Theorem ssdifim 3862

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 3858	. . 3 ⊢ (𝐴 ⊆ 𝑉 ↔ (𝑉 ∖ (𝑉 ∖ 𝐴)) = 𝐴)
2		eqcom 2629	. . 3 ⊢ ((𝑉 ∖ (𝑉 ∖ 𝐴)) = 𝐴 ↔ 𝐴 = (𝑉 ∖ (𝑉 ∖ 𝐴)))
3	1, 2	sylbb 209	. 2 ⊢ (𝐴 ⊆ 𝑉 → 𝐴 = (𝑉 ∖ (𝑉 ∖ 𝐴)))
4		difeq2 3722	. . 3 ⊢ (𝐵 = (𝑉 ∖ 𝐴) → (𝑉 ∖ 𝐵) = (𝑉 ∖ (𝑉 ∖ 𝐴)))
5	4	eqcomd 2628	. 2 ⊢ (𝐵 = (𝑉 ∖ 𝐴) → (𝑉 ∖ (𝑉 ∖ 𝐴)) = (𝑉 ∖ 𝐵))
6	3, 5	sylan9eq 2676	1 ⊢ ((𝐴 ⊆ 𝑉 ∧ 𝐵 = (𝑉 ∖ 𝐴)) → 𝐴 = (𝑉 ∖ 𝐵))

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∖ cdif 3571   ⊆ wss 3574

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rab 2921  df-v 3202  df-dif 3577  df-in 3581  df-ss 3588

This theorem is referenced by:  ssdifsym  3863  frgrwopregbsn  27181