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Theorem tfindsg 7850
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 4009 . . . . . . 7 (𝑥 = ∅ → (𝐵𝑥𝐵 ⊆ ∅))
21adantl 483 . . . . . 6 ((𝐵 = ∅ ∧ 𝑥 = ∅) → (𝐵𝑥𝐵 ⊆ ∅))
3 eqeq2 2745 . . . . . . . 8 (𝐵 = ∅ → (𝑥 = 𝐵𝑥 = ∅))
4 tfindsg.1 . . . . . . . 8 (𝑥 = 𝐵 → (𝜑𝜓))
53, 4syl6bir 254 . . . . . . 7 (𝐵 = ∅ → (𝑥 = ∅ → (𝜑𝜓)))
65imp 408 . . . . . 6 ((𝐵 = ∅ ∧ 𝑥 = ∅) → (𝜑𝜓))
72, 6imbi12d 345 . . . . 5 ((𝐵 = ∅ ∧ 𝑥 = ∅) → ((𝐵𝑥𝜑) ↔ (𝐵 ⊆ ∅ → 𝜓)))
81imbi1d 342 . . . . . 6 (𝑥 = ∅ → ((𝐵𝑥𝜑) ↔ (𝐵 ⊆ ∅ → 𝜑)))
9 ss0 4399 . . . . . . . . 9 (𝐵 ⊆ ∅ → 𝐵 = ∅)
109con3i 154 . . . . . . . 8 𝐵 = ∅ → ¬ 𝐵 ⊆ ∅)
1110pm2.21d 121 . . . . . . 7 𝐵 = ∅ → (𝐵 ⊆ ∅ → (𝜑𝜓)))
1211pm5.74d 273 . . . . . 6 𝐵 = ∅ → ((𝐵 ⊆ ∅ → 𝜑) ↔ (𝐵 ⊆ ∅ → 𝜓)))
138, 12sylan9bbr 512 . . . . 5 ((¬ 𝐵 = ∅ ∧ 𝑥 = ∅) → ((𝐵𝑥𝜑) ↔ (𝐵 ⊆ ∅ → 𝜓)))
147, 13pm2.61ian 811 . . . 4 (𝑥 = ∅ → ((𝐵𝑥𝜑) ↔ (𝐵 ⊆ ∅ → 𝜓)))
1514imbi2d 341 . . 3 (𝑥 = ∅ → ((𝐵 ∈ On → (𝐵𝑥𝜑)) ↔ (𝐵 ∈ On → (𝐵 ⊆ ∅ → 𝜓))))
16 sseq2 4009 . . . . 5 (𝑥 = 𝑦 → (𝐵𝑥𝐵𝑦))
17 tfindsg.2 . . . . 5 (𝑥 = 𝑦 → (𝜑𝜒))
1816, 17imbi12d 345 . . . 4 (𝑥 = 𝑦 → ((𝐵𝑥𝜑) ↔ (𝐵𝑦𝜒)))
1918imbi2d 341 . . 3 (𝑥 = 𝑦 → ((𝐵 ∈ On → (𝐵𝑥𝜑)) ↔ (𝐵 ∈ On → (𝐵𝑦𝜒))))
20 sseq2 4009 . . . . 5 (𝑥 = suc 𝑦 → (𝐵𝑥𝐵 ⊆ suc 𝑦))
21 tfindsg.3 . . . . 5 (𝑥 = suc 𝑦 → (𝜑𝜃))
2220, 21imbi12d 345 . . . 4 (𝑥 = suc 𝑦 → ((𝐵𝑥𝜑) ↔ (𝐵 ⊆ suc 𝑦𝜃)))
2322imbi2d 341 . . 3 (𝑥 = suc 𝑦 → ((𝐵 ∈ On → (𝐵𝑥𝜑)) ↔ (𝐵 ∈ On → (𝐵 ⊆ suc 𝑦𝜃))))
24 sseq2 4009 . . . . 5 (𝑥 = 𝐴 → (𝐵𝑥𝐵𝐴))
25 tfindsg.4 . . . . 5 (𝑥 = 𝐴 → (𝜑𝜏))
2624, 25imbi12d 345 . . . 4 (𝑥 = 𝐴 → ((𝐵𝑥𝜑) ↔ (𝐵𝐴𝜏)))
2726imbi2d 341 . . 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 3479 . . . . . . . . . . . . . 14 𝑦 ∈ V
3130sucex 7794 . . . . . . . . . . . . 13 suc 𝑦 ∈ V
3231eqvinc 3638 . . . . . . . . . . . 12 (suc 𝑦 = 𝐵 ↔ ∃𝑥(𝑥 = suc 𝑦𝑥 = 𝐵))
3328, 4imbitrrid 245 . . . . . . . . . . . . . 14 (𝑥 = 𝐵 → (𝐵 ∈ On → 𝜑))
3421biimpd 228 . . . . . . . . . . . . . 14 (𝑥 = suc 𝑦 → (𝜑𝜃))
3533, 34sylan9r 510 . . . . . . . . . . . . 13 ((𝑥 = suc 𝑦𝑥 = 𝐵) → (𝐵 ∈ On → 𝜃))
3635exlimiv 1934 . . . . . . . . . . . 12 (∃𝑥(𝑥 = suc 𝑦𝑥 = 𝐵) → (𝐵 ∈ On → 𝜃))
3732, 36sylbi 216 . . . . . . . . . . 11 (suc 𝑦 = 𝐵 → (𝐵 ∈ On → 𝜃))
3837eqcoms 2741 . . . . . . . . . 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 94 . . . . . . 7 (𝐵 ∈ On → ((𝐵 ⊆ suc 𝑦𝐵 = suc 𝑦) → ((𝐵𝑦𝜒) → (𝐵 ⊆ suc 𝑦𝜃))))
4241adantl 483 . . . . . 6 ((𝑦 ∈ On ∧ 𝐵 ∈ On) → ((𝐵 ⊆ suc 𝑦𝐵 = suc 𝑦) → ((𝐵𝑦𝜒) → (𝐵 ⊆ suc 𝑦𝜃))))
43 df-ne 2942 . . . . . . . . 9 (𝐵 ≠ suc 𝑦 ↔ ¬ 𝐵 = suc 𝑦)
4443anbi2i 624 . . . . . . . 8 ((𝐵 ⊆ suc 𝑦𝐵 ≠ suc 𝑦) ↔ (𝐵 ⊆ suc 𝑦 ∧ ¬ 𝐵 = suc 𝑦))
45 annim 405 . . . . . . . 8 ((𝐵 ⊆ suc 𝑦 ∧ ¬ 𝐵 = suc 𝑦) ↔ ¬ (𝐵 ⊆ suc 𝑦𝐵 = suc 𝑦))
4644, 45bitri 275 . . . . . . 7 ((𝐵 ⊆ suc 𝑦𝐵 ≠ suc 𝑦) ↔ ¬ (𝐵 ⊆ suc 𝑦𝐵 = suc 𝑦))
47 onsssuc 6455 . . . . . . . . . 10 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵𝑦𝐵 ∈ suc 𝑦))
48 onsuc 7799 . . . . . . . . . . 11 (𝑦 ∈ On → suc 𝑦 ∈ On)
49 onelpss 6405 . . . . . . . . . . 11 ((𝐵 ∈ On ∧ suc 𝑦 ∈ On) → (𝐵 ∈ suc 𝑦 ↔ (𝐵 ⊆ suc 𝑦𝐵 ≠ suc 𝑦)))
5048, 49sylan2 594 . . . . . . . . . 10 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 ∈ suc 𝑦 ↔ (𝐵 ⊆ suc 𝑦𝐵 ≠ suc 𝑦)))
5147, 50bitrd 279 . . . . . . . . 9 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵𝑦 ↔ (𝐵 ⊆ suc 𝑦𝐵 ≠ suc 𝑦)))
5251ancoms 460 . . . . . . . 8 ((𝑦 ∈ On ∧ 𝐵 ∈ On) → (𝐵𝑦 ↔ (𝐵 ⊆ suc 𝑦𝐵 ≠ suc 𝑦)))
53 tfindsg.6 . . . . . . . . . . . 12 (((𝑦 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐵𝑦) → (𝜒𝜃))
5453ex 414 . . . . . . . . . . 11 ((𝑦 ∈ On ∧ 𝐵 ∈ On) → (𝐵𝑦 → (𝜒𝜃)))
5554a1ddd 80 . . . . . . . . . 10 ((𝑦 ∈ On ∧ 𝐵 ∈ On) → (𝐵𝑦 → (𝜒 → (𝐵 ⊆ suc 𝑦𝜃))))
5655a2d 29 . . . . . . . . 9 ((𝑦 ∈ On ∧ 𝐵 ∈ On) → ((𝐵𝑦𝜒) → (𝐵𝑦 → (𝐵 ⊆ suc 𝑦𝜃))))
5756com23 86 . . . . . . . 8 ((𝑦 ∈ On ∧ 𝐵 ∈ On) → (𝐵𝑦 → ((𝐵𝑦𝜒) → (𝐵 ⊆ suc 𝑦𝜃))))
5852, 57sylbird 260 . . . . . . 7 ((𝑦 ∈ On ∧ 𝐵 ∈ On) → ((𝐵 ⊆ suc 𝑦𝐵 ≠ suc 𝑦) → ((𝐵𝑦𝜒) → (𝐵 ⊆ suc 𝑦𝜃))))
5946, 58biimtrrid 242 . . . . . 6 ((𝑦 ∈ On ∧ 𝐵 ∈ On) → (¬ (𝐵 ⊆ suc 𝑦𝐵 = suc 𝑦) → ((𝐵𝑦𝜒) → (𝐵 ⊆ suc 𝑦𝜃))))
6042, 59pm2.61d 179 . . . . 5 ((𝑦 ∈ On ∧ 𝐵 ∈ On) → ((𝐵𝑦𝜒) → (𝐵 ⊆ suc 𝑦𝜃)))
6160ex 414 . . . 4 (𝑦 ∈ On → (𝐵 ∈ On → ((𝐵𝑦𝜒) → (𝐵 ⊆ suc 𝑦𝜃))))
6261a2d 29 . . 3 (𝑦 ∈ On → ((𝐵 ∈ On → (𝐵𝑦𝜒)) → (𝐵 ∈ On → (𝐵 ⊆ suc 𝑦𝜃))))
63 pm2.27 42 . . . . . . . . 9 (𝐵 ∈ On → ((𝐵 ∈ On → (𝐵𝑦𝜒)) → (𝐵𝑦𝜒)))
6463ralimdv 3170 . . . . . . . 8 (𝐵 ∈ On → (∀𝑦𝑥 (𝐵 ∈ On → (𝐵𝑦𝜒)) → ∀𝑦𝑥 (𝐵𝑦𝜒)))
6564ad2antlr 726 . . . . . . 7 (((Lim 𝑥𝐵 ∈ On) ∧ 𝐵𝑥) → (∀𝑦𝑥 (𝐵 ∈ On → (𝐵𝑦𝜒)) → ∀𝑦𝑥 (𝐵𝑦𝜒)))
66 tfindsg.7 . . . . . . 7 (((Lim 𝑥𝐵 ∈ On) ∧ 𝐵𝑥) → (∀𝑦𝑥 (𝐵𝑦𝜒) → 𝜑))
6765, 66syld 47 . . . . . 6 (((Lim 𝑥𝐵 ∈ On) ∧ 𝐵𝑥) → (∀𝑦𝑥 (𝐵 ∈ On → (𝐵𝑦𝜒)) → 𝜑))
6867exp31 421 . . . . 5 (Lim 𝑥 → (𝐵 ∈ On → (𝐵𝑥 → (∀𝑦𝑥 (𝐵 ∈ On → (𝐵𝑦𝜒)) → 𝜑))))
6968com3l 89 . . . 4 (𝐵 ∈ On → (𝐵𝑥 → (Lim 𝑥 → (∀𝑦𝑥 (𝐵 ∈ On → (𝐵𝑦𝜒)) → 𝜑))))
7069com4t 93 . . 3 (Lim 𝑥 → (∀𝑦𝑥 (𝐵 ∈ On → (𝐵𝑦𝜒)) → (𝐵 ∈ On → (𝐵𝑥𝜑))))
7115, 19, 23, 27, 29, 62, 70tfinds 7849 . 2 (𝐴 ∈ On → (𝐵 ∈ On → (𝐵𝐴𝜏)))
7271imp31 419 1 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐵𝐴) → 𝜏)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 397   = wceq 1542  wex 1782  wcel 2107  wne 2941  wral 3062  wss 3949  c0 4323  Oncon0 6365  Lim wlim 6366  suc csuc 6367
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-tr 5267  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371
This theorem is referenced by:  tfindsg2  7851  oaordi  8546  infensuc  9155  r1ordg  9773
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