Theorem difeq 30766

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 4136	. . . 4 ⊢ ((𝐴 ∖ 𝐵) = 𝐶 → ((𝐴 ∖ 𝐵) ∩ 𝐵) = (𝐶 ∩ 𝐵))
2		disjdifr 4403	. . . 4 ⊢ ((𝐴 ∖ 𝐵) ∩ 𝐵) = ∅
3	1, 2	eqtr3di 2794	. . 3 ⊢ ((𝐴 ∖ 𝐵) = 𝐶 → (𝐶 ∩ 𝐵) = ∅)
4		uneq1 4086	. . . 4 ⊢ ((𝐴 ∖ 𝐵) = 𝐶 → ((𝐴 ∖ 𝐵) ∪ 𝐵) = (𝐶 ∪ 𝐵))
5		undif1 4406	. . . 4 ⊢ ((𝐴 ∖ 𝐵) ∪ 𝐵) = (𝐴 ∪ 𝐵)
6	4, 5	eqtr3di 2794	. . 3 ⊢ ((𝐴 ∖ 𝐵) = 𝐶 → (𝐶 ∪ 𝐵) = (𝐴 ∪ 𝐵))
7	3, 6	jca 511	. 2 ⊢ ((𝐴 ∖ 𝐵) = 𝐶 → ((𝐶 ∩ 𝐵) = ∅ ∧ (𝐶 ∪ 𝐵) = (𝐴 ∪ 𝐵)))
8		simpl 482	. . . 4 ⊢ (((𝐶 ∩ 𝐵) = ∅ ∧ (𝐶 ∪ 𝐵) = (𝐴 ∪ 𝐵)) → (𝐶 ∩ 𝐵) = ∅)
9		disj3 4384	. . . . 5 ⊢ ((𝐶 ∩ 𝐵) = ∅ ↔ 𝐶 = (𝐶 ∖ 𝐵))
10		eqcom 2745	. . . . 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 4046	. . . . . 6 ⊢ ((𝐶 ∪ 𝐵) = (𝐴 ∪ 𝐵) → ((𝐶 ∪ 𝐵) ∖ 𝐵) = ((𝐴 ∪ 𝐵) ∖ 𝐵))
14		difun2 4411	. . . . . 6 ⊢ ((𝐶 ∪ 𝐵) ∖ 𝐵) = (𝐶 ∖ 𝐵)
15		difun2 4411	. . . . . 6 ⊢ ((𝐴 ∪ 𝐵) ∖ 𝐵) = (𝐴 ∖ 𝐵)
16	13, 14, 15	3eqtr3g 2802	. . . . 5 ⊢ ((𝐶 ∪ 𝐵) = (𝐴 ∪ 𝐵) → (𝐶 ∖ 𝐵) = (𝐴 ∖ 𝐵))
17	16	eqeq1d 2740	. . . 4 ⊢ ((𝐶 ∪ 𝐵) = (𝐴 ∪ 𝐵) → ((𝐶 ∖ 𝐵) = 𝐶 ↔ (𝐴 ∖ 𝐵) = 𝐶))
18	17	adantl 481	. . 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 395   = wceq 1539   ∖ cdif 3880   ∪ cun 3881   ∩ cin 3882  ∅c0 4253

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-or 844  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254

This theorem is referenced by:  difioo  31005