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Theorem findcard 9188
Description: Schema for induction on the cardinality of a finite set. The inductive hypothesis is that the result is true on the given set with any one element removed. The result is then proven to be true for all finite sets. (Contributed by Jeff Madsen, 2-Sep-2009.)
Hypotheses
Ref Expression
findcard.1 (𝑥 = ∅ → (𝜑𝜓))
findcard.2 (𝑥 = (𝑦 ∖ {𝑧}) → (𝜑𝜒))
findcard.3 (𝑥 = 𝑦 → (𝜑𝜃))
findcard.4 (𝑥 = 𝐴 → (𝜑𝜏))
findcard.5 𝜓
findcard.6 (𝑦 ∈ Fin → (∀𝑧𝑦 𝜒𝜃))
Assertion
Ref Expression
findcard (𝐴 ∈ Fin → 𝜏)
Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝜓,𝑥   𝜒,𝑥   𝜃,𝑥   𝜏,𝑥   𝜑,𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦,𝑧)   𝜒(𝑦,𝑧)   𝜃(𝑦,𝑧)   𝜏(𝑦,𝑧)

Proof of Theorem findcard
Dummy variables 𝑤 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 findcard.4 . 2 (𝑥 = 𝐴 → (𝜑𝜏))
2 isfi 8997 . . 3 (𝑥 ∈ Fin ↔ ∃𝑤 ∈ ω 𝑥𝑤)
3 breq2 5153 . . . . . . . 8 (𝑤 = ∅ → (𝑥𝑤𝑥 ≈ ∅))
43imbi1d 340 . . . . . . 7 (𝑤 = ∅ → ((𝑥𝑤𝜑) ↔ (𝑥 ≈ ∅ → 𝜑)))
54albidv 1915 . . . . . 6 (𝑤 = ∅ → (∀𝑥(𝑥𝑤𝜑) ↔ ∀𝑥(𝑥 ≈ ∅ → 𝜑)))
6 breq2 5153 . . . . . . . 8 (𝑤 = 𝑣 → (𝑥𝑤𝑥𝑣))
76imbi1d 340 . . . . . . 7 (𝑤 = 𝑣 → ((𝑥𝑤𝜑) ↔ (𝑥𝑣𝜑)))
87albidv 1915 . . . . . 6 (𝑤 = 𝑣 → (∀𝑥(𝑥𝑤𝜑) ↔ ∀𝑥(𝑥𝑣𝜑)))
9 breq2 5153 . . . . . . . 8 (𝑤 = suc 𝑣 → (𝑥𝑤𝑥 ≈ suc 𝑣))
109imbi1d 340 . . . . . . 7 (𝑤 = suc 𝑣 → ((𝑥𝑤𝜑) ↔ (𝑥 ≈ suc 𝑣𝜑)))
1110albidv 1915 . . . . . 6 (𝑤 = suc 𝑣 → (∀𝑥(𝑥𝑤𝜑) ↔ ∀𝑥(𝑥 ≈ suc 𝑣𝜑)))
12 en0 9038 . . . . . . . 8 (𝑥 ≈ ∅ ↔ 𝑥 = ∅)
13 findcard.5 . . . . . . . . 9 𝜓
14 findcard.1 . . . . . . . . 9 (𝑥 = ∅ → (𝜑𝜓))
1513, 14mpbiri 257 . . . . . . . 8 (𝑥 = ∅ → 𝜑)
1612, 15sylbi 216 . . . . . . 7 (𝑥 ≈ ∅ → 𝜑)
1716ax-gen 1789 . . . . . 6 𝑥(𝑥 ≈ ∅ → 𝜑)
18 peano2 7897 . . . . . . . . . . . . 13 (𝑣 ∈ ω → suc 𝑣 ∈ ω)
19 breq2 5153 . . . . . . . . . . . . . 14 (𝑤 = suc 𝑣 → (𝑦𝑤𝑦 ≈ suc 𝑣))
2019rspcev 3606 . . . . . . . . . . . . 13 ((suc 𝑣 ∈ ω ∧ 𝑦 ≈ suc 𝑣) → ∃𝑤 ∈ ω 𝑦𝑤)
2118, 20sylan 578 . . . . . . . . . . . 12 ((𝑣 ∈ ω ∧ 𝑦 ≈ suc 𝑣) → ∃𝑤 ∈ ω 𝑦𝑤)
22 isfi 8997 . . . . . . . . . . . 12 (𝑦 ∈ Fin ↔ ∃𝑤 ∈ ω 𝑦𝑤)
2321, 22sylibr 233 . . . . . . . . . . 11 ((𝑣 ∈ ω ∧ 𝑦 ≈ suc 𝑣) → 𝑦 ∈ Fin)
24233adant2 1128 . . . . . . . . . 10 ((𝑣 ∈ ω ∧ ∀𝑥(𝑥𝑣𝜑) ∧ 𝑦 ≈ suc 𝑣) → 𝑦 ∈ Fin)
25 dif1ennn 9186 . . . . . . . . . . . . . . . 16 ((𝑣 ∈ ω ∧ 𝑦 ≈ suc 𝑣𝑧𝑦) → (𝑦 ∖ {𝑧}) ≈ 𝑣)
26253expa 1115 . . . . . . . . . . . . . . 15 (((𝑣 ∈ ω ∧ 𝑦 ≈ suc 𝑣) ∧ 𝑧𝑦) → (𝑦 ∖ {𝑧}) ≈ 𝑣)
27 vex 3465 . . . . . . . . . . . . . . . . 17 𝑦 ∈ V
2827difexi 5331 . . . . . . . . . . . . . . . 16 (𝑦 ∖ {𝑧}) ∈ V
29 breq1 5152 . . . . . . . . . . . . . . . . 17 (𝑥 = (𝑦 ∖ {𝑧}) → (𝑥𝑣 ↔ (𝑦 ∖ {𝑧}) ≈ 𝑣))
30 findcard.2 . . . . . . . . . . . . . . . . 17 (𝑥 = (𝑦 ∖ {𝑧}) → (𝜑𝜒))
3129, 30imbi12d 343 . . . . . . . . . . . . . . . 16 (𝑥 = (𝑦 ∖ {𝑧}) → ((𝑥𝑣𝜑) ↔ ((𝑦 ∖ {𝑧}) ≈ 𝑣𝜒)))
3228, 31spcv 3589 . . . . . . . . . . . . . . 15 (∀𝑥(𝑥𝑣𝜑) → ((𝑦 ∖ {𝑧}) ≈ 𝑣𝜒))
3326, 32syl5com 31 . . . . . . . . . . . . . 14 (((𝑣 ∈ ω ∧ 𝑦 ≈ suc 𝑣) ∧ 𝑧𝑦) → (∀𝑥(𝑥𝑣𝜑) → 𝜒))
3433ralrimdva 3143 . . . . . . . . . . . . 13 ((𝑣 ∈ ω ∧ 𝑦 ≈ suc 𝑣) → (∀𝑥(𝑥𝑣𝜑) → ∀𝑧𝑦 𝜒))
3534imp 405 . . . . . . . . . . . 12 (((𝑣 ∈ ω ∧ 𝑦 ≈ suc 𝑣) ∧ ∀𝑥(𝑥𝑣𝜑)) → ∀𝑧𝑦 𝜒)
3635an32s 650 . . . . . . . . . . 11 (((𝑣 ∈ ω ∧ ∀𝑥(𝑥𝑣𝜑)) ∧ 𝑦 ≈ suc 𝑣) → ∀𝑧𝑦 𝜒)
37363impa 1107 . . . . . . . . . 10 ((𝑣 ∈ ω ∧ ∀𝑥(𝑥𝑣𝜑) ∧ 𝑦 ≈ suc 𝑣) → ∀𝑧𝑦 𝜒)
38 findcard.6 . . . . . . . . . 10 (𝑦 ∈ Fin → (∀𝑧𝑦 𝜒𝜃))
3924, 37, 38sylc 65 . . . . . . . . 9 ((𝑣 ∈ ω ∧ ∀𝑥(𝑥𝑣𝜑) ∧ 𝑦 ≈ suc 𝑣) → 𝜃)
40393exp 1116 . . . . . . . 8 (𝑣 ∈ ω → (∀𝑥(𝑥𝑣𝜑) → (𝑦 ≈ suc 𝑣𝜃)))
4140alrimdv 1924 . . . . . . 7 (𝑣 ∈ ω → (∀𝑥(𝑥𝑣𝜑) → ∀𝑦(𝑦 ≈ suc 𝑣𝜃)))
42 breq1 5152 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑥 ≈ suc 𝑣𝑦 ≈ suc 𝑣))
43 findcard.3 . . . . . . . . 9 (𝑥 = 𝑦 → (𝜑𝜃))
4442, 43imbi12d 343 . . . . . . . 8 (𝑥 = 𝑦 → ((𝑥 ≈ suc 𝑣𝜑) ↔ (𝑦 ≈ suc 𝑣𝜃)))
4544cbvalvw 2031 . . . . . . 7 (∀𝑥(𝑥 ≈ suc 𝑣𝜑) ↔ ∀𝑦(𝑦 ≈ suc 𝑣𝜃))
4641, 45imbitrrdi 251 . . . . . 6 (𝑣 ∈ ω → (∀𝑥(𝑥𝑣𝜑) → ∀𝑥(𝑥 ≈ suc 𝑣𝜑)))
475, 8, 11, 17, 46finds1 7907 . . . . 5 (𝑤 ∈ ω → ∀𝑥(𝑥𝑤𝜑))
484719.21bi 2177 . . . 4 (𝑤 ∈ ω → (𝑥𝑤𝜑))
4948rexlimiv 3137 . . 3 (∃𝑤 ∈ ω 𝑥𝑤𝜑)
502, 49sylbi 216 . 2 (𝑥 ∈ Fin → 𝜑)
511, 50vtoclga 3556 1 (𝐴 ∈ Fin → 𝜏)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  w3a 1084  wal 1531   = wceq 1533  wcel 2098  wral 3050  wrex 3059  cdif 3941  c0 4322  {csn 4630   class class class wbr 5149  suc csuc 6373  ωcom 7871  cen 8961  Fincfn 8964
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429  ax-un 7741
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-om 7872  df-en 8965  df-fin 8968
This theorem is referenced by:  xpfiOLD  9344
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