Theorem uneqdifeq 3639

Description: Two ways to say that A and B partition C (when A and B don't overlap and A is a part of C). (Contributed by FL, 17-Nov-2008.)

Assertion

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

Proof of Theorem uneqdifeq

Step	Hyp	Ref	Expression
1		uncom 3409	. . . . 5 ⊢ (B ∪ A) = (A ∪ B)
2		eqtr 2370	. . . . . . 7 ⊢ (((B ∪ A) = (A ∪ B) ∧ (A ∪ B) = C) → (B ∪ A) = C)
3	2	eqcomd 2358	. . . . . 6 ⊢ (((B ∪ A) = (A ∪ B) ∧ (A ∪ B) = C) → C = (B ∪ A))
4		difeq1 3247	. . . . . . 7 ⊢ (C = (B ∪ A) → (C ∖ A) = ((B ∪ A) ∖ A))
5		difun2 3630	. . . . . . 7 ⊢ ((B ∪ A) ∖ A) = (B ∖ A)
6		eqtr 2370	. . . . . . . 8 ⊢ (((C ∖ A) = ((B ∪ A) ∖ A) ∧ ((B ∪ A) ∖ A) = (B ∖ A)) → (C ∖ A) = (B ∖ A))
7		incom 3449	. . . . . . . . . . 11 ⊢ (A ∩ B) = (B ∩ A)
8	7	eqeq1i 2360	. . . . . . . . . 10 ⊢ ((A ∩ B) = ∅ ↔ (B ∩ A) = ∅)
9		disj3 3596	. . . . . . . . . 10 ⊢ ((B ∩ A) = ∅ ↔ B = (B ∖ A))
10	8, 9	bitri 240	. . . . . . . . 9 ⊢ ((A ∩ B) = ∅ ↔ B = (B ∖ A))
11		eqtr 2370	. . . . . . . . . . 11 ⊢ (((C ∖ A) = (B ∖ A) ∧ (B ∖ A) = B) → (C ∖ A) = B)
12	11	expcom 424	. . . . . . . . . 10 ⊢ ((B ∖ A) = B → ((C ∖ A) = (B ∖ A) → (C ∖ A) = B))
13	12	eqcoms 2356	. . . . . . . . 9 ⊢ (B = (B ∖ A) → ((C ∖ A) = (B ∖ A) → (C ∖ A) = B))
14	10, 13	sylbi 187	. . . . . . . 8 ⊢ ((A ∩ B) = ∅ → ((C ∖ A) = (B ∖ A) → (C ∖ A) = B))
15	6, 14	syl5com 26	. . . . . . 7 ⊢ (((C ∖ A) = ((B ∪ A) ∖ A) ∧ ((B ∪ A) ∖ A) = (B ∖ A)) → ((A ∩ B) = ∅ → (C ∖ A) = B))
16	4, 5, 15	sylancl 643	. . . . . 6 ⊢ (C = (B ∪ A) → ((A ∩ B) = ∅ → (C ∖ A) = B))
17	3, 16	syl 15	. . . . 5 ⊢ (((B ∪ A) = (A ∪ B) ∧ (A ∪ B) = C) → ((A ∩ B) = ∅ → (C ∖ A) = B))
18	1, 17	mpan 651	. . . 4 ⊢ ((A ∪ B) = C → ((A ∩ B) = ∅ → (C ∖ A) = B))
19	18	com12 27	. . 3 ⊢ ((A ∩ B) = ∅ → ((A ∪ B) = C → (C ∖ A) = B))
20	19	adantl 452	. 2 ⊢ ((A ⊆ C ∧ (A ∩ B) = ∅) → ((A ∪ B) = C → (C ∖ A) = B))
21		difss 3394	. . . . . . . 8 ⊢ (C ∖ A) ⊆ C
22		sseq1 3293	. . . . . . . . 9 ⊢ ((C ∖ A) = B → ((C ∖ A) ⊆ C ↔ B ⊆ C))
23		unss 3438	. . . . . . . . . . 11 ⊢ ((A ⊆ C ∧ B ⊆ C) ↔ (A ∪ B) ⊆ C)
24	23	biimpi 186	. . . . . . . . . 10 ⊢ ((A ⊆ C ∧ B ⊆ C) → (A ∪ B) ⊆ C)
25	24	expcom 424	. . . . . . . . 9 ⊢ (B ⊆ C → (A ⊆ C → (A ∪ B) ⊆ C))
26	22, 25	syl6bi 219	. . . . . . . 8 ⊢ ((C ∖ A) = B → ((C ∖ A) ⊆ C → (A ⊆ C → (A ∪ B) ⊆ C)))
27	21, 26	mpi 16	. . . . . . 7 ⊢ ((C ∖ A) = B → (A ⊆ C → (A ∪ B) ⊆ C))
28	27	com12 27	. . . . . 6 ⊢ (A ⊆ C → ((C ∖ A) = B → (A ∪ B) ⊆ C))
29	28	adantr 451	. . . . 5 ⊢ ((A ⊆ C ∧ (A ∩ B) = ∅) → ((C ∖ A) = B → (A ∪ B) ⊆ C))
30	29	imp 418	. . . 4 ⊢ (((A ⊆ C ∧ (A ∩ B) = ∅) ∧ (C ∖ A) = B) → (A ∪ B) ⊆ C)
31		eqimss 3324	. . . . . . 7 ⊢ ((C ∖ A) = B → (C ∖ A) ⊆ B)
32	31	adantl 452	. . . . . 6 ⊢ ((A ⊆ C ∧ (C ∖ A) = B) → (C ∖ A) ⊆ B)
33		ssundif 3634	. . . . . 6 ⊢ (C ⊆ (A ∪ B) ↔ (C ∖ A) ⊆ B)
34	32, 33	sylibr 203	. . . . 5 ⊢ ((A ⊆ C ∧ (C ∖ A) = B) → C ⊆ (A ∪ B))
35	34	adantlr 695	. . . 4 ⊢ (((A ⊆ C ∧ (A ∩ B) = ∅) ∧ (C ∖ A) = B) → C ⊆ (A ∪ B))
36	30, 35	eqssd 3290	. . 3 ⊢ (((A ⊆ C ∧ (A ∩ B) = ∅) ∧ (C ∖ A) = B) → (A ∪ B) = C)
37	36	ex 423	. 2 ⊢ ((A ⊆ C ∧ (A ∩ B) = ∅) → ((C ∖ A) = B → (A ∪ B) = C))
38	20, 37	impbid 183	1 ⊢ ((A ⊆ C ∧ (A ∩ B) = ∅) → ((A ∪ B) = C ↔ (C ∖ A) = B))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∖ cdif 3207   ∪ cun 3208   ∩ cin 3209   ⊆ wss 3258  ∅c0 3551

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-ss 3260  df-nul 3552

This theorem is referenced by: (None)