Theorem undifdcss 6900

Description: Union of complementary parts into whole and decidability. (Contributed by Jim Kingdon, 17-Jun-2022.)

Assertion

Ref	Expression
undifdcss	⊢ (𝐴 = (𝐵 ∪ (𝐴 ∖ 𝐵)) ↔ (𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐴 DECID 𝑥 ∈ 𝐵))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem undifdcss

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqimss2 3202	. . . 4 ⊢ (𝐴 = (𝐵 ∪ (𝐴 ∖ 𝐵)) → (𝐵 ∪ (𝐴 ∖ 𝐵)) ⊆ 𝐴)
2		undifss 3495	. . . 4 ⊢ (𝐵 ⊆ 𝐴 ↔ (𝐵 ∪ (𝐴 ∖ 𝐵)) ⊆ 𝐴)
3	1, 2	sylibr 133	. . 3 ⊢ (𝐴 = (𝐵 ∪ (𝐴 ∖ 𝐵)) → 𝐵 ⊆ 𝐴)
4		eleq2 2234	. . . . . . . 8 ⊢ (𝐴 = (𝐵 ∪ (𝐴 ∖ 𝐵)) → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ (𝐵 ∪ (𝐴 ∖ 𝐵))))
5	4	biimpa 294	. . . . . . 7 ⊢ ((𝐴 = (𝐵 ∪ (𝐴 ∖ 𝐵)) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ (𝐵 ∪ (𝐴 ∖ 𝐵)))
6		elun 3268	. . . . . . 7 ⊢ (𝑥 ∈ (𝐵 ∪ (𝐴 ∖ 𝐵)) ↔ (𝑥 ∈ 𝐵 ∨ 𝑥 ∈ (𝐴 ∖ 𝐵)))
7	5, 6	sylib 121	. . . . . 6 ⊢ ((𝐴 = (𝐵 ∪ (𝐴 ∖ 𝐵)) ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ 𝐵 ∨ 𝑥 ∈ (𝐴 ∖ 𝐵)))
8		eldifn 3250	. . . . . . 7 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → ¬ 𝑥 ∈ 𝐵)
9	8	orim2i 756	. . . . . 6 ⊢ ((𝑥 ∈ 𝐵 ∨ 𝑥 ∈ (𝐴 ∖ 𝐵)) → (𝑥 ∈ 𝐵 ∨ ¬ 𝑥 ∈ 𝐵))
10	7, 9	syl 14	. . . . 5 ⊢ ((𝐴 = (𝐵 ∪ (𝐴 ∖ 𝐵)) ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ 𝐵 ∨ ¬ 𝑥 ∈ 𝐵))
11		df-dc 830	. . . . 5 ⊢ (DECID 𝑥 ∈ 𝐵 ↔ (𝑥 ∈ 𝐵 ∨ ¬ 𝑥 ∈ 𝐵))
12	10, 11	sylibr 133	. . . 4 ⊢ ((𝐴 = (𝐵 ∪ (𝐴 ∖ 𝐵)) ∧ 𝑥 ∈ 𝐴) → DECID 𝑥 ∈ 𝐵)
13	12	ralrimiva 2543	. . 3 ⊢ (𝐴 = (𝐵 ∪ (𝐴 ∖ 𝐵)) → ∀𝑥 ∈ 𝐴 DECID 𝑥 ∈ 𝐵)
14	3, 13	jca 304	. 2 ⊢ (𝐴 = (𝐵 ∪ (𝐴 ∖ 𝐵)) → (𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐴 DECID 𝑥 ∈ 𝐵))
15		elun1 3294	. . . . . . 7 ⊢ (𝑦 ∈ 𝐵 → 𝑦 ∈ (𝐵 ∪ (𝐴 ∖ 𝐵)))
16	15	adantl 275	. . . . . 6 ⊢ ((((𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐴 DECID 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ (𝐵 ∪ (𝐴 ∖ 𝐵)))
17		simplr 525	. . . . . . . 8 ⊢ ((((𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐴 DECID 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐴) ∧ ¬ 𝑦 ∈ 𝐵) → 𝑦 ∈ 𝐴)
18		simpr 109	. . . . . . . 8 ⊢ ((((𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐴 DECID 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐴) ∧ ¬ 𝑦 ∈ 𝐵) → ¬ 𝑦 ∈ 𝐵)
19	17, 18	eldifd 3131	. . . . . . 7 ⊢ ((((𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐴 DECID 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐴) ∧ ¬ 𝑦 ∈ 𝐵) → 𝑦 ∈ (𝐴 ∖ 𝐵))
20		elun2 3295	. . . . . . 7 ⊢ (𝑦 ∈ (𝐴 ∖ 𝐵) → 𝑦 ∈ (𝐵 ∪ (𝐴 ∖ 𝐵)))
21	19, 20	syl 14	. . . . . 6 ⊢ ((((𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐴 DECID 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐴) ∧ ¬ 𝑦 ∈ 𝐵) → 𝑦 ∈ (𝐵 ∪ (𝐴 ∖ 𝐵)))
22		eleq1 2233	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐵 ↔ 𝑦 ∈ 𝐵))
23	22	dcbid 833	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (DECID 𝑥 ∈ 𝐵 ↔ DECID 𝑦 ∈ 𝐵))
24		simplr 525	. . . . . . . 8 ⊢ (((𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐴 DECID 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐴) → ∀𝑥 ∈ 𝐴 DECID 𝑥 ∈ 𝐵)
25		simpr 109	. . . . . . . 8 ⊢ (((𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐴 DECID 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ 𝐴)
26	23, 24, 25	rspcdva 2839	. . . . . . 7 ⊢ (((𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐴 DECID 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐴) → DECID 𝑦 ∈ 𝐵)
27		exmiddc 831	. . . . . . 7 ⊢ (DECID 𝑦 ∈ 𝐵 → (𝑦 ∈ 𝐵 ∨ ¬ 𝑦 ∈ 𝐵))
28	26, 27	syl 14	. . . . . 6 ⊢ (((𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐴 DECID 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐴) → (𝑦 ∈ 𝐵 ∨ ¬ 𝑦 ∈ 𝐵))
29	16, 21, 28	mpjaodan 793	. . . . 5 ⊢ (((𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐴 DECID 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ (𝐵 ∪ (𝐴 ∖ 𝐵)))
30	29	ex 114	. . . 4 ⊢ ((𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐴 DECID 𝑥 ∈ 𝐵) → (𝑦 ∈ 𝐴 → 𝑦 ∈ (𝐵 ∪ (𝐴 ∖ 𝐵))))
31	30	ssrdv 3153	. . 3 ⊢ ((𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐴 DECID 𝑥 ∈ 𝐵) → 𝐴 ⊆ (𝐵 ∪ (𝐴 ∖ 𝐵)))
32	2	biimpi 119	. . . 4 ⊢ (𝐵 ⊆ 𝐴 → (𝐵 ∪ (𝐴 ∖ 𝐵)) ⊆ 𝐴)
33	32	adantr 274	. . 3 ⊢ ((𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐴 DECID 𝑥 ∈ 𝐵) → (𝐵 ∪ (𝐴 ∖ 𝐵)) ⊆ 𝐴)
34	31, 33	eqssd 3164	. 2 ⊢ ((𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐴 DECID 𝑥 ∈ 𝐵) → 𝐴 = (𝐵 ∪ (𝐴 ∖ 𝐵)))
35	14, 34	impbii 125	1 ⊢ (𝐴 = (𝐵 ∪ (𝐴 ∖ 𝐵)) ↔ (𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐴 DECID 𝑥 ∈ 𝐵))

Syntax hints:  ¬ wn 3   ∧ wa 103   ↔ wb 104   ∨ wo 703  DECID wdc 829   = wceq 1348   ∈ wcel 2141  ∀wral 2448   ∖ cdif 3118   ∪ cun 3119   ⊆ wss 3121

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-dc 830  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134

This theorem is referenced by:  sbthlemi5  6938  sbthlemi6  6939  exmidfodomrlemim  7178  bj-charfundcALT  13844