𝐴 = (𝑉 ∖ (𝑉 ∖ 𝐴)))
4		difeq2 4065	. . 3 ⊢ (𝐵 = (𝑉 ∖ 𝐴) → (𝑉 ∖ 𝐵) = (𝑉 ∖ (𝑉 ∖ 𝐴)))
5	4	eqcomd 2737	. 2 ⊢ (𝐵 = (𝑉 ∖ 𝐴) → (𝑉 ∖ (𝑉 ∖ 𝐴)) = (𝑉 ∖ 𝐵))
6	3, 5	sylan9eq 2786	1 ⊢ ((𝐴 ⊆ 𝑉 ∧ 𝐵 = (𝑉 ∖ 𝐴)) → 𝐴 = (𝑉 ∖ 𝐵))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∖ cdif 3894   ⊆ wss 3897

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-rab 3396  df-v 3438  df-dif 3900  df-in 3904  df-ss 3914

This theorem is referenced by:  ssdifsym  4219  frgrwopregbsn  30289