Theorem finds 4632

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 4156	. . . . . 6 ⊢ ∅ ∈ V
3		finds.1	. . . . . 6 ⊢ (𝑥 = ∅ → (𝜑 ↔ 𝜓))
4	2, 3	elab 2904	. . . . 5 ⊢ (∅ ∈ {𝑥 ∣ 𝜑} ↔ 𝜓)
5	1, 4	mpbir 146	. . . 4 ⊢ ∅ ∈ {𝑥 ∣ 𝜑}
6		finds.6	. . . . . 6 ⊢ (𝑦 ∈ ω → (𝜒 → 𝜃))
7		vex 2763	. . . . . . 7 ⊢ 𝑦 ∈ V
8		finds.2	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))
9	7, 8	elab 2904	. . . . . 6 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝜒)
10	7	sucex 4531	. . . . . . 7 ⊢ suc 𝑦 ∈ V
11		finds.3	. . . . . . 7 ⊢ (𝑥 = suc 𝑦 → (𝜑 ↔ 𝜃))
12	10, 11	elab 2904	. . . . . 6 ⊢ (suc 𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝜃)
13	6, 9, 12	3imtr4g 205	. . . . 5 ⊢ (𝑦 ∈ ω → (𝑦 ∈ {𝑥 ∣ 𝜑} → suc 𝑦 ∈ {𝑥 ∣ 𝜑}))
14	13	rgen 2547	. . . 4 ⊢ ∀𝑦 ∈ ω (𝑦 ∈ {𝑥 ∣ 𝜑} → suc 𝑦 ∈ {𝑥 ∣ 𝜑})
15		peano5 4630	. . . 4 ⊢ ((∅ ∈ {𝑥 ∣ 𝜑} ∧ ∀𝑦 ∈ ω (𝑦 ∈ {𝑥 ∣ 𝜑} → suc 𝑦 ∈ {𝑥 ∣ 𝜑})) → ω ⊆ {𝑥 ∣ 𝜑})
16	5, 14, 15	mp2an 426	. . 3 ⊢ ω ⊆ {𝑥 ∣ 𝜑}
17	16	sseli 3175	. 2 ⊢ (𝐴 ∈ ω → 𝐴 ∈ {𝑥 ∣ 𝜑})
18		finds.4	. . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏))
19	18	elabg 2906	. 2 ⊢ (𝐴 ∈ ω → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜏))
20	17, 19	mpbid 147	1 ⊢ (𝐴 ∈ ω → 𝜏)

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1364   ∈ wcel 2164  {cab 2179  ∀wral 2472   ⊆ wss 3153  ∅c0 3446  suc csuc 4396  ωcom 4622

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-iinf 4620

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-uni 3836  df-int 3871  df-suc 4402  df-iom 4623

This theorem is referenced by:  findes  4635  nn0suc  4636  elomssom  4637  ordom  4639  nndceq0  4650  0elnn  4651  omsinds  4654  nna0r  6531  nnm0r  6532  nnsucelsuc  6544  nneneq  6913  php5  6914  php5dom  6919  fidcenumlemrk  7013  fidcenumlemr  7014  nninfninc  7182  nnnninfeq  7187  nnnninfeq2  7188  frec2uzltd  10474  frecuzrdgg  10487  seq3val  10531  seqvalcd  10532  omgadd  10873  zfz1iso  10912  ennnfonelemhom  12572  nninfsellemdc  15505