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Theorem findcard2s 9167
Description: Variation of findcard2 9166 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 411 . . 3 (𝑦 ∈ Fin → (¬ 𝑧𝑦 → (𝜒𝜃)))
8 snssi 4810 . . . . . . . 8 (𝑧𝑦 → {𝑧} ⊆ 𝑦)
9 ssequn1 4179 . . . . . . . 8 ({𝑧} ⊆ 𝑦 ↔ ({𝑧} ∪ 𝑦) = 𝑦)
108, 9sylib 217 . . . . . . 7 (𝑧𝑦 → ({𝑧} ∪ 𝑦) = 𝑦)
11 uncom 4152 . . . . . . 7 ({𝑧} ∪ 𝑦) = (𝑦 ∪ {𝑧})
1210, 11eqtr3di 2785 . . . . . 6 (𝑧𝑦𝑦 = (𝑦 ∪ {𝑧}))
13 vex 3476 . . . . . . 7 𝑦 ∈ V
1413eqvinc 3636 . . . . . 6 (𝑦 = (𝑦 ∪ {𝑧}) ↔ ∃𝑥(𝑥 = 𝑦𝑥 = (𝑦 ∪ {𝑧})))
1512, 14sylib 217 . . . . 5 (𝑧𝑦 → ∃𝑥(𝑥 = 𝑦𝑥 = (𝑦 ∪ {𝑧})))
162bicomd 222 . . . . . . 7 (𝑥 = 𝑦 → (𝜒𝜑))
1716, 3sylan9bb 508 . . . . . 6 ((𝑥 = 𝑦𝑥 = (𝑦 ∪ {𝑧})) → (𝜒𝜃))
1817exlimiv 1931 . . . . 5 (∃𝑥(𝑥 = 𝑦𝑥 = (𝑦 ∪ {𝑧})) → (𝜒𝜃))
1915, 18syl 17 . . . 4 (𝑧𝑦 → (𝜒𝜃))
2019biimpd 228 . . 3 (𝑧𝑦 → (𝜒𝜃))
217, 20pm2.61d2 181 . 2 (𝑦 ∈ Fin → (𝜒𝜃))
221, 2, 3, 4, 5, 21findcard2 9166 1 (𝐴 ∈ Fin → 𝜏)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394   = wceq 1539  wex 1779  wcel 2104  cun 3945  wss 3947  c0 4321  {csn 4627  Fincfn 8941
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-om 7858  df-en 8942  df-fin 8945
This theorem is referenced by:  findcard2d  9168  unfi  9174  ac6sfi  9289  domunfican  9322  fodomfi  9327  hashxplem  14397  hashmap  14399  hashbc  14416  hashf1lem2  14421  hashf1  14422  fsum2d  15721  fsumabs  15751  fsumrlim  15761  fsumo1  15762  fsumiun  15771  incexclem  15786  fprod2d  15929  coprmprod  16602  coprmproddvds  16604  gsum2dlem2  19880  ablfac1eulem  19983  mplcoe1  21811  mplcoe5  21814  coe1fzgsumd  22046  evl1gsumd  22096  mdetunilem9  22342  ptcmpfi  23537  tmdgsum  23819  fsumcn  24608  ovolfiniun  25250  volfiniun  25296  itgfsum  25576  dvmptfsum  25727  jensen  26729  gsumle  32512  gsumvsca1  32641  gsumvsca2  32642  finixpnum  36776  matunitlindflem1  36787  pwslnm  42138  fnchoice  44015  dvmptfprod  44959
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