Theorem difeq 31615

Description: Rewriting an equation with class difference, without using quantifiers. (Contributed by Thierry Arnoux, 24-Sep-2017.)

Assertion

Ref	Expression
difeq	⊢ ((𝐴 ∖ 𝐵) = 𝐶 ↔ ((𝐶 ∩ 𝐵) = ∅ ∧ (𝐶 ∪ 𝐵) = (𝐴 ∪ 𝐵)))

Proof of Theorem difeq

Step	Hyp	Ref	Expression
1		ineq1 4198	. . . 4 ⊢ ((𝐴 ∖ 𝐵) = 𝐶 → ((𝐴 ∖ 𝐵) ∩ 𝐵) = (𝐶 ∩ 𝐵))
2		disjdifr 4465	. . . 4 ⊢ ((𝐴 ∖ 𝐵) ∩ 𝐵) = ∅
3	1, 2	eqtr3di 2786	. . 3 ⊢ ((𝐴 ∖ 𝐵) = 𝐶 → (𝐶 ∩ 𝐵) = ∅)
4		uneq1 4149	. . . 4 ⊢ ((𝐴 ∖ 𝐵) = 𝐶 → ((𝐴 ∖ 𝐵) ∪ 𝐵) = (𝐶 ∪ 𝐵))
5		undif1 4468	. . . 4 ⊢ ((𝐴 ∖ 𝐵) ∪ 𝐵) = (𝐴 ∪ 𝐵)
6	4, 5	eqtr3di 2786	. . 3 ⊢ ((𝐴 ∖ 𝐵) = 𝐶 → (𝐶 ∪ 𝐵) = (𝐴 ∪ 𝐵))
7	3, 6	jca 512	. 2 ⊢ ((𝐴 ∖ 𝐵) = 𝐶 → ((𝐶 ∩ 𝐵) = ∅ ∧ (𝐶 ∪ 𝐵) = (𝐴 ∪ 𝐵)))
8		simpl 483	. . . 4 ⊢ (((𝐶 ∩ 𝐵) = ∅ ∧ (𝐶 ∪ 𝐵) = (𝐴 ∪ 𝐵)) → (𝐶 ∩ 𝐵) = ∅)
9		disj3 4446	. . . . 5 ⊢ ((𝐶 ∩ 𝐵) = ∅ ↔ 𝐶 = (𝐶 ∖ 𝐵))
10		eqcom 2738	. . . . 5 ⊢ (𝐶 = (𝐶 ∖ 𝐵) ↔ (𝐶 ∖ 𝐵) = 𝐶)
11	9, 10	bitri 274	. . . 4 ⊢ ((𝐶 ∩ 𝐵) = ∅ ↔ (𝐶 ∖ 𝐵) = 𝐶)
12	8, 11	sylib 217	. . 3 ⊢ (((𝐶 ∩ 𝐵) = ∅ ∧ (𝐶 ∪ 𝐵) = (𝐴 ∪ 𝐵)) → (𝐶 ∖ 𝐵) = 𝐶)
13		difeq1 4108	. . . . . 6 ⊢ ((𝐶 ∪ 𝐵) = (𝐴 ∪ 𝐵) → ((𝐶 ∪ 𝐵) ∖ 𝐵) = ((𝐴 ∪ 𝐵) ∖ 𝐵))
14		difun2 4473	. . . . . 6 ⊢ ((𝐶 ∪ 𝐵) ∖ 𝐵) = (𝐶 ∖ 𝐵)
15		difun2 4473	. . . . . 6 ⊢ ((𝐴 ∪ 𝐵) ∖ 𝐵) = (𝐴 ∖ 𝐵)
16	13, 14, 15	3eqtr3g 2794	. . . . 5 ⊢ ((𝐶 ∪ 𝐵) = (𝐴 ∪ 𝐵) → (𝐶 ∖ 𝐵) = (𝐴 ∖ 𝐵))
17	16	eqeq1d 2733	. . . 4 ⊢ ((𝐶 ∪ 𝐵) = (𝐴 ∪ 𝐵) → ((𝐶 ∖ 𝐵) = 𝐶 ↔ (𝐴 ∖ 𝐵) = 𝐶))
18	17	adantl 482	. . 3 ⊢ (((𝐶 ∩ 𝐵) = ∅ ∧ (𝐶 ∪ 𝐵) = (𝐴 ∪ 𝐵)) → ((𝐶 ∖ 𝐵) = 𝐶 ↔ (𝐴 ∖ 𝐵) = 𝐶))
19	12, 18	mpbid 231	. 2 ⊢ (((𝐶 ∩ 𝐵) = ∅ ∧ (𝐶 ∪ 𝐵) = (𝐴 ∪ 𝐵)) → (𝐴 ∖ 𝐵) = 𝐶)
20	7, 19	impbii 208	1 ⊢ ((𝐴 ∖ 𝐵) = 𝐶 ↔ ((𝐶 ∩ 𝐵) = ∅ ∧ (𝐶 ∪ 𝐵) = (𝐴 ∪ 𝐵)))

Syntax hints:   ↔ wb 205   ∧ wa 396   = wceq 1541   ∖ cdif 3938   ∪ cun 3939   ∩ cin 3940  ∅c0 4315

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-rab 3430  df-v 3472  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316

This theorem is referenced by:  difioo  31859