Theorem ssdifsym 4194

Description: Symmetric class differences for subclasses. (Contributed by AV, 3-Jan-2022.)

Assertion

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

Proof of Theorem ssdifsym

Step	Hyp	Ref	Expression
1		ssdifim 4193	. . . 4 ⊢ ((𝐴 ⊆ 𝑉 ∧ 𝐵 = (𝑉 ∖ 𝐴)) → 𝐴 = (𝑉 ∖ 𝐵))
2	1	ex 412	. . 3 ⊢ (𝐴 ⊆ 𝑉 → (𝐵 = (𝑉 ∖ 𝐴) → 𝐴 = (𝑉 ∖ 𝐵)))
3	2	adantr 480	. 2 ⊢ ((𝐴 ⊆ 𝑉 ∧ 𝐵 ⊆ 𝑉) → (𝐵 = (𝑉 ∖ 𝐴) → 𝐴 = (𝑉 ∖ 𝐵)))
4		ssdifim 4193	. . . 4 ⊢ ((𝐵 ⊆ 𝑉 ∧ 𝐴 = (𝑉 ∖ 𝐵)) → 𝐵 = (𝑉 ∖ 𝐴))
5	4	ex 412	. . 3 ⊢ (𝐵 ⊆ 𝑉 → (𝐴 = (𝑉 ∖ 𝐵) → 𝐵 = (𝑉 ∖ 𝐴)))
6	5	adantl 481	. 2 ⊢ ((𝐴 ⊆ 𝑉 ∧ 𝐵 ⊆ 𝑉) → (𝐴 = (𝑉 ∖ 𝐵) → 𝐵 = (𝑉 ∖ 𝐴)))
7	3, 6	impbid 211	1 ⊢ ((𝐴 ⊆ 𝑉 ∧ 𝐵 ⊆ 𝑉) → (𝐵 = (𝑉 ∖ 𝐴) ↔ 𝐴 = (𝑉 ∖ 𝐵)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539   ∖ cdif 3880   ⊆ wss 3883

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1542  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-rab 3072  df-v 3424  df-dif 3886  df-in 3890  df-ss 3900

This theorem is referenced by:  zarcls  31726