Theorem dcdifsnid 6750

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 3845 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 3845	. . 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 3711	. . . . . . 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 3391	. . . . . 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 529	. . . . . . 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 684	. . . . . . 7 ⊢ ((((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝐵 ∈ 𝐴) ∧ 𝑧 ∈ 𝐴) ∧ ¬ 𝑧 = 𝐵) → ¬ 𝑧 ∈ {𝐵})
11	8, 10	eldifd 3224	. . . . . 6 ⊢ ((((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝐵 ∈ 𝐴) ∧ 𝑧 ∈ 𝐴) ∧ ¬ 𝑧 = 𝐵) → 𝑧 ∈ (𝐴 ∖ {𝐵}))
12		elun1 3390	. . . . . 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 529	. . . . . . . 8 ⊢ (((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝐵 ∈ 𝐴) ∧ 𝑧 ∈ 𝐴) → 𝐵 ∈ 𝐴)
17		equequ1 1760	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (𝑥 = 𝑦 ↔ 𝑧 = 𝑦))
18	17	dcbid 846	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → (DECID 𝑥 = 𝑦 ↔ DECID 𝑧 = 𝑦))
19		eqeq2 2244	. . . . . . . . . 10 ⊢ (𝑦 = 𝐵 → (𝑧 = 𝑦 ↔ 𝑧 = 𝐵))
20	19	dcbid 846	. . . . . . . . 9 ⊢ (𝑦 = 𝐵 → (DECID 𝑧 = 𝑦 ↔ DECID 𝑧 = 𝐵))
21	18, 20	rspc2v 2937	. . . . . . . 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 844	. . . . . 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 806	. . . 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 3248	. 2 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝐵 ∈ 𝐴) → 𝐴 ⊆ ((𝐴 ∖ {𝐵}) ∪ {𝐵}))
29	2, 28	eqssd 3259	1 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝐵 ∈ 𝐴) → ((𝐴 ∖ {𝐵}) ∪ {𝐵}) = 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∨ wo 716  DECID wdc 842   = wceq 1398   ∈ wcel 2205  ∀wral 2522   ∖ cdif 3211   ∪ cun 3212   ⊆ wss 3214  {csn 3694

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 2216

This theorem depends on definitions:  df-bi 117  df-dc 843  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-v 2817  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-sn 3700

This theorem is referenced by:  fnsnsplitdc  6751  nndifsnid  6753  fidifsnid  7139  undifdc  7197