Theorem dcdifsnid 6562

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 3768 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 3768	. . 3 ⊢ (𝐵 ∈ 𝐴 → ((𝐴 ∖ {𝐵}) ∪ {𝐵}) ⊆ 𝐴)
2	1	adantl 277	. 2 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝐵 ∈ 𝐴) → ((𝐴 ∖ {𝐵}) ∪ {𝐵}) ⊆ 𝐴)
3		simpr 110	. . . . . . 7 ⊢ ((((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝐵 ∈ 𝐴) ∧ 𝑧 ∈ 𝐴) ∧ 𝑧 = 𝐵) → 𝑧 = 𝐵)
4		velsn 3639	. . . . . . 7 ⊢ (𝑧 ∈ {𝐵} ↔ 𝑧 = 𝐵)
5	3, 4	sylibr 134	. . . . . 6 ⊢ ((((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝐵 ∈ 𝐴) ∧ 𝑧 ∈ 𝐴) ∧ 𝑧 = 𝐵) → 𝑧 ∈ {𝐵})
6		elun2 3331	. . . . . 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 528	. . . . . . 7 ⊢ ((((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝐵 ∈ 𝐴) ∧ 𝑧 ∈ 𝐴) ∧ ¬ 𝑧 = 𝐵) → 𝑧 ∈ 𝐴)
9		simpr 110	. . . . . . . 8 ⊢ ((((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝐵 ∈ 𝐴) ∧ 𝑧 ∈ 𝐴) ∧ ¬ 𝑧 = 𝐵) → ¬ 𝑧 = 𝐵)
10	9, 4	sylnibr 678	. . . . . . 7 ⊢ ((((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝐵 ∈ 𝐴) ∧ 𝑧 ∈ 𝐴) ∧ ¬ 𝑧 = 𝐵) → ¬ 𝑧 ∈ {𝐵})
11	8, 10	eldifd 3167	. . . . . 6 ⊢ ((((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝐵 ∈ 𝐴) ∧ 𝑧 ∈ 𝐴) ∧ ¬ 𝑧 = 𝐵) → 𝑧 ∈ (𝐴 ∖ {𝐵}))
12		elun1 3330	. . . . . 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 527	. . . . . . 7 ⊢ (((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝐵 ∈ 𝐴) ∧ 𝑧 ∈ 𝐴) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦)
15		simpr 110	. . . . . . . 8 ⊢ (((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝐵 ∈ 𝐴) ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ 𝐴)
16		simplr 528	. . . . . . . 8 ⊢ (((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝐵 ∈ 𝐴) ∧ 𝑧 ∈ 𝐴) → 𝐵 ∈ 𝐴)
17		equequ1 1726	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (𝑥 = 𝑦 ↔ 𝑧 = 𝑦))
18	17	dcbid 839	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → (DECID 𝑥 = 𝑦 ↔ DECID 𝑧 = 𝑦))
19		eqeq2 2206	. . . . . . . . . 10 ⊢ (𝑦 = 𝐵 → (𝑧 = 𝑦 ↔ 𝑧 = 𝐵))
20	19	dcbid 839	. . . . . . . . 9 ⊢ (𝑦 = 𝐵 → (DECID 𝑧 = 𝑦 ↔ DECID 𝑧 = 𝐵))
21	18, 20	rspc2v 2881	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝐵 ∈ 𝐴) → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 → DECID 𝑧 = 𝐵))
22	15, 16, 21	syl2anc 411	. . . . . . 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 837	. . . . . 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 799	. . . 4 ⊢ (((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝐵 ∈ 𝐴) ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ ((𝐴 ∖ {𝐵}) ∪ {𝐵}))
27	26	ex 115	. . 3 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝐵 ∈ 𝐴) → (𝑧 ∈ 𝐴 → 𝑧 ∈ ((𝐴 ∖ {𝐵}) ∪ {𝐵})))
28	27	ssrdv 3189	. 2 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝐵 ∈ 𝐴) → 𝐴 ⊆ ((𝐴 ∖ {𝐵}) ∪ {𝐵}))
29	2, 28	eqssd 3200	1 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝐵 ∈ 𝐴) → ((𝐴 ∖ {𝐵}) ∪ {𝐵}) = 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∨ wo 709  DECID wdc 835   = wceq 1364   ∈ wcel 2167  ∀wral 2475   ∖ cdif 3154   ∪ cun 3155   ⊆ wss 3157  {csn 3622

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-dc 836  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-v 2765  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-sn 3628

This theorem is referenced by:  fnsnsplitdc  6563  nndifsnid  6565  fidifsnid  6932  undifdc  6985