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Theorem findcard2s 8747
Description: Variation of findcard2 8746 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 416 . . 3 (𝑦 ∈ Fin → (¬ 𝑧𝑦 → (𝜒𝜃)))
8 uncom 4083 . . . . . . 7 ({𝑧} ∪ 𝑦) = (𝑦 ∪ {𝑧})
9 snssi 4704 . . . . . . . 8 (𝑧𝑦 → {𝑧} ⊆ 𝑦)
10 ssequn1 4110 . . . . . . . 8 ({𝑧} ⊆ 𝑦 ↔ ({𝑧} ∪ 𝑦) = 𝑦)
119, 10sylib 221 . . . . . . 7 (𝑧𝑦 → ({𝑧} ∪ 𝑦) = 𝑦)
128, 11syl5reqr 2851 . . . . . 6 (𝑧𝑦𝑦 = (𝑦 ∪ {𝑧}))
13 vex 3447 . . . . . . 7 𝑦 ∈ V
1413eqvinc 3593 . . . . . 6 (𝑦 = (𝑦 ∪ {𝑧}) ↔ ∃𝑥(𝑥 = 𝑦𝑥 = (𝑦 ∪ {𝑧})))
1512, 14sylib 221 . . . . 5 (𝑧𝑦 → ∃𝑥(𝑥 = 𝑦𝑥 = (𝑦 ∪ {𝑧})))
162bicomd 226 . . . . . . 7 (𝑥 = 𝑦 → (𝜒𝜑))
1716, 3sylan9bb 513 . . . . . 6 ((𝑥 = 𝑦𝑥 = (𝑦 ∪ {𝑧})) → (𝜒𝜃))
1817exlimiv 1931 . . . . 5 (∃𝑥(𝑥 = 𝑦𝑥 = (𝑦 ∪ {𝑧})) → (𝜒𝜃))
1915, 18syl 17 . . . 4 (𝑧𝑦 → (𝜒𝜃))
2019biimpd 232 . . 3 (𝑧𝑦 → (𝜒𝜃))
217, 20pm2.61d2 184 . 2 (𝑦 ∈ Fin → (𝜒𝜃))
221, 2, 3, 4, 5, 21findcard2 8746 1 (𝐴 ∈ Fin → 𝜏)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399   = wceq 1538  wex 1781  wcel 2112  cun 3882  wss 3884  c0 4246  {csn 4528  Fincfn 8496
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-om 7565  df-1o 8089  df-er 8276  df-en 8497  df-fin 8500
This theorem is referenced by:  findcard2d  8748  ac6sfi  8750  domunfican  8779  fodomfi  8785  hashxplem  13794  hashmap  13796  hashbc  13811  hashf1lem2  13814  hashf1  13815  fsum2d  15121  fsumabs  15151  fsumrlim  15161  fsumo1  15162  fsumiun  15171  incexclem  15186  fprod2d  15330  coprmprod  15998  coprmproddvds  16000  gsum2dlem2  19087  ablfac1eulem  19190  mplcoe1  20708  mplcoe5  20711  coe1fzgsumd  20934  evl1gsumd  20984  mdetunilem9  21228  ptcmpfi  22421  tmdgsum  22703  fsumcn  23478  ovolfiniun  24108  volfiniun  24154  itgfsum  24433  dvmptfsum  24581  jensen  25577  gsumle  30778  gsumvsca1  30907  gsumvsca2  30908  finixpnum  35035  matunitlindflem1  35046  pwslnm  40025  fnchoice  41645  dvmptfprod  42574
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