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Theorem findcard2s 9203
Description: Variation of findcard2 9202 requiring that the element added in the induction step not be a member of the original set. (Contributed by Paul Chapman, 30-Nov-2012.)
Hypotheses
Ref Expression
findcard2s.1 (𝑥 = ∅ → (𝜑𝜓))
findcard2s.2 (𝑥 = 𝑦 → (𝜑𝜒))
findcard2s.3 (𝑥 = (𝑦 ∪ {𝑧}) → (𝜑𝜃))
findcard2s.4 (𝑥 = 𝐴 → (𝜑𝜏))
findcard2s.5 𝜓
findcard2s.6 ((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) → (𝜒𝜃))
Assertion
Ref Expression
findcard2s (𝐴 ∈ Fin → 𝜏)
Distinct variable groups:   𝑥,𝐴,𝑦,𝑧   𝜒,𝑥   𝜑,𝑦,𝑧   𝜓,𝑥   𝜏,𝑥   𝜃,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦,𝑧)   𝜒(𝑦,𝑧)   𝜃(𝑦,𝑧)   𝜏(𝑦,𝑧)

Proof of Theorem findcard2s
StepHypRef Expression
1 findcard2s.1 . 2 (𝑥 = ∅ → (𝜑𝜓))
2 findcard2s.2 . 2 (𝑥 = 𝑦 → (𝜑𝜒))
3 findcard2s.3 . 2 (𝑥 = (𝑦 ∪ {𝑧}) → (𝜑𝜃))
4 findcard2s.4 . 2 (𝑥 = 𝐴 → (𝜑𝜏))
5 findcard2s.5 . 2 𝜓
6 findcard2s.6 . . . 4 ((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) → (𝜒𝜃))
76ex 412 . . 3 (𝑦 ∈ Fin → (¬ 𝑧𝑦 → (𝜒𝜃)))
8 snssi 4812 . . . . . . . 8 (𝑧𝑦 → {𝑧} ⊆ 𝑦)
9 ssequn1 4195 . . . . . . . 8 ({𝑧} ⊆ 𝑦 ↔ ({𝑧} ∪ 𝑦) = 𝑦)
108, 9sylib 218 . . . . . . 7 (𝑧𝑦 → ({𝑧} ∪ 𝑦) = 𝑦)
11 uncom 4167 . . . . . . 7 ({𝑧} ∪ 𝑦) = (𝑦 ∪ {𝑧})
1210, 11eqtr3di 2789 . . . . . 6 (𝑧𝑦𝑦 = (𝑦 ∪ {𝑧}))
13 vex 3481 . . . . . . 7 𝑦 ∈ V
1413eqvinc 3648 . . . . . 6 (𝑦 = (𝑦 ∪ {𝑧}) ↔ ∃𝑥(𝑥 = 𝑦𝑥 = (𝑦 ∪ {𝑧})))
1512, 14sylib 218 . . . . 5 (𝑧𝑦 → ∃𝑥(𝑥 = 𝑦𝑥 = (𝑦 ∪ {𝑧})))
162bicomd 223 . . . . . . 7 (𝑥 = 𝑦 → (𝜒𝜑))
1716, 3sylan9bb 509 . . . . . 6 ((𝑥 = 𝑦𝑥 = (𝑦 ∪ {𝑧})) → (𝜒𝜃))
1817exlimiv 1927 . . . . 5 (∃𝑥(𝑥 = 𝑦𝑥 = (𝑦 ∪ {𝑧})) → (𝜒𝜃))
1915, 18syl 17 . . . 4 (𝑧𝑦 → (𝜒𝜃))
2019biimpd 229 . . 3 (𝑧𝑦 → (𝜒𝜃))
217, 20pm2.61d2 181 . 2 (𝑦 ∈ Fin → (𝜒𝜃))
221, 2, 3, 4, 5, 21findcard2 9202 1 (𝐴 ∈ Fin → 𝜏)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395   = wceq 1536  wex 1775  wcel 2105  cun 3960  wss 3962  c0 4338  {csn 4630  Fincfn 8983
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-om 7887  df-en 8984  df-fin 8987
This theorem is referenced by:  findcard2d  9204  unfi  9209  ac6sfi  9317  fodomfi  9347  domunfican  9358  fodomfiOLD  9367  hashxplem  14468  hashmap  14470  hashbc  14488  hashf1lem2  14491  hashf1  14492  fsum2d  15803  fsumabs  15833  fsumrlim  15843  fsumo1  15844  fsumiun  15853  incexclem  15868  fprod2d  16013  coprmprod  16694  coprmproddvds  16696  gsum2dlem2  20003  ablfac1eulem  20106  mplcoe1  22072  mplcoe5  22075  coe1fzgsumd  22323  evl1gsumd  22376  mdetunilem9  22641  ptcmpfi  23836  tmdgsum  24118  fsumcn  24907  ovolfiniun  25549  volfiniun  25595  itgfsum  25876  dvmptfsum  26027  jensen  27046  gsumle  33083  gsumvsca1  33214  gsumvsca2  33215  finixpnum  37591  matunitlindflem1  37602  pwslnm  43082  fnchoice  44966  dvmptfprod  45900
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