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Theorem nndifsnid 6333
Description: If we remove a single element from a natural number then put it back in, we end up with the original natural number. This strengthens difsnss 3613 from subset to equality but the proof relies on equality being decidable. (Contributed by Jim Kingdon, 31-Aug-2021.)
Assertion
Ref Expression
nndifsnid ((𝐴 ∈ ω ∧ 𝐵𝐴) → ((𝐴 ∖ {𝐵}) ∪ {𝐵}) = 𝐴)

Proof of Theorem nndifsnid
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elnn 4457 . . . . . 6 ((𝑥𝐴𝐴 ∈ ω) → 𝑥 ∈ ω)
21expcom 115 . . . . 5 (𝐴 ∈ ω → (𝑥𝐴𝑥 ∈ ω))
3 elnn 4457 . . . . . 6 ((𝑦𝐴𝐴 ∈ ω) → 𝑦 ∈ ω)
43expcom 115 . . . . 5 (𝐴 ∈ ω → (𝑦𝐴𝑦 ∈ ω))
52, 4anim12d 331 . . . 4 (𝐴 ∈ ω → ((𝑥𝐴𝑦𝐴) → (𝑥 ∈ ω ∧ 𝑦 ∈ ω)))
6 nndceq 6325 . . . 4 ((𝑥 ∈ ω ∧ 𝑦 ∈ ω) → DECID 𝑥 = 𝑦)
75, 6syl6 33 . . 3 (𝐴 ∈ ω → ((𝑥𝐴𝑦𝐴) → DECID 𝑥 = 𝑦))
87ralrimivv 2472 . 2 (𝐴 ∈ ω → ∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦)
9 dcdifsnid 6330 . 2 ((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵𝐴) → ((𝐴 ∖ {𝐵}) ∪ {𝐵}) = 𝐴)
108, 9sylan 279 1 ((𝐴 ∈ ω ∧ 𝐵𝐴) → ((𝐴 ∖ {𝐵}) ∪ {𝐵}) = 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  DECID wdc 786   = wceq 1299  wcel 1448  wral 2375  cdif 3018  cun 3019  {csn 3474  ωcom 4442
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-nul 3994  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390  ax-iinf 4440
This theorem depends on definitions:  df-bi 116  df-dc 787  df-3or 931  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-ral 2380  df-rex 2381  df-v 2643  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-nul 3311  df-pw 3459  df-sn 3480  df-pr 3481  df-uni 3684  df-int 3719  df-tr 3967  df-iord 4226  df-on 4228  df-suc 4231  df-iom 4443
This theorem is referenced by:  phplem2  6676
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