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Theorem finds 4323
 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 3884 . . . . . 6 ∅ ∈ V
3 finds.1 . . . . . 6 (𝑥 = ∅ → (𝜑𝜓))
42, 3elab 2687 . . . . 5 (∅ ∈ {𝑥𝜑} ↔ 𝜓)
51, 4mpbir 134 . . . 4 ∅ ∈ {𝑥𝜑}
6 finds.6 . . . . . 6 (𝑦 ∈ ω → (𝜒𝜃))
7 vex 2560 . . . . . . 7 𝑦 ∈ V
8 finds.2 . . . . . . 7 (𝑥 = 𝑦 → (𝜑𝜒))
97, 8elab 2687 . . . . . 6 (𝑦 ∈ {𝑥𝜑} ↔ 𝜒)
107sucex 4225 . . . . . . 7 suc 𝑦 ∈ V
11 finds.3 . . . . . . 7 (𝑥 = suc 𝑦 → (𝜑𝜃))
1210, 11elab 2687 . . . . . 6 (suc 𝑦 ∈ {𝑥𝜑} ↔ 𝜃)
136, 9, 123imtr4g 194 . . . . 5 (𝑦 ∈ ω → (𝑦 ∈ {𝑥𝜑} → suc 𝑦 ∈ {𝑥𝜑}))
1413rgen 2374 . . . 4 𝑦 ∈ ω (𝑦 ∈ {𝑥𝜑} → suc 𝑦 ∈ {𝑥𝜑})
15 peano5 4321 . . . 4 ((∅ ∈ {𝑥𝜑} ∧ ∀𝑦 ∈ ω (𝑦 ∈ {𝑥𝜑} → suc 𝑦 ∈ {𝑥𝜑})) → ω ⊆ {𝑥𝜑})
165, 14, 15mp2an 402 . . 3 ω ⊆ {𝑥𝜑}
1716sseli 2941 . 2 (𝐴 ∈ ω → 𝐴 ∈ {𝑥𝜑})
18 finds.4 . . 3 (𝑥 = 𝐴 → (𝜑𝜏))
1918elabg 2688 . 2 (𝐴 ∈ ω → (𝐴 ∈ {𝑥𝜑} ↔ 𝜏))
2017, 19mpbid 135 1 (𝐴 ∈ ω → 𝜏)
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98   = wceq 1243   ∈ wcel 1393  {cab 2026  ∀wral 2306   ⊆ wss 2917  ∅c0 3224  suc csuc 4102  ωcom 4313 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-iinf 4311 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-uni 3581  df-int 3616  df-suc 4108  df-iom 4314 This theorem is referenced by:  findes  4326  nn0suc  4327  elnn  4328  ordom  4329  nndceq0  4339  0elnn  4340  nna0r  6057  nnm0r  6058  nnsucelsuc  6070  nneneq  6320  php5  6321  php5dom  6325  frec2uzltd  9189
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