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Theorem tfinds 7562
Description: Principle of Transfinite Induction (inference schema), using implicit substitutions. The first four hypotheses establish the substitutions we need. The last three are the basis, the induction step for successors, and the induction step for limit ordinals. Theorem Schema 4 of [Suppes] p. 197. (Contributed by NM, 16-Apr-1995.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Hypotheses
Ref Expression
tfinds.1 (𝑥 = ∅ → (𝜑𝜓))
tfinds.2 (𝑥 = 𝑦 → (𝜑𝜒))
tfinds.3 (𝑥 = suc 𝑦 → (𝜑𝜃))
tfinds.4 (𝑥 = 𝐴 → (𝜑𝜏))
tfinds.5 𝜓
tfinds.6 (𝑦 ∈ On → (𝜒𝜃))
tfinds.7 (Lim 𝑥 → (∀𝑦𝑥 𝜒𝜑))
Assertion
Ref Expression
tfinds (𝐴 ∈ On → 𝜏)
Distinct variable groups:   𝑥,𝑦   𝑥,𝐴   𝜒,𝑥   𝜏,𝑥   𝜑,𝑦
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥,𝑦)   𝜒(𝑦)   𝜃(𝑥,𝑦)   𝜏(𝑦)   𝐴(𝑦)

Proof of Theorem tfinds
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 tfinds.2 . 2 (𝑥 = 𝑦 → (𝜑𝜒))
2 tfinds.4 . 2 (𝑥 = 𝐴 → (𝜑𝜏))
3 dflim3 7550 . . . . 5 (Lim 𝑥 ↔ (Ord 𝑥 ∧ ¬ (𝑥 = ∅ ∨ ∃𝑦 ∈ On 𝑥 = suc 𝑦)))
43notbii 321 . . . 4 (¬ Lim 𝑥 ↔ ¬ (Ord 𝑥 ∧ ¬ (𝑥 = ∅ ∨ ∃𝑦 ∈ On 𝑥 = suc 𝑦)))
5 iman 402 . . . . 5 ((Ord 𝑥 → (𝑥 = ∅ ∨ ∃𝑦 ∈ On 𝑥 = suc 𝑦)) ↔ ¬ (Ord 𝑥 ∧ ¬ (𝑥 = ∅ ∨ ∃𝑦 ∈ On 𝑥 = suc 𝑦)))
6 eloni 6195 . . . . . . 7 (𝑥 ∈ On → Ord 𝑥)
7 pm2.27 42 . . . . . . 7 (Ord 𝑥 → ((Ord 𝑥 → (𝑥 = ∅ ∨ ∃𝑦 ∈ On 𝑥 = suc 𝑦)) → (𝑥 = ∅ ∨ ∃𝑦 ∈ On 𝑥 = suc 𝑦)))
86, 7syl 17 . . . . . 6 (𝑥 ∈ On → ((Ord 𝑥 → (𝑥 = ∅ ∨ ∃𝑦 ∈ On 𝑥 = suc 𝑦)) → (𝑥 = ∅ ∨ ∃𝑦 ∈ On 𝑥 = suc 𝑦)))
9 tfinds.5 . . . . . . . . 9 𝜓
10 tfinds.1 . . . . . . . . 9 (𝑥 = ∅ → (𝜑𝜓))
119, 10mpbiri 259 . . . . . . . 8 (𝑥 = ∅ → 𝜑)
1211a1d 25 . . . . . . 7 (𝑥 = ∅ → (∀𝑦𝑥 𝜒𝜑))
13 nfra1 3219 . . . . . . . . 9 𝑦𝑦𝑥 𝜒
14 nfv 1906 . . . . . . . . 9 𝑦𝜑
1513, 14nfim 1888 . . . . . . . 8 𝑦(∀𝑦𝑥 𝜒𝜑)
16 vex 3498 . . . . . . . . . . . . 13 𝑦 ∈ V
1716sucid 6264 . . . . . . . . . . . 12 𝑦 ∈ suc 𝑦
181rspcv 3617 . . . . . . . . . . . 12 (𝑦 ∈ suc 𝑦 → (∀𝑥 ∈ suc 𝑦𝜑𝜒))
1917, 18ax-mp 5 . . . . . . . . . . 11 (∀𝑥 ∈ suc 𝑦𝜑𝜒)
20 tfinds.6 . . . . . . . . . . 11 (𝑦 ∈ On → (𝜒𝜃))
2119, 20syl5 34 . . . . . . . . . 10 (𝑦 ∈ On → (∀𝑥 ∈ suc 𝑦𝜑𝜃))
22 raleq 3406 . . . . . . . . . . . 12 (𝑥 = suc 𝑦 → (∀𝑧𝑥 [𝑧 / 𝑥]𝜑 ↔ ∀𝑧 ∈ suc 𝑦[𝑧 / 𝑥]𝜑))
23 nfv 1906 . . . . . . . . . . . . . . 15 𝑥𝜒
2423, 1sbiev 2322 . . . . . . . . . . . . . 14 ([𝑦 / 𝑥]𝜑𝜒)
25 sbequ 2081 . . . . . . . . . . . . . 14 (𝑦 = 𝑧 → ([𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥]𝜑))
2624, 25syl5bbr 286 . . . . . . . . . . . . 13 (𝑦 = 𝑧 → (𝜒 ↔ [𝑧 / 𝑥]𝜑))
2726cbvralvw 3450 . . . . . . . . . . . 12 (∀𝑦𝑥 𝜒 ↔ ∀𝑧𝑥 [𝑧 / 𝑥]𝜑)
28 cbvralsvw 3468 . . . . . . . . . . . 12 (∀𝑥 ∈ suc 𝑦𝜑 ↔ ∀𝑧 ∈ suc 𝑦[𝑧 / 𝑥]𝜑)
2922, 27, 283bitr4g 315 . . . . . . . . . . 11 (𝑥 = suc 𝑦 → (∀𝑦𝑥 𝜒 ↔ ∀𝑥 ∈ suc 𝑦𝜑))
3029imbi1d 343 . . . . . . . . . 10 (𝑥 = suc 𝑦 → ((∀𝑦𝑥 𝜒𝜃) ↔ (∀𝑥 ∈ suc 𝑦𝜑𝜃)))
3121, 30syl5ibrcom 248 . . . . . . . . 9 (𝑦 ∈ On → (𝑥 = suc 𝑦 → (∀𝑦𝑥 𝜒𝜃)))
32 tfinds.3 . . . . . . . . . . 11 (𝑥 = suc 𝑦 → (𝜑𝜃))
3332biimprd 249 . . . . . . . . . 10 (𝑥 = suc 𝑦 → (𝜃𝜑))
3433a1i 11 . . . . . . . . 9 (𝑦 ∈ On → (𝑥 = suc 𝑦 → (𝜃𝜑)))
3531, 34syldd 72 . . . . . . . 8 (𝑦 ∈ On → (𝑥 = suc 𝑦 → (∀𝑦𝑥 𝜒𝜑)))
3615, 35rexlimi 3315 . . . . . . 7 (∃𝑦 ∈ On 𝑥 = suc 𝑦 → (∀𝑦𝑥 𝜒𝜑))
3712, 36jaoi 851 . . . . . 6 ((𝑥 = ∅ ∨ ∃𝑦 ∈ On 𝑥 = suc 𝑦) → (∀𝑦𝑥 𝜒𝜑))
388, 37syl6 35 . . . . 5 (𝑥 ∈ On → ((Ord 𝑥 → (𝑥 = ∅ ∨ ∃𝑦 ∈ On 𝑥 = suc 𝑦)) → (∀𝑦𝑥 𝜒𝜑)))
395, 38syl5bir 244 . . . 4 (𝑥 ∈ On → (¬ (Ord 𝑥 ∧ ¬ (𝑥 = ∅ ∨ ∃𝑦 ∈ On 𝑥 = suc 𝑦)) → (∀𝑦𝑥 𝜒𝜑)))
404, 39syl5bi 243 . . 3 (𝑥 ∈ On → (¬ Lim 𝑥 → (∀𝑦𝑥 𝜒𝜑)))
41 tfinds.7 . . 3 (Lim 𝑥 → (∀𝑦𝑥 𝜒𝜑))
4240, 41pm2.61d2 182 . 2 (𝑥 ∈ On → (∀𝑦𝑥 𝜒𝜑))
431, 2, 42tfis3 7560 1 (𝐴 ∈ On → 𝜏)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  wo 841   = wceq 1528  [wsb 2060  wcel 2105  wral 3138  wrex 3139  c0 4290  Ord word 6184  Oncon0 6185  Lim wlim 6186  suc csuc 6187
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pr 5321  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3497  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4833  df-br 5059  df-opab 5121  df-tr 5165  df-eprel 5459  df-po 5468  df-so 5469  df-fr 5508  df-we 5510  df-ord 6188  df-on 6189  df-lim 6190  df-suc 6191
This theorem is referenced by:  tfindsg  7563  tfindes  7565  tfinds3  7567  oa0r  8154  om0r  8155  om1r  8159  oe1m  8161  oeoalem  8212  r1sdom  9192  r1tr  9194  alephon  9484  alephcard  9485  alephordi  9489  rdgprc  32937
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