Theorem uneqdifeqOLD 4010

Description: Obsolete proof of uneqdifeq 4009 as of 14-Jul-2021. (Contributed by FL, 17-Nov-2008.) (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

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

Proof of Theorem uneqdifeqOLD

Step	Hyp	Ref	Expression
1		uncom 3719	. . . . 5 ⊢ (𝐵 ∪ 𝐴) = (𝐴 ∪ 𝐵)
2		eqtr 2629	. . . . . . 7 ⊢ (((𝐵 ∪ 𝐴) = (𝐴 ∪ 𝐵) ∧ (𝐴 ∪ 𝐵) = 𝐶) → (𝐵 ∪ 𝐴) = 𝐶)
3	2	eqcomd 2616	. . . . . 6 ⊢ (((𝐵 ∪ 𝐴) = (𝐴 ∪ 𝐵) ∧ (𝐴 ∪ 𝐵) = 𝐶) → 𝐶 = (𝐵 ∪ 𝐴))
4		difeq1 3683	. . . . . . 7 ⊢ (𝐶 = (𝐵 ∪ 𝐴) → (𝐶 ∖ 𝐴) = ((𝐵 ∪ 𝐴) ∖ 𝐴))
5		difun2 4000	. . . . . . 7 ⊢ ((𝐵 ∪ 𝐴) ∖ 𝐴) = (𝐵 ∖ 𝐴)
6		eqtr 2629	. . . . . . . 8 ⊢ (((𝐶 ∖ 𝐴) = ((𝐵 ∪ 𝐴) ∖ 𝐴) ∧ ((𝐵 ∪ 𝐴) ∖ 𝐴) = (𝐵 ∖ 𝐴)) → (𝐶 ∖ 𝐴) = (𝐵 ∖ 𝐴))
7		incom 3767	. . . . . . . . . . 11 ⊢ (𝐴 ∩ 𝐵) = (𝐵 ∩ 𝐴)
8	7	eqeq1i 2615	. . . . . . . . . 10 ⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ (𝐵 ∩ 𝐴) = ∅)
9		disj3 3973	. . . . . . . . . 10 ⊢ ((𝐵 ∩ 𝐴) = ∅ ↔ 𝐵 = (𝐵 ∖ 𝐴))
10	8, 9	bitri 263	. . . . . . . . 9 ⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ 𝐵 = (𝐵 ∖ 𝐴))
11		eqtr 2629	. . . . . . . . . . 11 ⊢ (((𝐶 ∖ 𝐴) = (𝐵 ∖ 𝐴) ∧ (𝐵 ∖ 𝐴) = 𝐵) → (𝐶 ∖ 𝐴) = 𝐵)
12	11	expcom 450	. . . . . . . . . 10 ⊢ ((𝐵 ∖ 𝐴) = 𝐵 → ((𝐶 ∖ 𝐴) = (𝐵 ∖ 𝐴) → (𝐶 ∖ 𝐴) = 𝐵))
13	12	eqcoms 2618	. . . . . . . . 9 ⊢ (𝐵 = (𝐵 ∖ 𝐴) → ((𝐶 ∖ 𝐴) = (𝐵 ∖ 𝐴) → (𝐶 ∖ 𝐴) = 𝐵))
14	10, 13	sylbi 206	. . . . . . . 8 ⊢ ((𝐴 ∩ 𝐵) = ∅ → ((𝐶 ∖ 𝐴) = (𝐵 ∖ 𝐴) → (𝐶 ∖ 𝐴) = 𝐵))
15	6, 14	syl5com 31	. . . . . . 7 ⊢ (((𝐶 ∖ 𝐴) = ((𝐵 ∪ 𝐴) ∖ 𝐴) ∧ ((𝐵 ∪ 𝐴) ∖ 𝐴) = (𝐵 ∖ 𝐴)) → ((𝐴 ∩ 𝐵) = ∅ → (𝐶 ∖ 𝐴) = 𝐵))
16	4, 5, 15	sylancl 693	. . . . . 6 ⊢ (𝐶 = (𝐵 ∪ 𝐴) → ((𝐴 ∩ 𝐵) = ∅ → (𝐶 ∖ 𝐴) = 𝐵))
17	3, 16	syl 17	. . . . 5 ⊢ (((𝐵 ∪ 𝐴) = (𝐴 ∪ 𝐵) ∧ (𝐴 ∪ 𝐵) = 𝐶) → ((𝐴 ∩ 𝐵) = ∅ → (𝐶 ∖ 𝐴) = 𝐵))
18	1, 17	mpan 702	. . . 4 ⊢ ((𝐴 ∪ 𝐵) = 𝐶 → ((𝐴 ∩ 𝐵) = ∅ → (𝐶 ∖ 𝐴) = 𝐵))
19	18	com12 32	. . 3 ⊢ ((𝐴 ∩ 𝐵) = ∅ → ((𝐴 ∪ 𝐵) = 𝐶 → (𝐶 ∖ 𝐴) = 𝐵))
20	19	adantl 481	. 2 ⊢ ((𝐴 ⊆ 𝐶 ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝐴 ∪ 𝐵) = 𝐶 → (𝐶 ∖ 𝐴) = 𝐵))
21		difss 3699	. . . . . . . 8 ⊢ (𝐶 ∖ 𝐴) ⊆ 𝐶
22		sseq1 3589	. . . . . . . . 9 ⊢ ((𝐶 ∖ 𝐴) = 𝐵 → ((𝐶 ∖ 𝐴) ⊆ 𝐶 ↔ 𝐵 ⊆ 𝐶))
23		unss 3749	. . . . . . . . . . 11 ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) ↔ (𝐴 ∪ 𝐵) ⊆ 𝐶)
24	23	biimpi 205	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) → (𝐴 ∪ 𝐵) ⊆ 𝐶)
25	24	expcom 450	. . . . . . . . 9 ⊢ (𝐵 ⊆ 𝐶 → (𝐴 ⊆ 𝐶 → (𝐴 ∪ 𝐵) ⊆ 𝐶))
26	22, 25	syl6bi 242	. . . . . . . 8 ⊢ ((𝐶 ∖ 𝐴) = 𝐵 → ((𝐶 ∖ 𝐴) ⊆ 𝐶 → (𝐴 ⊆ 𝐶 → (𝐴 ∪ 𝐵) ⊆ 𝐶)))
27	21, 26	mpi 20	. . . . . . 7 ⊢ ((𝐶 ∖ 𝐴) = 𝐵 → (𝐴 ⊆ 𝐶 → (𝐴 ∪ 𝐵) ⊆ 𝐶))
28	27	com12 32	. . . . . 6 ⊢ (𝐴 ⊆ 𝐶 → ((𝐶 ∖ 𝐴) = 𝐵 → (𝐴 ∪ 𝐵) ⊆ 𝐶))
29	28	adantr 480	. . . . 5 ⊢ ((𝐴 ⊆ 𝐶 ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝐶 ∖ 𝐴) = 𝐵 → (𝐴 ∪ 𝐵) ⊆ 𝐶))
30	29	imp 444	. . . 4 ⊢ (((𝐴 ⊆ 𝐶 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ (𝐶 ∖ 𝐴) = 𝐵) → (𝐴 ∪ 𝐵) ⊆ 𝐶)
31		eqimss 3620	. . . . . . 7 ⊢ ((𝐶 ∖ 𝐴) = 𝐵 → (𝐶 ∖ 𝐴) ⊆ 𝐵)
32	31	adantl 481	. . . . . 6 ⊢ ((𝐴 ⊆ 𝐶 ∧ (𝐶 ∖ 𝐴) = 𝐵) → (𝐶 ∖ 𝐴) ⊆ 𝐵)
33		ssundif 4004	. . . . . 6 ⊢ (𝐶 ⊆ (𝐴 ∪ 𝐵) ↔ (𝐶 ∖ 𝐴) ⊆ 𝐵)
34	32, 33	sylibr 223	. . . . 5 ⊢ ((𝐴 ⊆ 𝐶 ∧ (𝐶 ∖ 𝐴) = 𝐵) → 𝐶 ⊆ (𝐴 ∪ 𝐵))
35	34	adantlr 747	. . . 4 ⊢ (((𝐴 ⊆ 𝐶 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ (𝐶 ∖ 𝐴) = 𝐵) → 𝐶 ⊆ (𝐴 ∪ 𝐵))
36	30, 35	eqssd 3585	. . 3 ⊢ (((𝐴 ⊆ 𝐶 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ (𝐶 ∖ 𝐴) = 𝐵) → (𝐴 ∪ 𝐵) = 𝐶)
37	36	ex 449	. 2 ⊢ ((𝐴 ⊆ 𝐶 ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝐶 ∖ 𝐴) = 𝐵 → (𝐴 ∪ 𝐵) = 𝐶))
38	20, 37	impbid 201	1 ⊢ ((𝐴 ⊆ 𝐶 ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝐴 ∪ 𝐵) = 𝐶 ↔ (𝐶 ∖ 𝐴) = 𝐵))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∖ cdif 3537   ∪ cun 3538   ∩ cin 3539   ⊆ wss 3540  ∅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: (None)