Theorem dcdifsnid 6440

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 3698 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 3698	. . 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 3573	. . . . . . 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 3271	. . . . . 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 520	. . . . . . 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 667	. . . . . . 7 ⊢ ((((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝐵 ∈ 𝐴) ∧ 𝑧 ∈ 𝐴) ∧ ¬ 𝑧 = 𝐵) → ¬ 𝑧 ∈ {𝐵})
11	8, 10	eldifd 3108	. . . . . 6 ⊢ ((((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝐵 ∈ 𝐴) ∧ 𝑧 ∈ 𝐴) ∧ ¬ 𝑧 = 𝐵) → 𝑧 ∈ (𝐴 ∖ {𝐵}))
12		elun1 3270	. . . . . 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 519	. . . . . . 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 520	. . . . . . . 8 ⊢ (((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝐵 ∈ 𝐴) ∧ 𝑧 ∈ 𝐴) → 𝐵 ∈ 𝐴)
17		equequ1 1689	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (𝑥 = 𝑦 ↔ 𝑧 = 𝑦))
18	17	dcbid 824	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → (DECID 𝑥 = 𝑦 ↔ DECID 𝑧 = 𝑦))
19		eqeq2 2164	. . . . . . . . . 10 ⊢ (𝑦 = 𝐵 → (𝑧 = 𝑦 ↔ 𝑧 = 𝐵))
20	19	dcbid 824	. . . . . . . . 9 ⊢ (𝑦 = 𝐵 → (DECID 𝑧 = 𝑦 ↔ DECID 𝑧 = 𝐵))
21	18, 20	rspc2v 2826	. . . . . . . 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 822	. . . . . 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 788	. . . 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 3130	. 2 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝐵 ∈ 𝐴) → 𝐴 ⊆ ((𝐴 ∖ {𝐵}) ∪ {𝐵}))
29	2, 28	eqssd 3141	1 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝐵 ∈ 𝐴) → ((𝐴 ∖ {𝐵}) ∪ {𝐵}) = 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ∨ wo 698  DECID wdc 820   = wceq 1332   ∈ wcel 2125  ∀wral 2432   ∖ cdif 3095   ∪ cun 3096   ⊆ wss 3098  {csn 3556

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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-ext 2136

This theorem depends on definitions:  df-bi 116  df-dc 821  df-tru 1335  df-nf 1438  df-sb 1740  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-ral 2437  df-v 2711  df-dif 3100  df-un 3102  df-in 3104  df-ss 3111  df-sn 3562

This theorem is referenced by:  fnsnsplitdc  6441  nndifsnid  6443  fidifsnid  6805  undifdc  6857