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Theorem findcard 8084
 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 7865 . . 3 (𝑥 ∈ Fin ↔ ∃𝑤 ∈ ω 𝑥𝑤)
3 breq2 4587 . . . . . . . 8 (𝑤 = ∅ → (𝑥𝑤𝑥 ≈ ∅))
43imbi1d 330 . . . . . . 7 (𝑤 = ∅ → ((𝑥𝑤𝜑) ↔ (𝑥 ≈ ∅ → 𝜑)))
54albidv 1836 . . . . . 6 (𝑤 = ∅ → (∀𝑥(𝑥𝑤𝜑) ↔ ∀𝑥(𝑥 ≈ ∅ → 𝜑)))
6 breq2 4587 . . . . . . . 8 (𝑤 = 𝑣 → (𝑥𝑤𝑥𝑣))
76imbi1d 330 . . . . . . 7 (𝑤 = 𝑣 → ((𝑥𝑤𝜑) ↔ (𝑥𝑣𝜑)))
87albidv 1836 . . . . . 6 (𝑤 = 𝑣 → (∀𝑥(𝑥𝑤𝜑) ↔ ∀𝑥(𝑥𝑣𝜑)))
9 breq2 4587 . . . . . . . 8 (𝑤 = suc 𝑣 → (𝑥𝑤𝑥 ≈ suc 𝑣))
109imbi1d 330 . . . . . . 7 (𝑤 = suc 𝑣 → ((𝑥𝑤𝜑) ↔ (𝑥 ≈ suc 𝑣𝜑)))
1110albidv 1836 . . . . . 6 (𝑤 = suc 𝑣 → (∀𝑥(𝑥𝑤𝜑) ↔ ∀𝑥(𝑥 ≈ suc 𝑣𝜑)))
12 en0 7905 . . . . . . . 8 (𝑥 ≈ ∅ ↔ 𝑥 = ∅)
13 findcard.5 . . . . . . . . 9 𝜓
14 findcard.1 . . . . . . . . 9 (𝑥 = ∅ → (𝜑𝜓))
1513, 14mpbiri 247 . . . . . . . 8 (𝑥 = ∅ → 𝜑)
1612, 15sylbi 206 . . . . . . 7 (𝑥 ≈ ∅ → 𝜑)
1716ax-gen 1713 . . . . . 6 𝑥(𝑥 ≈ ∅ → 𝜑)
18 peano2 6978 . . . . . . . . . . . . 13 (𝑣 ∈ ω → suc 𝑣 ∈ ω)
19 breq2 4587 . . . . . . . . . . . . . 14 (𝑤 = suc 𝑣 → (𝑦𝑤𝑦 ≈ suc 𝑣))
2019rspcev 3282 . . . . . . . . . . . . 13 ((suc 𝑣 ∈ ω ∧ 𝑦 ≈ suc 𝑣) → ∃𝑤 ∈ ω 𝑦𝑤)
2118, 20sylan 487 . . . . . . . . . . . 12 ((𝑣 ∈ ω ∧ 𝑦 ≈ suc 𝑣) → ∃𝑤 ∈ ω 𝑦𝑤)
22 isfi 7865 . . . . . . . . . . . 12 (𝑦 ∈ Fin ↔ ∃𝑤 ∈ ω 𝑦𝑤)
2321, 22sylibr 223 . . . . . . . . . . 11 ((𝑣 ∈ ω ∧ 𝑦 ≈ suc 𝑣) → 𝑦 ∈ Fin)
24233adant2 1073 . . . . . . . . . 10 ((𝑣 ∈ ω ∧ ∀𝑥(𝑥𝑣𝜑) ∧ 𝑦 ≈ suc 𝑣) → 𝑦 ∈ Fin)
25 dif1en 8078 . . . . . . . . . . . . . . . 16 ((𝑣 ∈ ω ∧ 𝑦 ≈ suc 𝑣𝑧𝑦) → (𝑦 ∖ {𝑧}) ≈ 𝑣)
26253expa 1257 . . . . . . . . . . . . . . 15 (((𝑣 ∈ ω ∧ 𝑦 ≈ suc 𝑣) ∧ 𝑧𝑦) → (𝑦 ∖ {𝑧}) ≈ 𝑣)
27 vex 3176 . . . . . . . . . . . . . . . . 17 𝑦 ∈ V
28 difexg 4735 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ V → (𝑦 ∖ {𝑧}) ∈ V)
2927, 28ax-mp 5 . . . . . . . . . . . . . . . 16 (𝑦 ∖ {𝑧}) ∈ V
30 breq1 4586 . . . . . . . . . . . . . . . . 17 (𝑥 = (𝑦 ∖ {𝑧}) → (𝑥𝑣 ↔ (𝑦 ∖ {𝑧}) ≈ 𝑣))
31 findcard.2 . . . . . . . . . . . . . . . . 17 (𝑥 = (𝑦 ∖ {𝑧}) → (𝜑𝜒))
3230, 31imbi12d 333 . . . . . . . . . . . . . . . 16 (𝑥 = (𝑦 ∖ {𝑧}) → ((𝑥𝑣𝜑) ↔ ((𝑦 ∖ {𝑧}) ≈ 𝑣𝜒)))
3329, 32spcv 3272 . . . . . . . . . . . . . . 15 (∀𝑥(𝑥𝑣𝜑) → ((𝑦 ∖ {𝑧}) ≈ 𝑣𝜒))
3426, 33syl5com 31 . . . . . . . . . . . . . 14 (((𝑣 ∈ ω ∧ 𝑦 ≈ suc 𝑣) ∧ 𝑧𝑦) → (∀𝑥(𝑥𝑣𝜑) → 𝜒))
3534ralrimdva 2952 . . . . . . . . . . . . 13 ((𝑣 ∈ ω ∧ 𝑦 ≈ suc 𝑣) → (∀𝑥(𝑥𝑣𝜑) → ∀𝑧𝑦 𝜒))
3635imp 444 . . . . . . . . . . . 12 (((𝑣 ∈ ω ∧ 𝑦 ≈ suc 𝑣) ∧ ∀𝑥(𝑥𝑣𝜑)) → ∀𝑧𝑦 𝜒)
3736an32s 842 . . . . . . . . . . 11 (((𝑣 ∈ ω ∧ ∀𝑥(𝑥𝑣𝜑)) ∧ 𝑦 ≈ suc 𝑣) → ∀𝑧𝑦 𝜒)
38373impa 1251 . . . . . . . . . 10 ((𝑣 ∈ ω ∧ ∀𝑥(𝑥𝑣𝜑) ∧ 𝑦 ≈ suc 𝑣) → ∀𝑧𝑦 𝜒)
39 findcard.6 . . . . . . . . . 10 (𝑦 ∈ Fin → (∀𝑧𝑦 𝜒𝜃))
4024, 38, 39sylc 63 . . . . . . . . 9 ((𝑣 ∈ ω ∧ ∀𝑥(𝑥𝑣𝜑) ∧ 𝑦 ≈ suc 𝑣) → 𝜃)
41403exp 1256 . . . . . . . 8 (𝑣 ∈ ω → (∀𝑥(𝑥𝑣𝜑) → (𝑦 ≈ suc 𝑣𝜃)))
4241alrimdv 1844 . . . . . . 7 (𝑣 ∈ ω → (∀𝑥(𝑥𝑣𝜑) → ∀𝑦(𝑦 ≈ suc 𝑣𝜃)))
43 breq1 4586 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑥 ≈ suc 𝑣𝑦 ≈ suc 𝑣))
44 findcard.3 . . . . . . . . 9 (𝑥 = 𝑦 → (𝜑𝜃))
4543, 44imbi12d 333 . . . . . . . 8 (𝑥 = 𝑦 → ((𝑥 ≈ suc 𝑣𝜑) ↔ (𝑦 ≈ suc 𝑣𝜃)))
4645cbvalv 2261 . . . . . . 7 (∀𝑥(𝑥 ≈ suc 𝑣𝜑) ↔ ∀𝑦(𝑦 ≈ suc 𝑣𝜃))
4742, 46syl6ibr 241 . . . . . 6 (𝑣 ∈ ω → (∀𝑥(𝑥𝑣𝜑) → ∀𝑥(𝑥 ≈ suc 𝑣𝜑)))
485, 8, 11, 17, 47finds1 6987 . . . . 5 (𝑤 ∈ ω → ∀𝑥(𝑥𝑤𝜑))
494819.21bi 2047 . . . 4 (𝑤 ∈ ω → (𝑥𝑤𝜑))
5049rexlimiv 3009 . . 3 (∃𝑤 ∈ ω 𝑥𝑤𝜑)
512, 50sylbi 206 . 2 (𝑥 ∈ Fin → 𝜑)
521, 51vtoclga 3245 1 (𝐴 ∈ Fin → 𝜏)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031  ∀wal 1473   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  Vcvv 3173   ∖ cdif 3537  ∅c0 3874  {csn 4125   class class class wbr 4583  suc csuc 5642  ωcom 6957   ≈ cen 7838  Fincfn 7841 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-om 6958  df-1o 7447  df-er 7629  df-en 7842  df-fin 7845 This theorem is referenced by:  xpfi  8116
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