Theorem finds 2397

Description: Principle of Finite Induction (inference schema) with implicit substitutions. The first four hypotheses establish the substitutions we need. The last two are the basis and the induction hypothesis. Theorem Schema 22 of [Suppes] p. 136.

Hypotheses

Ref	Expression
finds.1	⊢ (x = ∅ → (φ ↔ ψ))
finds.2	⊢ (x = y → (φ ↔ χ))
finds.3	⊢ (x = suc y → (φ ↔ θ))
finds.4	⊢ (x = A → (φ ↔ τ))
finds.5	⊢ ψ
finds.6	⊢ (y ∈ ω → (χ → θ))

Assertion

Ref	Expression
finds	⊢ (A ∈ ω → τ)

Distinct variable group(s):   x,y   x,A   ψ,x   χ,x   θ,x   τ,x   φ,y

Proof of Theorem finds

Step	Hyp	Ref	Expression
1		finds.5	. . . . 5 ⊢ ψ
2		0ex 1745	. . . . . 6 ⊢ ∅ ∈ V
3		finds.1	. . . . . 6 ⊢ (x = ∅ → (φ ↔ ψ))
4	2, 3	elab 1415	. . . . 5 ⊢ (∅ ∈ {x∣φ} ↔ ψ)
5	1, 4	mpbir 165	. . . 4 ⊢ ∅ ∈ {x∣φ}
6		finds.6	. . . . . 6 ⊢ (y ∈ ω → (χ → θ))
7		visset 1350	. . . . . . 7 ⊢ y ∈ V
8		finds.2	. . . . . . 7 ⊢ (x = y → (φ ↔ χ))
9	7, 8	elab 1415	. . . . . 6 ⊢ (y ∈ {x∣φ} ↔ χ)
10	7	sucex 2303	. . . . . . 7 ⊢ suc y ∈ V
11		finds.3	. . . . . . 7 ⊢ (x = suc y → (φ ↔ θ))
12	10, 11	elab 1415	. . . . . 6 ⊢ (suc y ∈ {x∣φ} ↔ θ)
13	6, 9, 12	3imtr4g 426	. . . . 5 ⊢ (y ∈ ω → (y ∈ {x∣φ} → suc y ∈ {x∣φ}))
14	13	rgen 1247	. . . 4 ⊢ ∀y ∈ ω (y ∈ {x∣φ} → suc y ∈ {x∣φ})
15		peano5 2394	. . . 4 ⊢ ((∅ ∈ {x∣φ} ∧ ∀y ∈ ω (y ∈ {x∣φ} → suc y ∈ {x∣φ})) → ω ⊆ {x∣φ})
16	5, 14, 15	mp2an 520	. . 3 ⊢ ω ⊆ {x∣φ}
17	16	sseli 1504	. 2 ⊢ (A ∈ ω → A ∈ {x∣φ})
18		finds.4	. . 3 ⊢ (x = A → (φ ↔ τ))
19	18	elabg 1417	. 2 ⊢ (A ∈ ω → (A ∈ {x∣φ} ↔ τ))
20	17, 19	mpbid 170	1 ⊢ (A ∈ ω → τ)

Syntax hints:   → wi 2   ↔ wb 127   = weq 797  {cab 1090   = wceq 1091   ∈ wcel 1092  ∀wral 1201   ⊆ wss 1487  ∅c0 1707  suc csuc 2201  ωcom 2372

This theorem is referenced by:  findsg 2398  findes 2400  nnacl 3172  nnmcl 3173  nnacom 3175  nndi 3180  nnmass 3181  nnmsucr 3182  nnmcom 3183  nneneq 3408  pssnn 3428  inf3lem1 3464  inf3lem2 3465  om2uzuz 4653  om2uzlt 4654

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-om 2373

metamath.org