Theorem findcard 9196

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 = ∅ → (𝜑 ↔ 𝜓))
findcard.2	⊢ (𝑥 = (𝑦 ∖ {𝑧}) → (𝜑 ↔ 𝜒))
findcard.3	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜃))
findcard.4	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏))
findcard.5	⊢ 𝜓
findcard.6	⊢ (𝑦 ∈ Fin → (∀𝑧 ∈ 𝑦 𝜒 → 𝜃))

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.

Step	Hyp	Ref	Expression
1		findcard.4	. 2 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏))
2		isfi 9005	. . 3 ⊢ (𝑥 ∈ Fin ↔ ∃𝑤 ∈ ω 𝑥 ≈ 𝑤)
3		breq2 5156	. . . . . . . 8 ⊢ (𝑤 = ∅ → (𝑥 ≈ 𝑤 ↔ 𝑥 ≈ ∅))
4	3	imbi1d 340	. . . . . . 7 ⊢ (𝑤 = ∅ → ((𝑥 ≈ 𝑤 → 𝜑) ↔ (𝑥 ≈ ∅ → 𝜑)))
5	4	albidv 1915	. . . . . 6 ⊢ (𝑤 = ∅ → (∀𝑥(𝑥 ≈ 𝑤 → 𝜑) ↔ ∀𝑥(𝑥 ≈ ∅ → 𝜑)))
6		breq2 5156	. . . . . . . 8 ⊢ (𝑤 = 𝑣 → (𝑥 ≈ 𝑤 ↔ 𝑥 ≈ 𝑣))
7	6	imbi1d 340	. . . . . . 7 ⊢ (𝑤 = 𝑣 → ((𝑥 ≈ 𝑤 → 𝜑) ↔ (𝑥 ≈ 𝑣 → 𝜑)))
8	7	albidv 1915	. . . . . 6 ⊢ (𝑤 = 𝑣 → (∀𝑥(𝑥 ≈ 𝑤 → 𝜑) ↔ ∀𝑥(𝑥 ≈ 𝑣 → 𝜑)))
9		breq2 5156	. . . . . . . 8 ⊢ (𝑤 = suc 𝑣 → (𝑥 ≈ 𝑤 ↔ 𝑥 ≈ suc 𝑣))
10	9	imbi1d 340	. . . . . . 7 ⊢ (𝑤 = suc 𝑣 → ((𝑥 ≈ 𝑤 → 𝜑) ↔ (𝑥 ≈ suc 𝑣 → 𝜑)))
11	10	albidv 1915	. . . . . 6 ⊢ (𝑤 = suc 𝑣 → (∀𝑥(𝑥 ≈ 𝑤 → 𝜑) ↔ ∀𝑥(𝑥 ≈ suc 𝑣 → 𝜑)))
12		en0 9046	. . . . . . . 8 ⊢ (𝑥 ≈ ∅ ↔ 𝑥 = ∅)
13		findcard.5	. . . . . . . . 9 ⊢ 𝜓
14		findcard.1	. . . . . . . . 9 ⊢ (𝑥 = ∅ → (𝜑 ↔ 𝜓))
15	13, 14	mpbiri 257	. . . . . . . 8 ⊢ (𝑥 = ∅ → 𝜑)
16	12, 15	sylbi 216	. . . . . . 7 ⊢ (𝑥 ≈ ∅ → 𝜑)
17	16	ax-gen 1789	. . . . . 6 ⊢ ∀𝑥(𝑥 ≈ ∅ → 𝜑)
18		peano2 7904	. . . . . . . . . . . . 13 ⊢ (𝑣 ∈ ω → suc 𝑣 ∈ ω)
19		breq2 5156	. . . . . . . . . . . . . 14 ⊢ (𝑤 = suc 𝑣 → (𝑦 ≈ 𝑤 ↔ 𝑦 ≈ suc 𝑣))
20	19	rspcev 3611	. . . . . . . . . . . . 13 ⊢ ((suc 𝑣 ∈ ω ∧ 𝑦 ≈ suc 𝑣) → ∃𝑤 ∈ ω 𝑦 ≈ 𝑤)
21	18, 20	sylan 578	. . . . . . . . . . . 12 ⊢ ((𝑣 ∈ ω ∧ 𝑦 ≈ suc 𝑣) → ∃𝑤 ∈ ω 𝑦 ≈ 𝑤)
22		isfi 9005	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ Fin ↔ ∃𝑤 ∈ ω 𝑦 ≈ 𝑤)
23	21, 22	sylibr 233	. . . . . . . . . . 11 ⊢ ((𝑣 ∈ ω ∧ 𝑦 ≈ suc 𝑣) → 𝑦 ∈ Fin)
24	23	3adant2 1128	. . . . . . . . . 10 ⊢ ((𝑣 ∈ ω ∧ ∀𝑥(𝑥 ≈ 𝑣 → 𝜑) ∧ 𝑦 ≈ suc 𝑣) → 𝑦 ∈ Fin)
25		dif1ennn 9194	. . . . . . . . . . . . . . . 16 ⊢ ((𝑣 ∈ ω ∧ 𝑦 ≈ suc 𝑣 ∧ 𝑧 ∈ 𝑦) → (𝑦 ∖ {𝑧}) ≈ 𝑣)
26	25	3expa 1115	. . . . . . . . . . . . . . 15 ⊢ (((𝑣 ∈ ω ∧ 𝑦 ≈ suc 𝑣) ∧ 𝑧 ∈ 𝑦) → (𝑦 ∖ {𝑧}) ≈ 𝑣)
27		vex 3477	. . . . . . . . . . . . . . . . 17 ⊢ 𝑦 ∈ V
28	27	difexi 5334	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∖ {𝑧}) ∈ V
29		breq1 5155	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = (𝑦 ∖ {𝑧}) → (𝑥 ≈ 𝑣 ↔ (𝑦 ∖ {𝑧}) ≈ 𝑣))
30		findcard.2	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = (𝑦 ∖ {𝑧}) → (𝜑 ↔ 𝜒))
31	29, 30	imbi12d 343	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = (𝑦 ∖ {𝑧}) → ((𝑥 ≈ 𝑣 → 𝜑) ↔ ((𝑦 ∖ {𝑧}) ≈ 𝑣 → 𝜒)))
32	28, 31	spcv 3594	. . . . . . . . . . . . . . 15 ⊢ (∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → ((𝑦 ∖ {𝑧}) ≈ 𝑣 → 𝜒))
33	26, 32	syl5com 31	. . . . . . . . . . . . . 14 ⊢ (((𝑣 ∈ ω ∧ 𝑦 ≈ suc 𝑣) ∧ 𝑧 ∈ 𝑦) → (∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → 𝜒))
34	33	ralrimdva 3151	. . . . . . . . . . . . 13 ⊢ ((𝑣 ∈ ω ∧ 𝑦 ≈ suc 𝑣) → (∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → ∀𝑧 ∈ 𝑦 𝜒))
35	34	imp 405	. . . . . . . . . . . 12 ⊢ (((𝑣 ∈ ω ∧ 𝑦 ≈ suc 𝑣) ∧ ∀𝑥(𝑥 ≈ 𝑣 → 𝜑)) → ∀𝑧 ∈ 𝑦 𝜒)
36	35	an32s 650	. . . . . . . . . . 11 ⊢ (((𝑣 ∈ ω ∧ ∀𝑥(𝑥 ≈ 𝑣 → 𝜑)) ∧ 𝑦 ≈ suc 𝑣) → ∀𝑧 ∈ 𝑦 𝜒)
37	36	3impa 1107	. . . . . . . . . 10 ⊢ ((𝑣 ∈ ω ∧ ∀𝑥(𝑥 ≈ 𝑣 → 𝜑) ∧ 𝑦 ≈ suc 𝑣) → ∀𝑧 ∈ 𝑦 𝜒)
38		findcard.6	. . . . . . . . . 10 ⊢ (𝑦 ∈ Fin → (∀𝑧 ∈ 𝑦 𝜒 → 𝜃))
39	24, 37, 38	sylc 65	. . . . . . . . 9 ⊢ ((𝑣 ∈ ω ∧ ∀𝑥(𝑥 ≈ 𝑣 → 𝜑) ∧ 𝑦 ≈ suc 𝑣) → 𝜃)
40	39	3exp 1116	. . . . . . . 8 ⊢ (𝑣 ∈ ω → (∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → (𝑦 ≈ suc 𝑣 → 𝜃)))
41	40	alrimdv 1924	. . . . . . 7 ⊢ (𝑣 ∈ ω → (∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → ∀𝑦(𝑦 ≈ suc 𝑣 → 𝜃)))
42		breq1 5155	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑥 ≈ suc 𝑣 ↔ 𝑦 ≈ suc 𝑣))
43		findcard.3	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜃))
44	42, 43	imbi12d 343	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ((𝑥 ≈ suc 𝑣 → 𝜑) ↔ (𝑦 ≈ suc 𝑣 → 𝜃)))
45	44	cbvalvw 2031	. . . . . . 7 ⊢ (∀𝑥(𝑥 ≈ suc 𝑣 → 𝜑) ↔ ∀𝑦(𝑦 ≈ suc 𝑣 → 𝜃))
46	41, 45	imbitrrdi 251	. . . . . 6 ⊢ (𝑣 ∈ ω → (∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → ∀𝑥(𝑥 ≈ suc 𝑣 → 𝜑)))
47	5, 8, 11, 17, 46	finds1 7915	. . . . 5 ⊢ (𝑤 ∈ ω → ∀𝑥(𝑥 ≈ 𝑤 → 𝜑))
48	47	19.21bi 2177	. . . 4 ⊢ (𝑤 ∈ ω → (𝑥 ≈ 𝑤 → 𝜑))
49	48	rexlimiv 3145	. . 3 ⊢ (∃𝑤 ∈ ω 𝑥 ≈ 𝑤 → 𝜑)
50	2, 49	sylbi 216	. 2 ⊢ (𝑥 ∈ Fin → 𝜑)
51	1, 50	vtoclga 3565	1 ⊢ (𝐴 ∈ Fin → 𝜏)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1084  ∀wal 1531   = wceq 1533   ∈ wcel 2098  ∀wral 3058  ∃wrex 3067   ∖ cdif 3946  ∅c0 4326  {csn 4632   class class class wbr 5152  suc csuc 6376  ωcom 7878   ≈ cen 8969  Fincfn 8972

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 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433  ax-un 7748

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 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-om 7879  df-en 8973  df-fin 8976

This theorem is referenced by:  xpfiOLD  9352