ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  finds GIF version

Theorem finds 4630
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 = ∅ → (𝜑𝜓))
finds.2 (𝑥 = 𝑦 → (𝜑𝜒))
finds.3 (𝑥 = suc 𝑦 → (𝜑𝜃))
finds.4 (𝑥 = 𝐴 → (𝜑𝜏))
finds.5 𝜓
finds.6 (𝑦 ∈ ω → (𝜒𝜃))
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
StepHypRef Expression
1 finds.5 . . . . 5 𝜓
2 0ex 4156 . . . . . 6 ∅ ∈ V
3 finds.1 . . . . . 6 (𝑥 = ∅ → (𝜑𝜓))
42, 3elab 2904 . . . . 5 (∅ ∈ {𝑥𝜑} ↔ 𝜓)
51, 4mpbir 146 . . . 4 ∅ ∈ {𝑥𝜑}
6 finds.6 . . . . . 6 (𝑦 ∈ ω → (𝜒𝜃))
7 vex 2763 . . . . . . 7 𝑦 ∈ V
8 finds.2 . . . . . . 7 (𝑥 = 𝑦 → (𝜑𝜒))
97, 8elab 2904 . . . . . 6 (𝑦 ∈ {𝑥𝜑} ↔ 𝜒)
107sucex 4529 . . . . . . 7 suc 𝑦 ∈ V
11 finds.3 . . . . . . 7 (𝑥 = suc 𝑦 → (𝜑𝜃))
1210, 11elab 2904 . . . . . 6 (suc 𝑦 ∈ {𝑥𝜑} ↔ 𝜃)
136, 9, 123imtr4g 205 . . . . 5 (𝑦 ∈ ω → (𝑦 ∈ {𝑥𝜑} → suc 𝑦 ∈ {𝑥𝜑}))
1413rgen 2547 . . . 4 𝑦 ∈ ω (𝑦 ∈ {𝑥𝜑} → suc 𝑦 ∈ {𝑥𝜑})
15 peano5 4628 . . . 4 ((∅ ∈ {𝑥𝜑} ∧ ∀𝑦 ∈ ω (𝑦 ∈ {𝑥𝜑} → suc 𝑦 ∈ {𝑥𝜑})) → ω ⊆ {𝑥𝜑})
165, 14, 15mp2an 426 . . 3 ω ⊆ {𝑥𝜑}
1716sseli 3175 . 2 (𝐴 ∈ ω → 𝐴 ∈ {𝑥𝜑})
18 finds.4 . . 3 (𝑥 = 𝐴 → (𝜑𝜏))
1918elabg 2906 . 2 (𝐴 ∈ ω → (𝐴 ∈ {𝑥𝜑} ↔ 𝜏))
2017, 19mpbid 147 1 (𝐴 ∈ ω → 𝜏)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105   = wceq 1364  wcel 2164  {cab 2179  wral 2472  wss 3153  c0 3446  suc csuc 4394  ωcom 4620
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 4462  ax-iinf 4618
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 4400  df-iom 4621
This theorem is referenced by:  findes  4633  nn0suc  4634  elomssom  4635  ordom  4637  nndceq0  4648  0elnn  4649  omsinds  4652  nna0r  6526  nnm0r  6527  nnsucelsuc  6539  nneneq  6908  php5  6909  php5dom  6914  fidcenumlemrk  7007  fidcenumlemr  7008  nninfninc  7176  nnnninfeq  7181  nnnninfeq2  7182  frec2uzltd  10468  frecuzrdgg  10481  seq3val  10525  seqvalcd  10526  omgadd  10867  zfz1iso  10906  ennnfonelemhom  12566  nninfsellemdc  15478
  Copyright terms: Public domain W3C validator