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Theorem dcdifsnid 6400
 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 3666 from subset to equality but the proof relies on equality being decidable. (Contributed by Jim Kingdon, 17-Jun-2022.)
Assertion
Ref Expression
dcdifsnid ((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵𝐴) → ((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.
StepHypRef Expression
1 difsnss 3666 . . 3 (𝐵𝐴 → ((𝐴 ∖ {𝐵}) ∪ {𝐵}) ⊆ 𝐴)
21adantl 275 . 2 ((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵𝐴) → ((𝐴 ∖ {𝐵}) ∪ {𝐵}) ⊆ 𝐴)
3 simpr 109 . . . . . . 7 ((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵𝐴) ∧ 𝑧𝐴) ∧ 𝑧 = 𝐵) → 𝑧 = 𝐵)
4 velsn 3544 . . . . . . 7 (𝑧 ∈ {𝐵} ↔ 𝑧 = 𝐵)
53, 4sylibr 133 . . . . . 6 ((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵𝐴) ∧ 𝑧𝐴) ∧ 𝑧 = 𝐵) → 𝑧 ∈ {𝐵})
6 elun2 3244 . . . . . 6 (𝑧 ∈ {𝐵} → 𝑧 ∈ ((𝐴 ∖ {𝐵}) ∪ {𝐵}))
75, 6syl 14 . . . . 5 ((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵𝐴) ∧ 𝑧𝐴) ∧ 𝑧 = 𝐵) → 𝑧 ∈ ((𝐴 ∖ {𝐵}) ∪ {𝐵}))
8 simplr 519 . . . . . . 7 ((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵𝐴) ∧ 𝑧𝐴) ∧ ¬ 𝑧 = 𝐵) → 𝑧𝐴)
9 simpr 109 . . . . . . . 8 ((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵𝐴) ∧ 𝑧𝐴) ∧ ¬ 𝑧 = 𝐵) → ¬ 𝑧 = 𝐵)
109, 4sylnibr 666 . . . . . . 7 ((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵𝐴) ∧ 𝑧𝐴) ∧ ¬ 𝑧 = 𝐵) → ¬ 𝑧 ∈ {𝐵})
118, 10eldifd 3081 . . . . . 6 ((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵𝐴) ∧ 𝑧𝐴) ∧ ¬ 𝑧 = 𝐵) → 𝑧 ∈ (𝐴 ∖ {𝐵}))
12 elun1 3243 . . . . . 6 (𝑧 ∈ (𝐴 ∖ {𝐵}) → 𝑧 ∈ ((𝐴 ∖ {𝐵}) ∪ {𝐵}))
1311, 12syl 14 . . . . 5 ((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵𝐴) ∧ 𝑧𝐴) ∧ ¬ 𝑧 = 𝐵) → 𝑧 ∈ ((𝐴 ∖ {𝐵}) ∪ {𝐵}))
14 simpll 518 . . . . . . 7 (((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵𝐴) ∧ 𝑧𝐴) → ∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦)
15 simpr 109 . . . . . . . 8 (((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵𝐴) ∧ 𝑧𝐴) → 𝑧𝐴)
16 simplr 519 . . . . . . . 8 (((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵𝐴) ∧ 𝑧𝐴) → 𝐵𝐴)
17 equequ1 1688 . . . . . . . . . 10 (𝑥 = 𝑧 → (𝑥 = 𝑦𝑧 = 𝑦))
1817dcbid 823 . . . . . . . . 9 (𝑥 = 𝑧 → (DECID 𝑥 = 𝑦DECID 𝑧 = 𝑦))
19 eqeq2 2149 . . . . . . . . . 10 (𝑦 = 𝐵 → (𝑧 = 𝑦𝑧 = 𝐵))
2019dcbid 823 . . . . . . . . 9 (𝑦 = 𝐵 → (DECID 𝑧 = 𝑦DECID 𝑧 = 𝐵))
2118, 20rspc2v 2802 . . . . . . . 8 ((𝑧𝐴𝐵𝐴) → (∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦DECID 𝑧 = 𝐵))
2215, 16, 21syl2anc 408 . . . . . . 7 (((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵𝐴) ∧ 𝑧𝐴) → (∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦DECID 𝑧 = 𝐵))
2314, 22mpd 13 . . . . . 6 (((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵𝐴) ∧ 𝑧𝐴) → DECID 𝑧 = 𝐵)
24 exmiddc 821 . . . . . 6 (DECID 𝑧 = 𝐵 → (𝑧 = 𝐵 ∨ ¬ 𝑧 = 𝐵))
2523, 24syl 14 . . . . 5 (((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵𝐴) ∧ 𝑧𝐴) → (𝑧 = 𝐵 ∨ ¬ 𝑧 = 𝐵))
267, 13, 25mpjaodan 787 . . . 4 (((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵𝐴) ∧ 𝑧𝐴) → 𝑧 ∈ ((𝐴 ∖ {𝐵}) ∪ {𝐵}))
2726ex 114 . . 3 ((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵𝐴) → (𝑧𝐴𝑧 ∈ ((𝐴 ∖ {𝐵}) ∪ {𝐵})))
2827ssrdv 3103 . 2 ((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵𝐴) → 𝐴 ⊆ ((𝐴 ∖ {𝐵}) ∪ {𝐵}))
292, 28eqssd 3114 1 ((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵𝐴) → ((𝐴 ∖ {𝐵}) ∪ {𝐵}) = 𝐴)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ∨ wo 697  DECID wdc 819   = wceq 1331   ∈ wcel 1480  ∀wral 2416   ∖ cdif 3068   ∪ cun 3069   ⊆ wss 3071  {csn 3527 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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121 This theorem depends on definitions:  df-bi 116  df-dc 820  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-v 2688  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-sn 3533 This theorem is referenced by:  fnsnsplitdc  6401  nndifsnid  6403  fidifsnid  6765  undifdc  6812
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