Theorem finds 7942

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 5267	. . . . . 6 ⊢ ∅ ∈ V
3		finds.1	. . . . . 6 ⊢ (𝑥 = ∅ → (𝜑 ↔ 𝜓))
4	2, 3	elab 3636	. . . . 5 ⊢ (∅ ∈ {𝑥 ∣ 𝜑} ↔ 𝜓)
5	1, 4	mpbir 234	. . . 4 ⊢ ∅ ∈ {𝑥 ∣ 𝜑}
6		finds.6	. . . . . 6 ⊢ (𝑦 ∈ ω → (𝜒 → 𝜃))
7		vex 3458	. . . . . . 7 ⊢ 𝑦 ∈ V
8		finds.2	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))
9	7, 8	elab 3636	. . . . . 6 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝜒)
10	7	sucex 7854	. . . . . . 7 ⊢ suc 𝑦 ∈ V
11		finds.3	. . . . . . 7 ⊢ (𝑥 = suc 𝑦 → (𝜑 ↔ 𝜃))
12	10, 11	elab 3636	. . . . . 6 ⊢ (suc 𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝜃)
13	6, 9, 12	3imtr4g 299	. . . . 5 ⊢ (𝑦 ∈ ω → (𝑦 ∈ {𝑥 ∣ 𝜑} → suc 𝑦 ∈ {𝑥 ∣ 𝜑}))
14	13	rgen 3082	. . . 4 ⊢ ∀𝑦 ∈ ω (𝑦 ∈ {𝑥 ∣ 𝜑} → suc 𝑦 ∈ {𝑥 ∣ 𝜑})
15		peano5 7939	. . . 4 ⊢ ((∅ ∈ {𝑥 ∣ 𝜑} ∧ ∀𝑦 ∈ ω (𝑦 ∈ {𝑥 ∣ 𝜑} → suc 𝑦 ∈ {𝑥 ∣ 𝜑})) → ω ⊆ {𝑥 ∣ 𝜑})
16	5, 14, 15	mp2an 705	. . 3 ⊢ ω ⊆ {𝑥 ∣ 𝜑}
17	16	sseli 3909	. 2 ⊢ (𝐴 ∈ ω → 𝐴 ∈ {𝑥 ∣ 𝜑})
18		finds.4	. . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏))
19	18	elabg 3633	. 2 ⊢ (𝐴 ∈ ω → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜏))
20	17, 19	mpbid 235	1 ⊢ (𝐴 ∈ ω → 𝜏)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570   ∈ wcel 2147  {cab 2741  ∀wral 3080   ⊆ wss 3900  ∅c0 4281  suc csuc 6390  ωcom 7911

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-tr 5219  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-om 7912

This theorem is used by:  findsg  7943  findes  7946  seqomlem1  8498  nna0r  8656  nnm0r  8657  nnawordi  8668  nneob  8703  naddoa  8750  enrefnn  9112  pssnn  9222  nneneq  9259  inf3lem1  9666  inf3lem2  9667  cantnfval2  9707  cantnfp1lem3  9718  ttrclss  9758  ttrclselem2  9764  r1fin  9817  ackbij1lem14  10349  ackbij1lem16  10351  ackbij1  10354  ackbij2lem2  10356  ackbij2lem3  10357  infpssrlem4  10423  fin23lem14  10450  fin23lem34  10463  itunitc1  10537  ituniiun  10539  om2uzuzi  14132  om2uzlti  14133  om2uzrdg  14139  uzrdgxfr  14150  hashgadd  14562  mreexexd  17872  precsexlem8  28657  precsexlem9  28658  om2noseqrdg  28747  bdayn0sf1o  28813  dfnns2  28815  constrfin  34489  constrextdg2  34492  satfrel  36340  satfdm  36342  satfrnmapom  36343  satf0op  36350  satf0n0  36351  sat1el2xp  36352  fmlafvel  36358  fmlaomn0  36363  gonar  36368  goalr  36370  satffun  36382  findfvcl  37455  finxp00  38593  onmcl  44594  naddonnn  44658  omhf  46285