Theorem findcard 8552

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 8330	. . 3 ⊢ (𝑥 ∈ Fin ↔ ∃𝑤 ∈ ω 𝑥 ≈ 𝑤)
3		breq2 4933	. . . . . . . 8 ⊢ (𝑤 = ∅ → (𝑥 ≈ 𝑤 ↔ 𝑥 ≈ ∅))
4	3	imbi1d 334	. . . . . . 7 ⊢ (𝑤 = ∅ → ((𝑥 ≈ 𝑤 → 𝜑) ↔ (𝑥 ≈ ∅ → 𝜑)))
5	4	albidv 1879	. . . . . 6 ⊢ (𝑤 = ∅ → (∀𝑥(𝑥 ≈ 𝑤 → 𝜑) ↔ ∀𝑥(𝑥 ≈ ∅ → 𝜑)))
6		breq2 4933	. . . . . . . 8 ⊢ (𝑤 = 𝑣 → (𝑥 ≈ 𝑤 ↔ 𝑥 ≈ 𝑣))
7	6	imbi1d 334	. . . . . . 7 ⊢ (𝑤 = 𝑣 → ((𝑥 ≈ 𝑤 → 𝜑) ↔ (𝑥 ≈ 𝑣 → 𝜑)))
8	7	albidv 1879	. . . . . 6 ⊢ (𝑤 = 𝑣 → (∀𝑥(𝑥 ≈ 𝑤 → 𝜑) ↔ ∀𝑥(𝑥 ≈ 𝑣 → 𝜑)))
9		breq2 4933	. . . . . . . 8 ⊢ (𝑤 = suc 𝑣 → (𝑥 ≈ 𝑤 ↔ 𝑥 ≈ suc 𝑣))
10	9	imbi1d 334	. . . . . . 7 ⊢ (𝑤 = suc 𝑣 → ((𝑥 ≈ 𝑤 → 𝜑) ↔ (𝑥 ≈ suc 𝑣 → 𝜑)))
11	10	albidv 1879	. . . . . 6 ⊢ (𝑤 = suc 𝑣 → (∀𝑥(𝑥 ≈ 𝑤 → 𝜑) ↔ ∀𝑥(𝑥 ≈ suc 𝑣 → 𝜑)))
12		en0 8369	. . . . . . . 8 ⊢ (𝑥 ≈ ∅ ↔ 𝑥 = ∅)
13		findcard.5	. . . . . . . . 9 ⊢ 𝜓
14		findcard.1	. . . . . . . . 9 ⊢ (𝑥 = ∅ → (𝜑 ↔ 𝜓))
15	13, 14	mpbiri 250	. . . . . . . 8 ⊢ (𝑥 = ∅ → 𝜑)
16	12, 15	sylbi 209	. . . . . . 7 ⊢ (𝑥 ≈ ∅ → 𝜑)
17	16	ax-gen 1758	. . . . . 6 ⊢ ∀𝑥(𝑥 ≈ ∅ → 𝜑)
18		peano2 7417	. . . . . . . . . . . . 13 ⊢ (𝑣 ∈ ω → suc 𝑣 ∈ ω)
19		breq2 4933	. . . . . . . . . . . . . 14 ⊢ (𝑤 = suc 𝑣 → (𝑦 ≈ 𝑤 ↔ 𝑦 ≈ suc 𝑣))
20	19	rspcev 3535	. . . . . . . . . . . . 13 ⊢ ((suc 𝑣 ∈ ω ∧ 𝑦 ≈ suc 𝑣) → ∃𝑤 ∈ ω 𝑦 ≈ 𝑤)
21	18, 20	sylan 572	. . . . . . . . . . . 12 ⊢ ((𝑣 ∈ ω ∧ 𝑦 ≈ suc 𝑣) → ∃𝑤 ∈ ω 𝑦 ≈ 𝑤)
22		isfi 8330	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ Fin ↔ ∃𝑤 ∈ ω 𝑦 ≈ 𝑤)
23	21, 22	sylibr 226	. . . . . . . . . . 11 ⊢ ((𝑣 ∈ ω ∧ 𝑦 ≈ suc 𝑣) → 𝑦 ∈ Fin)
24	23	3adant2 1111	. . . . . . . . . 10 ⊢ ((𝑣 ∈ ω ∧ ∀𝑥(𝑥 ≈ 𝑣 → 𝜑) ∧ 𝑦 ≈ suc 𝑣) → 𝑦 ∈ Fin)
25		dif1en 8546	. . . . . . . . . . . . . . . 16 ⊢ ((𝑣 ∈ ω ∧ 𝑦 ≈ suc 𝑣 ∧ 𝑧 ∈ 𝑦) → (𝑦 ∖ {𝑧}) ≈ 𝑣)
26	25	3expa 1098	. . . . . . . . . . . . . . 15 ⊢ (((𝑣 ∈ ω ∧ 𝑦 ≈ suc 𝑣) ∧ 𝑧 ∈ 𝑦) → (𝑦 ∖ {𝑧}) ≈ 𝑣)
27		vex 3418	. . . . . . . . . . . . . . . . 17 ⊢ 𝑦 ∈ V
28	27	difexi 5088	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∖ {𝑧}) ∈ V
29		breq1 4932	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = (𝑦 ∖ {𝑧}) → (𝑥 ≈ 𝑣 ↔ (𝑦 ∖ {𝑧}) ≈ 𝑣))
30		findcard.2	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = (𝑦 ∖ {𝑧}) → (𝜑 ↔ 𝜒))
31	29, 30	imbi12d 337	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = (𝑦 ∖ {𝑧}) → ((𝑥 ≈ 𝑣 → 𝜑) ↔ ((𝑦 ∖ {𝑧}) ≈ 𝑣 → 𝜒)))
32	28, 31	spcv 3524	. . . . . . . . . . . . . . 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 3139	. . . . . . . . . . . . 13 ⊢ ((𝑣 ∈ ω ∧ 𝑦 ≈ suc 𝑣) → (∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → ∀𝑧 ∈ 𝑦 𝜒))
35	34	imp 398	. . . . . . . . . . . 12 ⊢ (((𝑣 ∈ ω ∧ 𝑦 ≈ suc 𝑣) ∧ ∀𝑥(𝑥 ≈ 𝑣 → 𝜑)) → ∀𝑧 ∈ 𝑦 𝜒)
36	35	an32s 639	. . . . . . . . . . 11 ⊢ (((𝑣 ∈ ω ∧ ∀𝑥(𝑥 ≈ 𝑣 → 𝜑)) ∧ 𝑦 ≈ suc 𝑣) → ∀𝑧 ∈ 𝑦 𝜒)
37	36	3impa 1090	. . . . . . . . . 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 1099	. . . . . . . 8 ⊢ (𝑣 ∈ ω → (∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → (𝑦 ≈ suc 𝑣 → 𝜃)))
41	40	alrimdv 1888	. . . . . . 7 ⊢ (𝑣 ∈ ω → (∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → ∀𝑦(𝑦 ≈ suc 𝑣 → 𝜃)))
42		breq1 4932	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑥 ≈ suc 𝑣 ↔ 𝑦 ≈ suc 𝑣))
43		findcard.3	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜃))
44	42, 43	imbi12d 337	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ((𝑥 ≈ suc 𝑣 → 𝜑) ↔ (𝑦 ≈ suc 𝑣 → 𝜃)))
45	44	cbvalvw 1993	. . . . . . 7 ⊢ (∀𝑥(𝑥 ≈ suc 𝑣 → 𝜑) ↔ ∀𝑦(𝑦 ≈ suc 𝑣 → 𝜃))
46	41, 45	syl6ibr 244	. . . . . 6 ⊢ (𝑣 ∈ ω → (∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → ∀𝑥(𝑥 ≈ suc 𝑣 → 𝜑)))
47	5, 8, 11, 17, 46	finds1 7426	. . . . 5 ⊢ (𝑤 ∈ ω → ∀𝑥(𝑥 ≈ 𝑤 → 𝜑))
48	47	19.21bi 2117	. . . 4 ⊢ (𝑤 ∈ ω → (𝑥 ≈ 𝑤 → 𝜑))
49	48	rexlimiv 3225	. . 3 ⊢ (∃𝑤 ∈ ω 𝑥 ≈ 𝑤 → 𝜑)
50	2, 49	sylbi 209	. 2 ⊢ (𝑥 ∈ Fin → 𝜑)
51	1, 50	vtoclga 3493	1 ⊢ (𝐴 ∈ Fin → 𝜏)

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 387   ∧ w3a 1068  ∀wal 1505   = wceq 1507   ∈ wcel 2050  ∀wral 3088  ∃wrex 3089   ∖ cdif 3826  ∅c0 4178  {csn 4441   class class class wbr 4929  suc csuc 6031  ωcom 7396   ≈ cen 8303  Fincfn 8306

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-sep 5060  ax-nul 5067  ax-pow 5119  ax-pr 5186  ax-un 7279

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-ral 3093  df-rex 3094  df-rab 3097  df-v 3417  df-sbc 3682  df-dif 3832  df-un 3834  df-in 3836  df-ss 3843  df-pss 3845  df-nul 4179  df-if 4351  df-pw 4424  df-sn 4442  df-pr 4444  df-tp 4446  df-op 4448  df-uni 4713  df-br 4930  df-opab 4992  df-tr 5031  df-id 5312  df-eprel 5317  df-po 5326  df-so 5327  df-fr 5366  df-we 5368  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-res 5419  df-ima 5420  df-ord 6032  df-on 6033  df-lim 6034  df-suc 6035  df-iota 6152  df-fun 6190  df-fn 6191  df-f 6192  df-f1 6193  df-fo 6194  df-f1o 6195  df-fv 6196  df-om 7397  df-1o 7905  df-er 8089  df-en 8307  df-fin 8310

This theorem is referenced by:  xpfi  8584