Theorem difeq 28739

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		incom 3767	. . . . 5 ⊢ (𝐵 ∩ (𝐴 ∖ 𝐵)) = ((𝐴 ∖ 𝐵) ∩ 𝐵)
2		disjdif 3992	. . . . 5 ⊢ (𝐵 ∩ (𝐴 ∖ 𝐵)) = ∅
3	1, 2	eqtr3i 2634	. . . 4 ⊢ ((𝐴 ∖ 𝐵) ∩ 𝐵) = ∅
4		ineq1 3769	. . . 4 ⊢ ((𝐴 ∖ 𝐵) = 𝐶 → ((𝐴 ∖ 𝐵) ∩ 𝐵) = (𝐶 ∩ 𝐵))
5	3, 4	syl5reqr 2659	. . 3 ⊢ ((𝐴 ∖ 𝐵) = 𝐶 → (𝐶 ∩ 𝐵) = ∅)
6		undif1 3995	. . . 4 ⊢ ((𝐴 ∖ 𝐵) ∪ 𝐵) = (𝐴 ∪ 𝐵)
7		uneq1 3722	. . . 4 ⊢ ((𝐴 ∖ 𝐵) = 𝐶 → ((𝐴 ∖ 𝐵) ∪ 𝐵) = (𝐶 ∪ 𝐵))
8	6, 7	syl5reqr 2659	. . 3 ⊢ ((𝐴 ∖ 𝐵) = 𝐶 → (𝐶 ∪ 𝐵) = (𝐴 ∪ 𝐵))
9	5, 8	jca 553	. 2 ⊢ ((𝐴 ∖ 𝐵) = 𝐶 → ((𝐶 ∩ 𝐵) = ∅ ∧ (𝐶 ∪ 𝐵) = (𝐴 ∪ 𝐵)))
10		simpl 472	. . . 4 ⊢ (((𝐶 ∩ 𝐵) = ∅ ∧ (𝐶 ∪ 𝐵) = (𝐴 ∪ 𝐵)) → (𝐶 ∩ 𝐵) = ∅)
11		disj3 3973	. . . . 5 ⊢ ((𝐶 ∩ 𝐵) = ∅ ↔ 𝐶 = (𝐶 ∖ 𝐵))
12		eqcom 2617	. . . . 5 ⊢ (𝐶 = (𝐶 ∖ 𝐵) ↔ (𝐶 ∖ 𝐵) = 𝐶)
13	11, 12	bitri 263	. . . 4 ⊢ ((𝐶 ∩ 𝐵) = ∅ ↔ (𝐶 ∖ 𝐵) = 𝐶)
14	10, 13	sylib 207	. . 3 ⊢ (((𝐶 ∩ 𝐵) = ∅ ∧ (𝐶 ∪ 𝐵) = (𝐴 ∪ 𝐵)) → (𝐶 ∖ 𝐵) = 𝐶)
15		difeq1 3683	. . . . . 6 ⊢ ((𝐶 ∪ 𝐵) = (𝐴 ∪ 𝐵) → ((𝐶 ∪ 𝐵) ∖ 𝐵) = ((𝐴 ∪ 𝐵) ∖ 𝐵))
16		difun2 4000	. . . . . 6 ⊢ ((𝐶 ∪ 𝐵) ∖ 𝐵) = (𝐶 ∖ 𝐵)
17		difun2 4000	. . . . . 6 ⊢ ((𝐴 ∪ 𝐵) ∖ 𝐵) = (𝐴 ∖ 𝐵)
18	15, 16, 17	3eqtr3g 2667	. . . . 5 ⊢ ((𝐶 ∪ 𝐵) = (𝐴 ∪ 𝐵) → (𝐶 ∖ 𝐵) = (𝐴 ∖ 𝐵))
19	18	eqeq1d 2612	. . . 4 ⊢ ((𝐶 ∪ 𝐵) = (𝐴 ∪ 𝐵) → ((𝐶 ∖ 𝐵) = 𝐶 ↔ (𝐴 ∖ 𝐵) = 𝐶))
20	19	adantl 481	. . 3 ⊢ (((𝐶 ∩ 𝐵) = ∅ ∧ (𝐶 ∪ 𝐵) = (𝐴 ∪ 𝐵)) → ((𝐶 ∖ 𝐵) = 𝐶 ↔ (𝐴 ∖ 𝐵) = 𝐶))
21	14, 20	mpbid 221	. 2 ⊢ (((𝐶 ∩ 𝐵) = ∅ ∧ (𝐶 ∪ 𝐵) = (𝐴 ∪ 𝐵)) → (𝐴 ∖ 𝐵) = 𝐶)
22	9, 21	impbii 198	1 ⊢ ((𝐴 ∖ 𝐵) = 𝐶 ↔ ((𝐶 ∩ 𝐵) = ∅ ∧ (𝐶 ∪ 𝐵) = (𝐴 ∪ 𝐵)))

Syntax hints:   ↔ wb 195   ∧ wa 383   = wceq 1475   ∖ cdif 3537   ∪ cun 3538   ∩ cin 3539  ∅c0 3874

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875

This theorem is referenced by:  difioo  28934