Theorem finds 4266

Description: Principle of Finite Induction (inference schema), using implicit substitutions. The first four hypotheses establish the substitutions we need. The last two are the basis and the induction step. Theorem Schema 22 of [Suppes] p. 136. This is Metamath 100 proof #74. (Contributed by NM, 14-Apr-1995.)

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 groups:   x,y   x,A   ψ,x   χ,x   θ,x   τ,x   φ,y

Allowed substitution hints:   φ(x)   ψ(y)   χ(y)   θ(y)   τ(y)   A(y)

Proof of Theorem finds

Step	Hyp	Ref	Expression
1		finds.5	. . . . 5 ⊢ ψ
2		0ex 3875	. . . . . 6 ⊢ ∅ ∈ V
3		finds.1	. . . . . 6 ⊢ (x = ∅ → (φ ↔ ψ))
4	2, 3	elab 2681	. . . . 5 ⊢ (∅ ∈ {x ∣ φ} ↔ ψ)
5	1, 4	mpbir 134	. . . 4 ⊢ ∅ ∈ {x ∣ φ}
6		finds.6	. . . . . 6 ⊢ (y ∈ 𝜔 → (χ → θ))
7		vex 2554	. . . . . . 7 ⊢ y ∈ V
8		finds.2	. . . . . . 7 ⊢ (x = y → (φ ↔ χ))
9	7, 8	elab 2681	. . . . . 6 ⊢ (y ∈ {x ∣ φ} ↔ χ)
10	7	sucex 4191	. . . . . . 7 ⊢ suc y ∈ V
11		finds.3	. . . . . . 7 ⊢ (x = suc y → (φ ↔ θ))
12	10, 11	elab 2681	. . . . . 6 ⊢ (suc y ∈ {x ∣ φ} ↔ θ)
13	6, 9, 12	3imtr4g 194	. . . . 5 ⊢ (y ∈ 𝜔 → (y ∈ {x ∣ φ} → suc y ∈ {x ∣ φ}))
14	13	rgen 2368	. . . 4 ⊢ ∀y ∈ 𝜔 (y ∈ {x ∣ φ} → suc y ∈ {x ∣ φ})
15		peano5 4264	. . . 4 ⊢ ((∅ ∈ {x ∣ φ} ∧ ∀y ∈ 𝜔 (y ∈ {x ∣ φ} → suc y ∈ {x ∣ φ})) → 𝜔 ⊆ {x ∣ φ})
16	5, 14, 15	mp2an 402	. . 3 ⊢ 𝜔 ⊆ {x ∣ φ}
17	16	sseli 2935	. 2 ⊢ (A ∈ 𝜔 → A ∈ {x ∣ φ})
18		finds.4	. . 3 ⊢ (x = A → (φ ↔ τ))
19	18	elabg 2682	. 2 ⊢ (A ∈ 𝜔 → (A ∈ {x ∣ φ} ↔ τ))
20	17, 19	mpbid 135	1 ⊢ (A ∈ 𝜔 → τ)

Syntax hints:   → wi 4   ↔ wb 98   = wceq 1242   ∈ wcel 1390  {cab 2023  ∀wral 2300   ⊆ wss 2911  ∅c0 3218  suc csuc 4068  𝜔com 4256

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-iinf 4254

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-uni 3572  df-int 3607  df-suc 4074  df-iom 4257

This theorem is referenced by:  findes  4269  nn0suc  4270  elnn  4271  ordom  4272  nndceq0  4282  0elnn  4283  nna0r  5996  nnm0r  5997  nnsucelsuc  6009  frec2uzltd  8870