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Theorem fidifsnid 6667
Description: If we remove a single element from a finite set then put it back in, we end up with the original finite set. This strengthens difsnss 3605 from subset to equality when the set is finite. (Contributed by Jim Kingdon, 9-Sep-2021.)
Assertion
Ref Expression
fidifsnid ((𝐴 ∈ Fin ∧ 𝐵𝐴) → ((𝐴 ∖ {𝐵}) ∪ {𝐵}) = 𝐴)

Proof of Theorem fidifsnid
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fidceq 6665 . . . 4 ((𝐴 ∈ Fin ∧ 𝑥𝐴𝑦𝐴) → DECID 𝑥 = 𝑦)
213expb 1147 . . 3 ((𝐴 ∈ Fin ∧ (𝑥𝐴𝑦𝐴)) → DECID 𝑥 = 𝑦)
32ralrimivva 2467 . 2 (𝐴 ∈ Fin → ∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦)
4 dcdifsnid 6303 . 2 ((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵𝐴) → ((𝐴 ∖ {𝐵}) ∪ {𝐵}) = 𝐴)
53, 4sylan 278 1 ((𝐴 ∈ Fin ∧ 𝐵𝐴) → ((𝐴 ∖ {𝐵}) ∪ {𝐵}) = 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  DECID wdc 783   = wceq 1296  wcel 1445  wral 2370  cdif 3010  cun 3011  {csn 3466  Fincfn 6537
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 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-nul 3986  ax-pow 4030  ax-pr 4060  ax-un 4284  ax-setind 4381  ax-iinf 4431
This theorem depends on definitions:  df-bi 116  df-dc 784  df-3or 928  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ne 2263  df-ral 2375  df-rex 2376  df-v 2635  df-sbc 2855  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-nul 3303  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-int 3711  df-br 3868  df-opab 3922  df-tr 3959  df-id 4144  df-iord 4217  df-on 4219  df-suc 4222  df-iom 4434  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-rn 4478  df-iota 5014  df-fun 5051  df-fn 5052  df-f 5053  df-f1 5054  df-fo 5055  df-f1o 5056  df-fv 5057  df-en 6538  df-fin 6540
This theorem is referenced by:  findcard2  6685  findcard2s  6686  xpfi  6720  fisseneq  6722  zfz1isolem1  10376
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