Theorem finds 7821

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 5240	. . . . . 6 ⊢ ∅ ∈ V
3		finds.1	. . . . . 6 ⊢ (𝑥 = ∅ → (𝜑 ↔ 𝜓))
4	2, 3	elab 3630	. . . . 5 ⊢ (∅ ∈ {𝑥 ∣ 𝜑} ↔ 𝜓)
5	1, 4	mpbir 231	. . . 4 ⊢ ∅ ∈ {𝑥 ∣ 𝜑}
6		finds.6	. . . . . 6 ⊢ (𝑦 ∈ ω → (𝜒 → 𝜃))
7		vex 3440	. . . . . . 7 ⊢ 𝑦 ∈ V
8		finds.2	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))
9	7, 8	elab 3630	. . . . . 6 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝜒)
10	7	sucex 7734	. . . . . . 7 ⊢ suc 𝑦 ∈ V
11		finds.3	. . . . . . 7 ⊢ (𝑥 = suc 𝑦 → (𝜑 ↔ 𝜃))
12	10, 11	elab 3630	. . . . . 6 ⊢ (suc 𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝜃)
13	6, 9, 12	3imtr4g 296	. . . . 5 ⊢ (𝑦 ∈ ω → (𝑦 ∈ {𝑥 ∣ 𝜑} → suc 𝑦 ∈ {𝑥 ∣ 𝜑}))
14	13	rgen 3049	. . . 4 ⊢ ∀𝑦 ∈ ω (𝑦 ∈ {𝑥 ∣ 𝜑} → suc 𝑦 ∈ {𝑥 ∣ 𝜑})
15		peano5 7818	. . . 4 ⊢ ((∅ ∈ {𝑥 ∣ 𝜑} ∧ ∀𝑦 ∈ ω (𝑦 ∈ {𝑥 ∣ 𝜑} → suc 𝑦 ∈ {𝑥 ∣ 𝜑})) → ω ⊆ {𝑥 ∣ 𝜑})
16	5, 14, 15	mp2an 692	. . 3 ⊢ ω ⊆ {𝑥 ∣ 𝜑}
17	16	sseli 3925	. 2 ⊢ (𝐴 ∈ ω → 𝐴 ∈ {𝑥 ∣ 𝜑})
18		finds.4	. . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏))
19	18	elabg 3627	. 2 ⊢ (𝐴 ∈ ω → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜏))
20	17, 19	mpbid 232	1 ⊢ (𝐴 ∈ ω → 𝜏)

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1541   ∈ wcel 2111  {cab 2709  ∀wral 3047   ⊆ wss 3897  ∅c0 4278  suc csuc 6303  ωcom 7791

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5229  ax-nul 5239  ax-pr 5365  ax-un 7663

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4279  df-if 4471  df-pw 4547  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-br 5087  df-opab 5149  df-tr 5194  df-eprel 5511  df-po 5519  df-so 5520  df-fr 5564  df-we 5566  df-ord 6304  df-on 6305  df-lim 6306  df-suc 6307  df-om 7792

This theorem is referenced by:  findsg  7822  findes  7825  seqomlem1  8364  nna0r  8519  nnm0r  8520  nnawordi  8531  nneob  8566  naddoa  8612  enrefnn  8963  pssnn  9073  nneneq  9110  inf3lem1  9513  inf3lem2  9514  cantnfval2  9554  cantnfp1lem3  9565  ttrclss  9605  ttrclselem2  9611  r1fin  9661  ackbij1lem14  10118  ackbij1lem16  10120  ackbij1  10123  ackbij2lem2  10125  ackbij2lem3  10126  infpssrlem4  10192  fin23lem14  10219  fin23lem34  10232  itunitc1  10306  ituniiun  10308  om2uzuzi  13851  om2uzlti  13852  om2uzrdg  13858  uzrdgxfr  13869  hashgadd  14279  mreexexd  17549  precsexlem8  28147  precsexlem9  28148  om2noseqrdg  28229  bdayn0sf1o  28290  dfnns2  28292  constrfin  33751  constrextdg2  33754  satfrel  35403  satfdm  35405  satfrnmapom  35406  satf0op  35413  satf0n0  35414  sat1el2xp  35415  fmlafvel  35421  fmlaomn0  35426  gonar  35431  goalr  35433  satffun  35445  findfvcl  36486  finxp00  37436  onmcl  43364  naddonnn  43428