Theorem finds 4698

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 4216	. . . . . 6 ⊢ ∅ ∈ V
3		finds.1	. . . . . 6 ⊢ (𝑥 = ∅ → (𝜑 ↔ 𝜓))
4	2, 3	elab 2950	. . . . 5 ⊢ (∅ ∈ {𝑥 ∣ 𝜑} ↔ 𝜓)
5	1, 4	mpbir 146	. . . 4 ⊢ ∅ ∈ {𝑥 ∣ 𝜑}
6		finds.6	. . . . . 6 ⊢ (𝑦 ∈ ω → (𝜒 → 𝜃))
7		vex 2805	. . . . . . 7 ⊢ 𝑦 ∈ V
8		finds.2	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))
9	7, 8	elab 2950	. . . . . 6 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝜒)
10	7	sucex 4597	. . . . . . 7 ⊢ suc 𝑦 ∈ V
11		finds.3	. . . . . . 7 ⊢ (𝑥 = suc 𝑦 → (𝜑 ↔ 𝜃))
12	10, 11	elab 2950	. . . . . 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 2585	. . . 4 ⊢ ∀𝑦 ∈ ω (𝑦 ∈ {𝑥 ∣ 𝜑} → suc 𝑦 ∈ {𝑥 ∣ 𝜑})
15		peano5 4696	. . . 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 3223	. 2 ⊢ (𝐴 ∈ ω → 𝐴 ∈ {𝑥 ∣ 𝜑})
18		finds.4	. . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏))
19	18	elabg 2952	. 2 ⊢ (𝐴 ∈ ω → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜏))
20	17, 19	mpbid 147	1 ⊢ (𝐴 ∈ ω → 𝜏)

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1397   ∈ wcel 2202  {cab 2217  ∀wral 2510   ⊆ wss 3200  ∅c0 3494  suc csuc 4462  ωcom 4688

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-iinf 4686

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-uni 3894  df-int 3929  df-suc 4468  df-iom 4689

This theorem is referenced by:  findes  4701  nn0suc  4702  elomssom  4703  ordom  4705  nndceq0  4716  0elnn  4717  omsinds  4720  nna0r  6646  nnm0r  6647  nnsucelsuc  6659  nneneq  7043  php5  7044  php5dom  7049  fidcenumlemrk  7153  fidcenumlemr  7154  nninfninc  7322  nnnninfeq  7327  nnnninfeq2  7328  frec2uzltd  10665  frecuzrdgg  10678  seq3val  10722  seqvalcd  10723  omgadd  11065  zfz1iso  11105  ennnfonelemhom  13037  nninfsellemdc  16615  nnnninfex  16627