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Theorem tfindsg 6925
Description: 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. The basis of this version is an arbitrary ordinal 𝐵 instead of zero. Remark in [TakeutiZaring] p. 57. (Contributed by NM, 5-Mar-2004.)
Hypotheses
Ref Expression
tfindsg.1 (𝑥 = 𝐵 → (𝜑𝜓))
tfindsg.2 (𝑥 = 𝑦 → (𝜑𝜒))
tfindsg.3 (𝑥 = suc 𝑦 → (𝜑𝜃))
tfindsg.4 (𝑥 = 𝐴 → (𝜑𝜏))
tfindsg.5 (𝐵 ∈ On → 𝜓)
tfindsg.6 (((𝑦 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐵𝑦) → (𝜒𝜃))
tfindsg.7 (((Lim 𝑥𝐵 ∈ On) ∧ 𝐵𝑥) → (∀𝑦𝑥 (𝐵𝑦𝜒) → 𝜑))
Assertion
Ref Expression
tfindsg (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐵𝐴) → 𝜏)
Distinct variable groups:   𝑥,𝐴   𝑥,𝑦,𝐵   𝜒,𝑥   𝜃,𝑥   𝜏,𝑥   𝜑,𝑦
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥,𝑦)   𝜒(𝑦)   𝜃(𝑦)   𝜏(𝑦)   𝐴(𝑦)

Proof of Theorem tfindsg
StepHypRef Expression
1 sseq2 3585 . . . . . . 7 (𝑥 = ∅ → (𝐵𝑥𝐵 ⊆ ∅))
21adantl 480 . . . . . 6 ((𝐵 = ∅ ∧ 𝑥 = ∅) → (𝐵𝑥𝐵 ⊆ ∅))
3 eqeq2 2616 . . . . . . . 8 (𝐵 = ∅ → (𝑥 = 𝐵𝑥 = ∅))
4 tfindsg.1 . . . . . . . 8 (𝑥 = 𝐵 → (𝜑𝜓))
53, 4syl6bir 242 . . . . . . 7 (𝐵 = ∅ → (𝑥 = ∅ → (𝜑𝜓)))
65imp 443 . . . . . 6 ((𝐵 = ∅ ∧ 𝑥 = ∅) → (𝜑𝜓))
72, 6imbi12d 332 . . . . 5 ((𝐵 = ∅ ∧ 𝑥 = ∅) → ((𝐵𝑥𝜑) ↔ (𝐵 ⊆ ∅ → 𝜓)))
81imbi1d 329 . . . . . 6 (𝑥 = ∅ → ((𝐵𝑥𝜑) ↔ (𝐵 ⊆ ∅ → 𝜑)))
9 ss0 3921 . . . . . . . . 9 (𝐵 ⊆ ∅ → 𝐵 = ∅)
109con3i 148 . . . . . . . 8 𝐵 = ∅ → ¬ 𝐵 ⊆ ∅)
1110pm2.21d 116 . . . . . . 7 𝐵 = ∅ → (𝐵 ⊆ ∅ → (𝜑𝜓)))
1211pm5.74d 260 . . . . . 6 𝐵 = ∅ → ((𝐵 ⊆ ∅ → 𝜑) ↔ (𝐵 ⊆ ∅ → 𝜓)))
138, 12sylan9bbr 732 . . . . 5 ((¬ 𝐵 = ∅ ∧ 𝑥 = ∅) → ((𝐵𝑥𝜑) ↔ (𝐵 ⊆ ∅ → 𝜓)))
147, 13pm2.61ian 826 . . . 4 (𝑥 = ∅ → ((𝐵𝑥𝜑) ↔ (𝐵 ⊆ ∅ → 𝜓)))
1514imbi2d 328 . . 3 (𝑥 = ∅ → ((𝐵 ∈ On → (𝐵𝑥𝜑)) ↔ (𝐵 ∈ On → (𝐵 ⊆ ∅ → 𝜓))))
16 sseq2 3585 . . . . 5 (𝑥 = 𝑦 → (𝐵𝑥𝐵𝑦))
17 tfindsg.2 . . . . 5 (𝑥 = 𝑦 → (𝜑𝜒))
1816, 17imbi12d 332 . . . 4 (𝑥 = 𝑦 → ((𝐵𝑥𝜑) ↔ (𝐵𝑦𝜒)))
1918imbi2d 328 . . 3 (𝑥 = 𝑦 → ((𝐵 ∈ On → (𝐵𝑥𝜑)) ↔ (𝐵 ∈ On → (𝐵𝑦𝜒))))
20 sseq2 3585 . . . . 5 (𝑥 = suc 𝑦 → (𝐵𝑥𝐵 ⊆ suc 𝑦))
21 tfindsg.3 . . . . 5 (𝑥 = suc 𝑦 → (𝜑𝜃))
2220, 21imbi12d 332 . . . 4 (𝑥 = suc 𝑦 → ((𝐵𝑥𝜑) ↔ (𝐵 ⊆ suc 𝑦𝜃)))
2322imbi2d 328 . . 3 (𝑥 = suc 𝑦 → ((𝐵 ∈ On → (𝐵𝑥𝜑)) ↔ (𝐵 ∈ On → (𝐵 ⊆ suc 𝑦𝜃))))
24 sseq2 3585 . . . . 5 (𝑥 = 𝐴 → (𝐵𝑥𝐵𝐴))
25 tfindsg.4 . . . . 5 (𝑥 = 𝐴 → (𝜑𝜏))
2624, 25imbi12d 332 . . . 4 (𝑥 = 𝐴 → ((𝐵𝑥𝜑) ↔ (𝐵𝐴𝜏)))
2726imbi2d 328 . . 3 (𝑥 = 𝐴 → ((𝐵 ∈ On → (𝐵𝑥𝜑)) ↔ (𝐵 ∈ On → (𝐵𝐴𝜏))))
28 tfindsg.5 . . . 4 (𝐵 ∈ On → 𝜓)
2928a1d 25 . . 3 (𝐵 ∈ On → (𝐵 ⊆ ∅ → 𝜓))
30 vex 3171 . . . . . . . . . . . . . 14 𝑦 ∈ V
3130sucex 6876 . . . . . . . . . . . . 13 suc 𝑦 ∈ V
3231eqvinc 3295 . . . . . . . . . . . 12 (suc 𝑦 = 𝐵 ↔ ∃𝑥(𝑥 = suc 𝑦𝑥 = 𝐵))
3328, 4syl5ibr 234 . . . . . . . . . . . . . 14 (𝑥 = 𝐵 → (𝐵 ∈ On → 𝜑))
3421biimpd 217 . . . . . . . . . . . . . 14 (𝑥 = suc 𝑦 → (𝜑𝜃))
3533, 34sylan9r 687 . . . . . . . . . . . . 13 ((𝑥 = suc 𝑦𝑥 = 𝐵) → (𝐵 ∈ On → 𝜃))
3635exlimiv 1843 . . . . . . . . . . . 12 (∃𝑥(𝑥 = suc 𝑦𝑥 = 𝐵) → (𝐵 ∈ On → 𝜃))
3732, 36sylbi 205 . . . . . . . . . . 11 (suc 𝑦 = 𝐵 → (𝐵 ∈ On → 𝜃))
3837eqcoms 2613 . . . . . . . . . 10 (𝐵 = suc 𝑦 → (𝐵 ∈ On → 𝜃))
3938imim2i 16 . . . . . . . . 9 ((𝐵 ⊆ suc 𝑦𝐵 = suc 𝑦) → (𝐵 ⊆ suc 𝑦 → (𝐵 ∈ On → 𝜃)))
4039a1d 25 . . . . . . . 8 ((𝐵 ⊆ suc 𝑦𝐵 = suc 𝑦) → ((𝐵𝑦𝜒) → (𝐵 ⊆ suc 𝑦 → (𝐵 ∈ On → 𝜃))))
4140com4r 91 . . . . . . 7 (𝐵 ∈ On → ((𝐵 ⊆ suc 𝑦𝐵 = suc 𝑦) → ((𝐵𝑦𝜒) → (𝐵 ⊆ suc 𝑦𝜃))))
4241adantl 480 . . . . . 6 ((𝑦 ∈ On ∧ 𝐵 ∈ On) → ((𝐵 ⊆ suc 𝑦𝐵 = suc 𝑦) → ((𝐵𝑦𝜒) → (𝐵 ⊆ suc 𝑦𝜃))))
43 df-ne 2777 . . . . . . . . 9 (𝐵 ≠ suc 𝑦 ↔ ¬ 𝐵 = suc 𝑦)
4443anbi2i 725 . . . . . . . 8 ((𝐵 ⊆ suc 𝑦𝐵 ≠ suc 𝑦) ↔ (𝐵 ⊆ suc 𝑦 ∧ ¬ 𝐵 = suc 𝑦))
45 annim 439 . . . . . . . 8 ((𝐵 ⊆ suc 𝑦 ∧ ¬ 𝐵 = suc 𝑦) ↔ ¬ (𝐵 ⊆ suc 𝑦𝐵 = suc 𝑦))
4644, 45bitri 262 . . . . . . 7 ((𝐵 ⊆ suc 𝑦𝐵 ≠ suc 𝑦) ↔ ¬ (𝐵 ⊆ suc 𝑦𝐵 = suc 𝑦))
47 onsssuc 5712 . . . . . . . . . 10 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵𝑦𝐵 ∈ suc 𝑦))
48 suceloni 6878 . . . . . . . . . . 11 (𝑦 ∈ On → suc 𝑦 ∈ On)
49 onelpss 5663 . . . . . . . . . . 11 ((𝐵 ∈ On ∧ suc 𝑦 ∈ On) → (𝐵 ∈ suc 𝑦 ↔ (𝐵 ⊆ suc 𝑦𝐵 ≠ suc 𝑦)))
5048, 49sylan2 489 . . . . . . . . . 10 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 ∈ suc 𝑦 ↔ (𝐵 ⊆ suc 𝑦𝐵 ≠ suc 𝑦)))
5147, 50bitrd 266 . . . . . . . . 9 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵𝑦 ↔ (𝐵 ⊆ suc 𝑦𝐵 ≠ suc 𝑦)))
5251ancoms 467 . . . . . . . 8 ((𝑦 ∈ On ∧ 𝐵 ∈ On) → (𝐵𝑦 ↔ (𝐵 ⊆ suc 𝑦𝐵 ≠ suc 𝑦)))
53 tfindsg.6 . . . . . . . . . . . 12 (((𝑦 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐵𝑦) → (𝜒𝜃))
5453ex 448 . . . . . . . . . . 11 ((𝑦 ∈ On ∧ 𝐵 ∈ On) → (𝐵𝑦 → (𝜒𝜃)))
55 ax-1 6 . . . . . . . . . . 11 (𝜃 → (𝐵 ⊆ suc 𝑦𝜃))
5654, 55syl8 73 . . . . . . . . . 10 ((𝑦 ∈ On ∧ 𝐵 ∈ On) → (𝐵𝑦 → (𝜒 → (𝐵 ⊆ suc 𝑦𝜃))))
5756a2d 29 . . . . . . . . 9 ((𝑦 ∈ On ∧ 𝐵 ∈ On) → ((𝐵𝑦𝜒) → (𝐵𝑦 → (𝐵 ⊆ suc 𝑦𝜃))))
5857com23 83 . . . . . . . 8 ((𝑦 ∈ On ∧ 𝐵 ∈ On) → (𝐵𝑦 → ((𝐵𝑦𝜒) → (𝐵 ⊆ suc 𝑦𝜃))))
5952, 58sylbird 248 . . . . . . 7 ((𝑦 ∈ On ∧ 𝐵 ∈ On) → ((𝐵 ⊆ suc 𝑦𝐵 ≠ suc 𝑦) → ((𝐵𝑦𝜒) → (𝐵 ⊆ suc 𝑦𝜃))))
6046, 59syl5bir 231 . . . . . 6 ((𝑦 ∈ On ∧ 𝐵 ∈ On) → (¬ (𝐵 ⊆ suc 𝑦𝐵 = suc 𝑦) → ((𝐵𝑦𝜒) → (𝐵 ⊆ suc 𝑦𝜃))))
6142, 60pm2.61d 168 . . . . 5 ((𝑦 ∈ On ∧ 𝐵 ∈ On) → ((𝐵𝑦𝜒) → (𝐵 ⊆ suc 𝑦𝜃)))
6261ex 448 . . . 4 (𝑦 ∈ On → (𝐵 ∈ On → ((𝐵𝑦𝜒) → (𝐵 ⊆ suc 𝑦𝜃))))
6362a2d 29 . . 3 (𝑦 ∈ On → ((𝐵 ∈ On → (𝐵𝑦𝜒)) → (𝐵 ∈ On → (𝐵 ⊆ suc 𝑦𝜃))))
64 pm2.27 40 . . . . . . . . 9 (𝐵 ∈ On → ((𝐵 ∈ On → (𝐵𝑦𝜒)) → (𝐵𝑦𝜒)))
6564ralimdv 2941 . . . . . . . 8 (𝐵 ∈ On → (∀𝑦𝑥 (𝐵 ∈ On → (𝐵𝑦𝜒)) → ∀𝑦𝑥 (𝐵𝑦𝜒)))
6665ad2antlr 758 . . . . . . 7 (((Lim 𝑥𝐵 ∈ On) ∧ 𝐵𝑥) → (∀𝑦𝑥 (𝐵 ∈ On → (𝐵𝑦𝜒)) → ∀𝑦𝑥 (𝐵𝑦𝜒)))
67 tfindsg.7 . . . . . . 7 (((Lim 𝑥𝐵 ∈ On) ∧ 𝐵𝑥) → (∀𝑦𝑥 (𝐵𝑦𝜒) → 𝜑))
6866, 67syld 45 . . . . . 6 (((Lim 𝑥𝐵 ∈ On) ∧ 𝐵𝑥) → (∀𝑦𝑥 (𝐵 ∈ On → (𝐵𝑦𝜒)) → 𝜑))
6968exp31 627 . . . . 5 (Lim 𝑥 → (𝐵 ∈ On → (𝐵𝑥 → (∀𝑦𝑥 (𝐵 ∈ On → (𝐵𝑦𝜒)) → 𝜑))))
7069com3l 86 . . . 4 (𝐵 ∈ On → (𝐵𝑥 → (Lim 𝑥 → (∀𝑦𝑥 (𝐵 ∈ On → (𝐵𝑦𝜒)) → 𝜑))))
7170com4t 90 . . 3 (Lim 𝑥 → (∀𝑦𝑥 (𝐵 ∈ On → (𝐵𝑦𝜒)) → (𝐵 ∈ On → (𝐵𝑥𝜑))))
7215, 19, 23, 27, 29, 63, 71tfinds 6924 . 2 (𝐴 ∈ On → (𝐵 ∈ On → (𝐵𝐴𝜏)))
7372imp31 446 1 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐵𝐴) → 𝜏)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wa 382   = wceq 1474  wex 1694  wcel 1975  wne 2775  wral 2891  wss 3535  c0 3869  Oncon0 5622  Lim wlim 5623  suc csuc 5624
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-sep 4699  ax-nul 4708  ax-pr 4824  ax-un 6820
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 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ne 2777  df-ral 2896  df-rex 2897  df-rab 2900  df-v 3170  df-sbc 3398  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-pss 3551  df-nul 3870  df-if 4032  df-pw 4105  df-sn 4121  df-pr 4123  df-tp 4125  df-op 4127  df-uni 4363  df-br 4574  df-opab 4634  df-tr 4671  df-eprel 4935  df-po 4945  df-so 4946  df-fr 4983  df-we 4985  df-ord 5625  df-on 5626  df-lim 5627  df-suc 5628
This theorem is referenced by:  tfindsg2  6926  oaordi  7486  infensuc  7996  r1ordg  8497
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