Theorem undifdcss 6925

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 3212	. . . 4 ⊢ (𝐴 = (𝐵 ∪ (𝐴 ∖ 𝐵)) → (𝐵 ∪ (𝐴 ∖ 𝐵)) ⊆ 𝐴)
2		undifss 3505	. . . 4 ⊢ (𝐵 ⊆ 𝐴 ↔ (𝐵 ∪ (𝐴 ∖ 𝐵)) ⊆ 𝐴)
3	1, 2	sylibr 134	. . 3 ⊢ (𝐴 = (𝐵 ∪ (𝐴 ∖ 𝐵)) → 𝐵 ⊆ 𝐴)
4		eleq2 2241	. . . . . . . 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 3278	. . . . . . 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 3260	. . . . . . 7 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → ¬ 𝑥 ∈ 𝐵)
9	8	orim2i 761	. . . . . 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 835	. . . . 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 2550	. . 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 3304	. . . . . . 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 528	. . . . . . . 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 3141	. . . . . . 7 ⊢ ((((𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐴 DECID 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐴) ∧ ¬ 𝑦 ∈ 𝐵) → 𝑦 ∈ (𝐴 ∖ 𝐵))
20		elun2 3305	. . . . . . 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 2240	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐵 ↔ 𝑦 ∈ 𝐵))
23	22	dcbid 838	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (DECID 𝑥 ∈ 𝐵 ↔ DECID 𝑦 ∈ 𝐵))
24		simplr 528	. . . . . . . 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 2848	. . . . . . 7 ⊢ (((𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐴 DECID 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐴) → DECID 𝑦 ∈ 𝐵)
27		exmiddc 836	. . . . . . 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 798	. . . . 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 3163	. . 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 3174	. 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 708  DECID wdc 834   = wceq 1353   ∈ wcel 2148  ∀wral 2455   ∖ cdif 3128   ∪ cun 3129   ⊆ wss 3131

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-dc 835  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-v 2741  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144

This theorem is referenced by:  sbthlemi5  6963  sbthlemi6  6964  exmidfodomrlemim  7203  bj-charfundcALT  14701