Theorem dcdifsnid 6483

Description: If we remove a single element from a set with decidable equality then put it back in, we end up with the original set. This strengthens difsnss 3726 from subset to equality but the proof relies on equality being decidable. (Contributed by Jim Kingdon, 17-Jun-2022.)

Assertion

Ref	Expression
dcdifsnid	⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝐵 ∈ 𝐴) → ((𝐴 ∖ {𝐵}) ∪ {𝐵}) = 𝐴)

Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦

Proof of Theorem dcdifsnid

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		difsnss 3726	. . 3 ⊢ (𝐵 ∈ 𝐴 → ((𝐴 ∖ {𝐵}) ∪ {𝐵}) ⊆ 𝐴)
2	1	adantl 275	. 2 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝐵 ∈ 𝐴) → ((𝐴 ∖ {𝐵}) ∪ {𝐵}) ⊆ 𝐴)
3		simpr 109	. . . . . . 7 ⊢ ((((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝐵 ∈ 𝐴) ∧ 𝑧 ∈ 𝐴) ∧ 𝑧 = 𝐵) → 𝑧 = 𝐵)
4		velsn 3600	. . . . . . 7 ⊢ (𝑧 ∈ {𝐵} ↔ 𝑧 = 𝐵)
5	3, 4	sylibr 133	. . . . . 6 ⊢ ((((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝐵 ∈ 𝐴) ∧ 𝑧 ∈ 𝐴) ∧ 𝑧 = 𝐵) → 𝑧 ∈ {𝐵})
6		elun2 3295	. . . . . 6 ⊢ (𝑧 ∈ {𝐵} → 𝑧 ∈ ((𝐴 ∖ {𝐵}) ∪ {𝐵}))
7	5, 6	syl 14	. . . . 5 ⊢ ((((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝐵 ∈ 𝐴) ∧ 𝑧 ∈ 𝐴) ∧ 𝑧 = 𝐵) → 𝑧 ∈ ((𝐴 ∖ {𝐵}) ∪ {𝐵}))
8		simplr 525	. . . . . . 7 ⊢ ((((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝐵 ∈ 𝐴) ∧ 𝑧 ∈ 𝐴) ∧ ¬ 𝑧 = 𝐵) → 𝑧 ∈ 𝐴)
9		simpr 109	. . . . . . . 8 ⊢ ((((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝐵 ∈ 𝐴) ∧ 𝑧 ∈ 𝐴) ∧ ¬ 𝑧 = 𝐵) → ¬ 𝑧 = 𝐵)
10	9, 4	sylnibr 672	. . . . . . 7 ⊢ ((((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝐵 ∈ 𝐴) ∧ 𝑧 ∈ 𝐴) ∧ ¬ 𝑧 = 𝐵) → ¬ 𝑧 ∈ {𝐵})
11	8, 10	eldifd 3131	. . . . . 6 ⊢ ((((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝐵 ∈ 𝐴) ∧ 𝑧 ∈ 𝐴) ∧ ¬ 𝑧 = 𝐵) → 𝑧 ∈ (𝐴 ∖ {𝐵}))
12		elun1 3294	. . . . . 6 ⊢ (𝑧 ∈ (𝐴 ∖ {𝐵}) → 𝑧 ∈ ((𝐴 ∖ {𝐵}) ∪ {𝐵}))
13	11, 12	syl 14	. . . . 5 ⊢ ((((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝐵 ∈ 𝐴) ∧ 𝑧 ∈ 𝐴) ∧ ¬ 𝑧 = 𝐵) → 𝑧 ∈ ((𝐴 ∖ {𝐵}) ∪ {𝐵}))
14		simpll 524	. . . . . . 7 ⊢ (((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝐵 ∈ 𝐴) ∧ 𝑧 ∈ 𝐴) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦)
15		simpr 109	. . . . . . . 8 ⊢ (((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝐵 ∈ 𝐴) ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ 𝐴)
16		simplr 525	. . . . . . . 8 ⊢ (((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝐵 ∈ 𝐴) ∧ 𝑧 ∈ 𝐴) → 𝐵 ∈ 𝐴)
17		equequ1 1705	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (𝑥 = 𝑦 ↔ 𝑧 = 𝑦))
18	17	dcbid 833	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → (DECID 𝑥 = 𝑦 ↔ DECID 𝑧 = 𝑦))
19		eqeq2 2180	. . . . . . . . . 10 ⊢ (𝑦 = 𝐵 → (𝑧 = 𝑦 ↔ 𝑧 = 𝐵))
20	19	dcbid 833	. . . . . . . . 9 ⊢ (𝑦 = 𝐵 → (DECID 𝑧 = 𝑦 ↔ DECID 𝑧 = 𝐵))
21	18, 20	rspc2v 2847	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝐵 ∈ 𝐴) → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 → DECID 𝑧 = 𝐵))
22	15, 16, 21	syl2anc 409	. . . . . . 7 ⊢ (((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝐵 ∈ 𝐴) ∧ 𝑧 ∈ 𝐴) → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 → DECID 𝑧 = 𝐵))
23	14, 22	mpd 13	. . . . . 6 ⊢ (((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝐵 ∈ 𝐴) ∧ 𝑧 ∈ 𝐴) → DECID 𝑧 = 𝐵)
24		exmiddc 831	. . . . . 6 ⊢ (DECID 𝑧 = 𝐵 → (𝑧 = 𝐵 ∨ ¬ 𝑧 = 𝐵))
25	23, 24	syl 14	. . . . 5 ⊢ (((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝐵 ∈ 𝐴) ∧ 𝑧 ∈ 𝐴) → (𝑧 = 𝐵 ∨ ¬ 𝑧 = 𝐵))
26	7, 13, 25	mpjaodan 793	. . . 4 ⊢ (((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝐵 ∈ 𝐴) ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ ((𝐴 ∖ {𝐵}) ∪ {𝐵}))
27	26	ex 114	. . 3 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝐵 ∈ 𝐴) → (𝑧 ∈ 𝐴 → 𝑧 ∈ ((𝐴 ∖ {𝐵}) ∪ {𝐵})))
28	27	ssrdv 3153	. 2 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝐵 ∈ 𝐴) → 𝐴 ⊆ ((𝐴 ∖ {𝐵}) ∪ {𝐵}))
29	2, 28	eqssd 3164	1 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝐵 ∈ 𝐴) → ((𝐴 ∖ {𝐵}) ∪ {𝐵}) = 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ∨ wo 703  DECID wdc 829   = wceq 1348   ∈ wcel 2141  ∀wral 2448   ∖ cdif 3118   ∪ cun 3119   ⊆ wss 3121  {csn 3583

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  df-sn 3589

This theorem is referenced by:  fnsnsplitdc  6484  nndifsnid  6486  fidifsnid  6849  undifdc  6901