Theorem finds 4656

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 4179	. . . . . 6 ⊢ ∅ ∈ V
3		finds.1	. . . . . 6 ⊢ (𝑥 = ∅ → (𝜑 ↔ 𝜓))
4	2, 3	elab 2921	. . . . 5 ⊢ (∅ ∈ {𝑥 ∣ 𝜑} ↔ 𝜓)
5	1, 4	mpbir 146	. . . 4 ⊢ ∅ ∈ {𝑥 ∣ 𝜑}
6		finds.6	. . . . . 6 ⊢ (𝑦 ∈ ω → (𝜒 → 𝜃))
7		vex 2776	. . . . . . 7 ⊢ 𝑦 ∈ V
8		finds.2	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))
9	7, 8	elab 2921	. . . . . 6 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝜒)
10	7	sucex 4555	. . . . . . 7 ⊢ suc 𝑦 ∈ V
11		finds.3	. . . . . . 7 ⊢ (𝑥 = suc 𝑦 → (𝜑 ↔ 𝜃))
12	10, 11	elab 2921	. . . . . 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 2560	. . . 4 ⊢ ∀𝑦 ∈ ω (𝑦 ∈ {𝑥 ∣ 𝜑} → suc 𝑦 ∈ {𝑥 ∣ 𝜑})
15		peano5 4654	. . . 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 3193	. 2 ⊢ (𝐴 ∈ ω → 𝐴 ∈ {𝑥 ∣ 𝜑})
18		finds.4	. . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏))
19	18	elabg 2923	. 2 ⊢ (𝐴 ∈ ω → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜏))
20	17, 19	mpbid 147	1 ⊢ (𝐴 ∈ ω → 𝜏)

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1373   ∈ wcel 2177  {cab 2192  ∀wral 2485   ⊆ wss 3170  ∅c0 3464  suc csuc 4420  ωcom 4646

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4170  ax-nul 4178  ax-pow 4226  ax-pr 4261  ax-un 4488  ax-iinf 4644

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-v 2775  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-pw 3623  df-sn 3644  df-pr 3645  df-uni 3857  df-int 3892  df-suc 4426  df-iom 4647

This theorem is referenced by:  findes  4659  nn0suc  4660  elomssom  4661  ordom  4663  nndceq0  4674  0elnn  4675  omsinds  4678  nna0r  6577  nnm0r  6578  nnsucelsuc  6590  nneneq  6969  php5  6970  php5dom  6975  fidcenumlemrk  7071  fidcenumlemr  7072  nninfninc  7240  nnnninfeq  7245  nnnninfeq2  7246  frec2uzltd  10570  frecuzrdgg  10583  seq3val  10627  seqvalcd  10628  omgadd  10969  zfz1iso  11008  ennnfonelemhom  12861  nninfsellemdc  16088  nnnninfex  16100