Theorem findcard 9203

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 9016	. . 3 ⊢ (𝑥 ∈ Fin ↔ ∃𝑤 ∈ ω 𝑥 ≈ 𝑤)
3		breq2 5147	. . . . . . . 8 ⊢ (𝑤 = ∅ → (𝑥 ≈ 𝑤 ↔ 𝑥 ≈ ∅))
4	3	imbi1d 341	. . . . . . 7 ⊢ (𝑤 = ∅ → ((𝑥 ≈ 𝑤 → 𝜑) ↔ (𝑥 ≈ ∅ → 𝜑)))
5	4	albidv 1920	. . . . . 6 ⊢ (𝑤 = ∅ → (∀𝑥(𝑥 ≈ 𝑤 → 𝜑) ↔ ∀𝑥(𝑥 ≈ ∅ → 𝜑)))
6		breq2 5147	. . . . . . . 8 ⊢ (𝑤 = 𝑣 → (𝑥 ≈ 𝑤 ↔ 𝑥 ≈ 𝑣))
7	6	imbi1d 341	. . . . . . 7 ⊢ (𝑤 = 𝑣 → ((𝑥 ≈ 𝑤 → 𝜑) ↔ (𝑥 ≈ 𝑣 → 𝜑)))
8	7	albidv 1920	. . . . . 6 ⊢ (𝑤 = 𝑣 → (∀𝑥(𝑥 ≈ 𝑤 → 𝜑) ↔ ∀𝑥(𝑥 ≈ 𝑣 → 𝜑)))
9		breq2 5147	. . . . . . . 8 ⊢ (𝑤 = suc 𝑣 → (𝑥 ≈ 𝑤 ↔ 𝑥 ≈ suc 𝑣))
10	9	imbi1d 341	. . . . . . 7 ⊢ (𝑤 = suc 𝑣 → ((𝑥 ≈ 𝑤 → 𝜑) ↔ (𝑥 ≈ suc 𝑣 → 𝜑)))
11	10	albidv 1920	. . . . . 6 ⊢ (𝑤 = suc 𝑣 → (∀𝑥(𝑥 ≈ 𝑤 → 𝜑) ↔ ∀𝑥(𝑥 ≈ suc 𝑣 → 𝜑)))
12		en0 9058	. . . . . . . 8 ⊢ (𝑥 ≈ ∅ ↔ 𝑥 = ∅)
13		findcard.5	. . . . . . . . 9 ⊢ 𝜓
14		findcard.1	. . . . . . . . 9 ⊢ (𝑥 = ∅ → (𝜑 ↔ 𝜓))
15	13, 14	mpbiri 258	. . . . . . . 8 ⊢ (𝑥 = ∅ → 𝜑)
16	12, 15	sylbi 217	. . . . . . 7 ⊢ (𝑥 ≈ ∅ → 𝜑)
17	16	ax-gen 1795	. . . . . 6 ⊢ ∀𝑥(𝑥 ≈ ∅ → 𝜑)
18		peano2 7912	. . . . . . . . . . . . 13 ⊢ (𝑣 ∈ ω → suc 𝑣 ∈ ω)
19		breq2 5147	. . . . . . . . . . . . . 14 ⊢ (𝑤 = suc 𝑣 → (𝑦 ≈ 𝑤 ↔ 𝑦 ≈ suc 𝑣))
20	19	rspcev 3622	. . . . . . . . . . . . 13 ⊢ ((suc 𝑣 ∈ ω ∧ 𝑦 ≈ suc 𝑣) → ∃𝑤 ∈ ω 𝑦 ≈ 𝑤)
21	18, 20	sylan 580	. . . . . . . . . . . 12 ⊢ ((𝑣 ∈ ω ∧ 𝑦 ≈ suc 𝑣) → ∃𝑤 ∈ ω 𝑦 ≈ 𝑤)
22		isfi 9016	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ Fin ↔ ∃𝑤 ∈ ω 𝑦 ≈ 𝑤)
23	21, 22	sylibr 234	. . . . . . . . . . 11 ⊢ ((𝑣 ∈ ω ∧ 𝑦 ≈ suc 𝑣) → 𝑦 ∈ Fin)
24	23	3adant2 1132	. . . . . . . . . 10 ⊢ ((𝑣 ∈ ω ∧ ∀𝑥(𝑥 ≈ 𝑣 → 𝜑) ∧ 𝑦 ≈ suc 𝑣) → 𝑦 ∈ Fin)
25		dif1ennn 9201	. . . . . . . . . . . . . . . 16 ⊢ ((𝑣 ∈ ω ∧ 𝑦 ≈ suc 𝑣 ∧ 𝑧 ∈ 𝑦) → (𝑦 ∖ {𝑧}) ≈ 𝑣)
26	25	3expa 1119	. . . . . . . . . . . . . . 15 ⊢ (((𝑣 ∈ ω ∧ 𝑦 ≈ suc 𝑣) ∧ 𝑧 ∈ 𝑦) → (𝑦 ∖ {𝑧}) ≈ 𝑣)
27		vex 3484	. . . . . . . . . . . . . . . . 17 ⊢ 𝑦 ∈ V
28	27	difexi 5330	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∖ {𝑧}) ∈ V
29		breq1 5146	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = (𝑦 ∖ {𝑧}) → (𝑥 ≈ 𝑣 ↔ (𝑦 ∖ {𝑧}) ≈ 𝑣))
30		findcard.2	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = (𝑦 ∖ {𝑧}) → (𝜑 ↔ 𝜒))
31	29, 30	imbi12d 344	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = (𝑦 ∖ {𝑧}) → ((𝑥 ≈ 𝑣 → 𝜑) ↔ ((𝑦 ∖ {𝑧}) ≈ 𝑣 → 𝜒)))
32	28, 31	spcv 3605	. . . . . . . . . . . . . . 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 3154	. . . . . . . . . . . . 13 ⊢ ((𝑣 ∈ ω ∧ 𝑦 ≈ suc 𝑣) → (∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → ∀𝑧 ∈ 𝑦 𝜒))
35	34	imp 406	. . . . . . . . . . . 12 ⊢ (((𝑣 ∈ ω ∧ 𝑦 ≈ suc 𝑣) ∧ ∀𝑥(𝑥 ≈ 𝑣 → 𝜑)) → ∀𝑧 ∈ 𝑦 𝜒)
36	35	an32s 652	. . . . . . . . . . 11 ⊢ (((𝑣 ∈ ω ∧ ∀𝑥(𝑥 ≈ 𝑣 → 𝜑)) ∧ 𝑦 ≈ suc 𝑣) → ∀𝑧 ∈ 𝑦 𝜒)
37	36	3impa 1110	. . . . . . . . . 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 1120	. . . . . . . 8 ⊢ (𝑣 ∈ ω → (∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → (𝑦 ≈ suc 𝑣 → 𝜃)))
41	40	alrimdv 1929	. . . . . . 7 ⊢ (𝑣 ∈ ω → (∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → ∀𝑦(𝑦 ≈ suc 𝑣 → 𝜃)))
42		breq1 5146	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑥 ≈ suc 𝑣 ↔ 𝑦 ≈ suc 𝑣))
43		findcard.3	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜃))
44	42, 43	imbi12d 344	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ((𝑥 ≈ suc 𝑣 → 𝜑) ↔ (𝑦 ≈ suc 𝑣 → 𝜃)))
45	44	cbvalvw 2035	. . . . . . 7 ⊢ (∀𝑥(𝑥 ≈ suc 𝑣 → 𝜑) ↔ ∀𝑦(𝑦 ≈ suc 𝑣 → 𝜃))
46	41, 45	imbitrrdi 252	. . . . . 6 ⊢ (𝑣 ∈ ω → (∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → ∀𝑥(𝑥 ≈ suc 𝑣 → 𝜑)))
47	5, 8, 11, 17, 46	finds1 7921	. . . . 5 ⊢ (𝑤 ∈ ω → ∀𝑥(𝑥 ≈ 𝑤 → 𝜑))
48	47	19.21bi 2189	. . . 4 ⊢ (𝑤 ∈ ω → (𝑥 ≈ 𝑤 → 𝜑))
49	48	rexlimiv 3148	. . 3 ⊢ (∃𝑤 ∈ ω 𝑥 ≈ 𝑤 → 𝜑)
50	2, 49	sylbi 217	. 2 ⊢ (𝑥 ∈ Fin → 𝜑)
51	1, 50	vtoclga 3577	1 ⊢ (𝐴 ∈ Fin → 𝜏)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087  ∀wal 1538   = wceq 1540   ∈ wcel 2108  ∀wral 3061  ∃wrex 3070   ∖ cdif 3948  ∅c0 4333  {csn 4626   class class class wbr 5143  suc csuc 6386  ωcom 7887   ≈ cen 8982  Fincfn 8985

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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-om 7888  df-en 8986  df-fin 8989

This theorem is referenced by:  xpfiOLD  9359