Theorem undifdcss 7158

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 3283	. . . 4 ⊢ (𝐴 = (𝐵 ∪ (𝐴 ∖ 𝐵)) → (𝐵 ∪ (𝐴 ∖ 𝐵)) ⊆ 𝐴)
2		undifss 3577	. . . 4 ⊢ (𝐵 ⊆ 𝐴 ↔ (𝐵 ∪ (𝐴 ∖ 𝐵)) ⊆ 𝐴)
3	1, 2	sylibr 134	. . 3 ⊢ (𝐴 = (𝐵 ∪ (𝐴 ∖ 𝐵)) → 𝐵 ⊆ 𝐴)
4		eleq2 2295	. . . . . . . 8 ⊢ (𝐴 = (𝐵 ∪ (𝐴 ∖ 𝐵)) → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ (𝐵 ∪ (𝐴 ∖ 𝐵))))
5	4	biimpa 296	. . . . . . 7 ⊢ ((𝐴 = (𝐵 ∪ (𝐴 ∖ 𝐵)) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ (𝐵 ∪ (𝐴 ∖ 𝐵)))
6		elun 3350	. . . . . . 7 ⊢ (𝑥 ∈ (𝐵 ∪ (𝐴 ∖ 𝐵)) ↔ (𝑥 ∈ 𝐵 ∨ 𝑥 ∈ (𝐴 ∖ 𝐵)))
7	5, 6	sylib 122	. . . . . 6 ⊢ ((𝐴 = (𝐵 ∪ (𝐴 ∖ 𝐵)) ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ 𝐵 ∨ 𝑥 ∈ (𝐴 ∖ 𝐵)))
8		eldifn 3332	. . . . . . 7 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → ¬ 𝑥 ∈ 𝐵)
9	8	orim2i 769	. . . . . 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 843	. . . . 5 ⊢ (DECID 𝑥 ∈ 𝐵 ↔ (𝑥 ∈ 𝐵 ∨ ¬ 𝑥 ∈ 𝐵))
12	10, 11	sylibr 134	. . . 4 ⊢ ((𝐴 = (𝐵 ∪ (𝐴 ∖ 𝐵)) ∧ 𝑥 ∈ 𝐴) → DECID 𝑥 ∈ 𝐵)
13	12	ralrimiva 2606	. . 3 ⊢ (𝐴 = (𝐵 ∪ (𝐴 ∖ 𝐵)) → ∀𝑥 ∈ 𝐴 DECID 𝑥 ∈ 𝐵)
14	3, 13	jca 306	. 2 ⊢ (𝐴 = (𝐵 ∪ (𝐴 ∖ 𝐵)) → (𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐴 DECID 𝑥 ∈ 𝐵))
15		elun1 3376	. . . . . . 7 ⊢ (𝑦 ∈ 𝐵 → 𝑦 ∈ (𝐵 ∪ (𝐴 ∖ 𝐵)))
16	15	adantl 277	. . . . . 6 ⊢ ((((𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐴 DECID 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ (𝐵 ∪ (𝐴 ∖ 𝐵)))
17		simplr 529	. . . . . . . 8 ⊢ ((((𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐴 DECID 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐴) ∧ ¬ 𝑦 ∈ 𝐵) → 𝑦 ∈ 𝐴)
18		simpr 110	. . . . . . . 8 ⊢ ((((𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐴 DECID 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐴) ∧ ¬ 𝑦 ∈ 𝐵) → ¬ 𝑦 ∈ 𝐵)
19	17, 18	eldifd 3211	. . . . . . 7 ⊢ ((((𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐴 DECID 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐴) ∧ ¬ 𝑦 ∈ 𝐵) → 𝑦 ∈ (𝐴 ∖ 𝐵))
20		elun2 3377	. . . . . . 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 2294	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐵 ↔ 𝑦 ∈ 𝐵))
23	22	dcbid 846	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (DECID 𝑥 ∈ 𝐵 ↔ DECID 𝑦 ∈ 𝐵))
24		simplr 529	. . . . . . . 8 ⊢ (((𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐴 DECID 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐴) → ∀𝑥 ∈ 𝐴 DECID 𝑥 ∈ 𝐵)
25		simpr 110	. . . . . . . 8 ⊢ (((𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐴 DECID 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ 𝐴)
26	23, 24, 25	rspcdva 2916	. . . . . . 7 ⊢ (((𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐴 DECID 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐴) → DECID 𝑦 ∈ 𝐵)
27		exmiddc 844	. . . . . . 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 806	. . . . 5 ⊢ (((𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐴 DECID 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ (𝐵 ∪ (𝐴 ∖ 𝐵)))
30	29	ex 115	. . . 4 ⊢ ((𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐴 DECID 𝑥 ∈ 𝐵) → (𝑦 ∈ 𝐴 → 𝑦 ∈ (𝐵 ∪ (𝐴 ∖ 𝐵))))
31	30	ssrdv 3234	. . 3 ⊢ ((𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐴 DECID 𝑥 ∈ 𝐵) → 𝐴 ⊆ (𝐵 ∪ (𝐴 ∖ 𝐵)))
32	2	biimpi 120	. . . 4 ⊢ (𝐵 ⊆ 𝐴 → (𝐵 ∪ (𝐴 ∖ 𝐵)) ⊆ 𝐴)
33	32	adantr 276	. . 3 ⊢ ((𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐴 DECID 𝑥 ∈ 𝐵) → (𝐵 ∪ (𝐴 ∖ 𝐵)) ⊆ 𝐴)
34	31, 33	eqssd 3245	. 2 ⊢ ((𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐴 DECID 𝑥 ∈ 𝐵) → 𝐴 = (𝐵 ∪ (𝐴 ∖ 𝐵)))
35	14, 34	impbii 126	1 ⊢ (𝐴 = (𝐵 ∪ (𝐴 ∖ 𝐵)) ↔ (𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐴 DECID 𝑥 ∈ 𝐵))

Syntax hints:  ¬ wn 3   ∧ wa 104   ↔ wb 105   ∨ wo 716  DECID wdc 842   = wceq 1398   ∈ wcel 2202  ∀wral 2511   ∖ cdif 3198   ∪ cun 3199   ⊆ wss 3201

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-dc 843  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-v 2805  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214

This theorem is referenced by:  sbthlemi5  7203  sbthlemi6  7204  exmidfodomrlemim  7455  bj-charfundcALT  16508