Theorem uneqdifeqim 3387

Description: Two ways that 𝐴 and 𝐵 can "partition" 𝐶 (when 𝐴 and 𝐵 don't overlap and 𝐴 is a part of 𝐶). In classical logic, the second implication would be a biconditional. (Contributed by Jim Kingdon, 4-Aug-2018.)

Assertion

Ref	Expression
uneqdifeqim	⊢ ((𝐴 ⊆ 𝐶 ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝐴 ∪ 𝐵) = 𝐶 → (𝐶 ∖ 𝐴) = 𝐵))

Proof of Theorem uneqdifeqim

Step	Hyp	Ref	Expression
1		uncom 3159	. . . 4 ⊢ (𝐵 ∪ 𝐴) = (𝐴 ∪ 𝐵)
2		eqtr 2112	. . . . . 6 ⊢ (((𝐵 ∪ 𝐴) = (𝐴 ∪ 𝐵) ∧ (𝐴 ∪ 𝐵) = 𝐶) → (𝐵 ∪ 𝐴) = 𝐶)
3	2	eqcomd 2100	. . . . 5 ⊢ (((𝐵 ∪ 𝐴) = (𝐴 ∪ 𝐵) ∧ (𝐴 ∪ 𝐵) = 𝐶) → 𝐶 = (𝐵 ∪ 𝐴))
4		difeq1 3126	. . . . . 6 ⊢ (𝐶 = (𝐵 ∪ 𝐴) → (𝐶 ∖ 𝐴) = ((𝐵 ∪ 𝐴) ∖ 𝐴))
5		difun2 3381	. . . . . 6 ⊢ ((𝐵 ∪ 𝐴) ∖ 𝐴) = (𝐵 ∖ 𝐴)
6		eqtr 2112	. . . . . . 7 ⊢ (((𝐶 ∖ 𝐴) = ((𝐵 ∪ 𝐴) ∖ 𝐴) ∧ ((𝐵 ∪ 𝐴) ∖ 𝐴) = (𝐵 ∖ 𝐴)) → (𝐶 ∖ 𝐴) = (𝐵 ∖ 𝐴))
7		incom 3207	. . . . . . . . . 10 ⊢ (𝐴 ∩ 𝐵) = (𝐵 ∩ 𝐴)
8	7	eqeq1i 2102	. . . . . . . . 9 ⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ (𝐵 ∩ 𝐴) = ∅)
9		disj3 3354	. . . . . . . . 9 ⊢ ((𝐵 ∩ 𝐴) = ∅ ↔ 𝐵 = (𝐵 ∖ 𝐴))
10	8, 9	bitri 183	. . . . . . . 8 ⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ 𝐵 = (𝐵 ∖ 𝐴))
11		eqtr 2112	. . . . . . . . . 10 ⊢ (((𝐶 ∖ 𝐴) = (𝐵 ∖ 𝐴) ∧ (𝐵 ∖ 𝐴) = 𝐵) → (𝐶 ∖ 𝐴) = 𝐵)
12	11	expcom 115	. . . . . . . . 9 ⊢ ((𝐵 ∖ 𝐴) = 𝐵 → ((𝐶 ∖ 𝐴) = (𝐵 ∖ 𝐴) → (𝐶 ∖ 𝐴) = 𝐵))
13	12	eqcoms 2098	. . . . . . . 8 ⊢ (𝐵 = (𝐵 ∖ 𝐴) → ((𝐶 ∖ 𝐴) = (𝐵 ∖ 𝐴) → (𝐶 ∖ 𝐴) = 𝐵))
14	10, 13	sylbi 120	. . . . . . 7 ⊢ ((𝐴 ∩ 𝐵) = ∅ → ((𝐶 ∖ 𝐴) = (𝐵 ∖ 𝐴) → (𝐶 ∖ 𝐴) = 𝐵))
15	6, 14	syl5com 29	. . . . . 6 ⊢ (((𝐶 ∖ 𝐴) = ((𝐵 ∪ 𝐴) ∖ 𝐴) ∧ ((𝐵 ∪ 𝐴) ∖ 𝐴) = (𝐵 ∖ 𝐴)) → ((𝐴 ∩ 𝐵) = ∅ → (𝐶 ∖ 𝐴) = 𝐵))
16	4, 5, 15	sylancl 405	. . . . 5 ⊢ (𝐶 = (𝐵 ∪ 𝐴) → ((𝐴 ∩ 𝐵) = ∅ → (𝐶 ∖ 𝐴) = 𝐵))
17	3, 16	syl 14	. . . 4 ⊢ (((𝐵 ∪ 𝐴) = (𝐴 ∪ 𝐵) ∧ (𝐴 ∪ 𝐵) = 𝐶) → ((𝐴 ∩ 𝐵) = ∅ → (𝐶 ∖ 𝐴) = 𝐵))
18	1, 17	mpan 416	. . 3 ⊢ ((𝐴 ∪ 𝐵) = 𝐶 → ((𝐴 ∩ 𝐵) = ∅ → (𝐶 ∖ 𝐴) = 𝐵))
19	18	com12 30	. 2 ⊢ ((𝐴 ∩ 𝐵) = ∅ → ((𝐴 ∪ 𝐵) = 𝐶 → (𝐶 ∖ 𝐴) = 𝐵))
20	19	adantl 272	1 ⊢ ((𝐴 ⊆ 𝐶 ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝐴 ∪ 𝐵) = 𝐶 → (𝐶 ∖ 𝐴) = 𝐵))

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1296   ∖ cdif 3010   ∪ cun 3011   ∩ cin 3012   ⊆ wss 3013  ∅c0 3302

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077

This theorem depends on definitions:  df-bi 116  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rab 2379  df-v 2635  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-nul 3303

This theorem is referenced by: (None)