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Theorem nndifsnid 6407
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 3670 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 4523 . . . . . 6 ((𝑥𝐴𝐴 ∈ ω) → 𝑥 ∈ ω)
21expcom 115 . . . . 5 (𝐴 ∈ ω → (𝑥𝐴𝑥 ∈ ω))
3 elnn 4523 . . . . . 6 ((𝑦𝐴𝐴 ∈ ω) → 𝑦 ∈ ω)
43expcom 115 . . . . 5 (𝐴 ∈ ω → (𝑦𝐴𝑦 ∈ ω))
52, 4anim12d 333 . . . 4 (𝐴 ∈ ω → ((𝑥𝐴𝑦𝐴) → (𝑥 ∈ ω ∧ 𝑦 ∈ ω)))
6 nndceq 6399 . . . 4 ((𝑥 ∈ ω ∧ 𝑦 ∈ ω) → DECID 𝑥 = 𝑦)
75, 6syl6 33 . . 3 (𝐴 ∈ ω → ((𝑥𝐴𝑦𝐴) → DECID 𝑥 = 𝑦))
87ralrimivv 2514 . 2 (𝐴 ∈ ω → ∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦)
9 dcdifsnid 6404 . 2 ((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵𝐴) → ((𝐴 ∖ {𝐵}) ∪ {𝐵}) = 𝐴)
108, 9sylan 281 1 ((𝐴 ∈ ω ∧ 𝐵𝐴) → ((𝐴 ∖ {𝐵}) ∪ {𝐵}) = 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  DECID wdc 820   = wceq 1332  wcel 1481  wral 2417  cdif 3069  cun 3070  {csn 3528  ωcom 4508
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 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4050  ax-nul 4058  ax-pow 4102  ax-pr 4135  ax-un 4359  ax-setind 4456  ax-iinf 4506
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-v 2689  df-dif 3074  df-un 3076  df-in 3078  df-ss 3085  df-nul 3365  df-pw 3513  df-sn 3534  df-pr 3535  df-uni 3741  df-int 3776  df-tr 4031  df-iord 4292  df-on 4294  df-suc 4297  df-iom 4509
This theorem is referenced by:  phplem2  6751
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