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Theorem tfinds 6925
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 6913 . . . . 5 (Lim 𝑥 ↔ (Ord 𝑥 ∧ ¬ (𝑥 = ∅ ∨ ∃𝑦 ∈ On 𝑥 = suc 𝑦)))
43notbii 308 . . . 4 (¬ Lim 𝑥 ↔ ¬ (Ord 𝑥 ∧ ¬ (𝑥 = ∅ ∨ ∃𝑦 ∈ On 𝑥 = suc 𝑦)))
5 iman 438 . . . . 5 ((Ord 𝑥 → (𝑥 = ∅ ∨ ∃𝑦 ∈ On 𝑥 = suc 𝑦)) ↔ ¬ (Ord 𝑥 ∧ ¬ (𝑥 = ∅ ∨ ∃𝑦 ∈ On 𝑥 = suc 𝑦)))
6 eloni 5633 . . . . . . 7 (𝑥 ∈ On → Ord 𝑥)
7 pm2.27 40 . . . . . . 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 246 . . . . . . . 8 (𝑥 = ∅ → 𝜑)
1211a1d 25 . . . . . . 7 (𝑥 = ∅ → (∀𝑦𝑥 𝜒𝜑))
13 nfra1 2921 . . . . . . . . 9 𝑦𝑦𝑥 𝜒
14 nfv 1829 . . . . . . . . 9 𝑦𝜑
1513, 14nfim 1812 . . . . . . . 8 𝑦(∀𝑦𝑥 𝜒𝜑)
16 vex 3172 . . . . . . . . . . . . 13 𝑦 ∈ V
1716sucid 5704 . . . . . . . . . . . 12 𝑦 ∈ suc 𝑦
181rspcv 3274 . . . . . . . . . . . 12 (𝑦 ∈ suc 𝑦 → (∀𝑥 ∈ suc 𝑦𝜑𝜒))
1917, 18ax-mp 5 . . . . . . . . . . 11 (∀𝑥 ∈ suc 𝑦𝜑𝜒)
20 tfinds.6 . . . . . . . . . . 11 (𝑦 ∈ On → (𝜒𝜃))
2119, 20syl5 33 . . . . . . . . . 10 (𝑦 ∈ On → (∀𝑥 ∈ suc 𝑦𝜑𝜃))
22 raleq 3111 . . . . . . . . . . . 12 (𝑥 = suc 𝑦 → (∀𝑧𝑥 [𝑧 / 𝑥]𝜑 ↔ ∀𝑧 ∈ suc 𝑦[𝑧 / 𝑥]𝜑))
23 nfv 1829 . . . . . . . . . . . . . . 15 𝑥𝜒
2423, 1sbie 2392 . . . . . . . . . . . . . 14 ([𝑦 / 𝑥]𝜑𝜒)
25 sbequ 2360 . . . . . . . . . . . . . 14 (𝑦 = 𝑧 → ([𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥]𝜑))
2624, 25syl5bbr 272 . . . . . . . . . . . . 13 (𝑦 = 𝑧 → (𝜒 ↔ [𝑧 / 𝑥]𝜑))
2726cbvralv 3143 . . . . . . . . . . . 12 (∀𝑦𝑥 𝜒 ↔ ∀𝑧𝑥 [𝑧 / 𝑥]𝜑)
28 cbvralsv 3154 . . . . . . . . . . . 12 (∀𝑥 ∈ suc 𝑦𝜑 ↔ ∀𝑧 ∈ suc 𝑦[𝑧 / 𝑥]𝜑)
2922, 27, 283bitr4g 301 . . . . . . . . . . 11 (𝑥 = suc 𝑦 → (∀𝑦𝑥 𝜒 ↔ ∀𝑥 ∈ suc 𝑦𝜑))
3029imbi1d 329 . . . . . . . . . 10 (𝑥 = suc 𝑦 → ((∀𝑦𝑥 𝜒𝜃) ↔ (∀𝑥 ∈ suc 𝑦𝜑𝜃)))
3121, 30syl5ibrcom 235 . . . . . . . . 9 (𝑦 ∈ On → (𝑥 = suc 𝑦 → (∀𝑦𝑥 𝜒𝜃)))
32 tfinds.3 . . . . . . . . . . 11 (𝑥 = suc 𝑦 → (𝜑𝜃))
3332biimprd 236 . . . . . . . . . 10 (𝑥 = suc 𝑦 → (𝜃𝜑))
3433a1i 11 . . . . . . . . 9 (𝑦 ∈ On → (𝑥 = suc 𝑦 → (𝜃𝜑)))
3531, 34syldd 69 . . . . . . . 8 (𝑦 ∈ On → (𝑥 = suc 𝑦 → (∀𝑦𝑥 𝜒𝜑)))
3615, 35rexlimi 3002 . . . . . . 7 (∃𝑦 ∈ On 𝑥 = suc 𝑦 → (∀𝑦𝑥 𝜒𝜑))
3712, 36jaoi 392 . . . . . 6 ((𝑥 = ∅ ∨ ∃𝑦 ∈ On 𝑥 = suc 𝑦) → (∀𝑦𝑥 𝜒𝜑))
388, 37syl6 34 . . . . 5 (𝑥 ∈ On → ((Ord 𝑥 → (𝑥 = ∅ ∨ ∃𝑦 ∈ On 𝑥 = suc 𝑦)) → (∀𝑦𝑥 𝜒𝜑)))
395, 38syl5bir 231 . . . 4 (𝑥 ∈ On → (¬ (Ord 𝑥 ∧ ¬ (𝑥 = ∅ ∨ ∃𝑦 ∈ On 𝑥 = suc 𝑦)) → (∀𝑦𝑥 𝜒𝜑)))
404, 39syl5bi 230 . . 3 (𝑥 ∈ On → (¬ Lim 𝑥 → (∀𝑦𝑥 𝜒𝜑)))
41 tfinds.7 . . 3 (Lim 𝑥 → (∀𝑦𝑥 𝜒𝜑))
4240, 41pm2.61d2 170 . 2 (𝑥 ∈ On → (∀𝑦𝑥 𝜒𝜑))
431, 2, 42tfis3 6923 1 (𝐴 ∈ On → 𝜏)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wo 381  wa 382   = wceq 1474  [wsb 1866  wcel 1976  wral 2892  wrex 2893  c0 3870  Ord word 5622  Oncon0 5623  Lim wlim 5624  suc csuc 5625
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2229  ax-ext 2586  ax-sep 4700  ax-nul 4709  ax-pr 4825  ax-un 6821
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2458  df-mo 2459  df-clab 2593  df-cleq 2599  df-clel 2602  df-nfc 2736  df-ne 2778  df-ral 2897  df-rex 2898  df-rab 2901  df-v 3171  df-sbc 3399  df-dif 3539  df-un 3541  df-in 3543  df-ss 3550  df-pss 3552  df-nul 3871  df-if 4033  df-pw 4106  df-sn 4122  df-pr 4124  df-tp 4126  df-op 4128  df-uni 4364  df-br 4575  df-opab 4635  df-tr 4672  df-eprel 4936  df-po 4946  df-so 4947  df-fr 4984  df-we 4986  df-ord 5626  df-on 5627  df-lim 5628  df-suc 5629
This theorem is referenced by:  tfindsg  6926  tfindes  6928  tfinds3  6930  oa0r  7479  om0r  7480  om1r  7484  oe1m  7486  oeoalem  7537  r1sdom  8494  r1tr  8496  alephon  8749  alephcard  8750  alephordi  8754  rdgprc  30747
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