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Theorem tfindsg 7559
 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 3944 . . . . . . 7 (𝑥 = ∅ → (𝐵𝑥𝐵 ⊆ ∅))
21adantl 485 . . . . . 6 ((𝐵 = ∅ ∧ 𝑥 = ∅) → (𝐵𝑥𝐵 ⊆ ∅))
3 eqeq2 2813 . . . . . . . 8 (𝐵 = ∅ → (𝑥 = 𝐵𝑥 = ∅))
4 tfindsg.1 . . . . . . . 8 (𝑥 = 𝐵 → (𝜑𝜓))
53, 4syl6bir 257 . . . . . . 7 (𝐵 = ∅ → (𝑥 = ∅ → (𝜑𝜓)))
65imp 410 . . . . . 6 ((𝐵 = ∅ ∧ 𝑥 = ∅) → (𝜑𝜓))
72, 6imbi12d 348 . . . . 5 ((𝐵 = ∅ ∧ 𝑥 = ∅) → ((𝐵𝑥𝜑) ↔ (𝐵 ⊆ ∅ → 𝜓)))
81imbi1d 345 . . . . . 6 (𝑥 = ∅ → ((𝐵𝑥𝜑) ↔ (𝐵 ⊆ ∅ → 𝜑)))
9 ss0 4309 . . . . . . . . 9 (𝐵 ⊆ ∅ → 𝐵 = ∅)
109con3i 157 . . . . . . . 8 𝐵 = ∅ → ¬ 𝐵 ⊆ ∅)
1110pm2.21d 121 . . . . . . 7 𝐵 = ∅ → (𝐵 ⊆ ∅ → (𝜑𝜓)))
1211pm5.74d 276 . . . . . 6 𝐵 = ∅ → ((𝐵 ⊆ ∅ → 𝜑) ↔ (𝐵 ⊆ ∅ → 𝜓)))
138, 12sylan9bbr 514 . . . . 5 ((¬ 𝐵 = ∅ ∧ 𝑥 = ∅) → ((𝐵𝑥𝜑) ↔ (𝐵 ⊆ ∅ → 𝜓)))
147, 13pm2.61ian 811 . . . 4 (𝑥 = ∅ → ((𝐵𝑥𝜑) ↔ (𝐵 ⊆ ∅ → 𝜓)))
1514imbi2d 344 . . 3 (𝑥 = ∅ → ((𝐵 ∈ On → (𝐵𝑥𝜑)) ↔ (𝐵 ∈ On → (𝐵 ⊆ ∅ → 𝜓))))
16 sseq2 3944 . . . . 5 (𝑥 = 𝑦 → (𝐵𝑥𝐵𝑦))
17 tfindsg.2 . . . . 5 (𝑥 = 𝑦 → (𝜑𝜒))
1816, 17imbi12d 348 . . . 4 (𝑥 = 𝑦 → ((𝐵𝑥𝜑) ↔ (𝐵𝑦𝜒)))
1918imbi2d 344 . . 3 (𝑥 = 𝑦 → ((𝐵 ∈ On → (𝐵𝑥𝜑)) ↔ (𝐵 ∈ On → (𝐵𝑦𝜒))))
20 sseq2 3944 . . . . 5 (𝑥 = suc 𝑦 → (𝐵𝑥𝐵 ⊆ suc 𝑦))
21 tfindsg.3 . . . . 5 (𝑥 = suc 𝑦 → (𝜑𝜃))
2220, 21imbi12d 348 . . . 4 (𝑥 = suc 𝑦 → ((𝐵𝑥𝜑) ↔ (𝐵 ⊆ suc 𝑦𝜃)))
2322imbi2d 344 . . 3 (𝑥 = suc 𝑦 → ((𝐵 ∈ On → (𝐵𝑥𝜑)) ↔ (𝐵 ∈ On → (𝐵 ⊆ suc 𝑦𝜃))))
24 sseq2 3944 . . . . 5 (𝑥 = 𝐴 → (𝐵𝑥𝐵𝐴))
25 tfindsg.4 . . . . 5 (𝑥 = 𝐴 → (𝜑𝜏))
2624, 25imbi12d 348 . . . 4 (𝑥 = 𝐴 → ((𝐵𝑥𝜑) ↔ (𝐵𝐴𝜏)))
2726imbi2d 344 . . 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 3447 . . . . . . . . . . . . . 14 𝑦 ∈ V
3130sucex 7510 . . . . . . . . . . . . 13 suc 𝑦 ∈ V
3231eqvinc 3593 . . . . . . . . . . . 12 (suc 𝑦 = 𝐵 ↔ ∃𝑥(𝑥 = suc 𝑦𝑥 = 𝐵))
3328, 4syl5ibr 249 . . . . . . . . . . . . . 14 (𝑥 = 𝐵 → (𝐵 ∈ On → 𝜑))
3421biimpd 232 . . . . . . . . . . . . . 14 (𝑥 = suc 𝑦 → (𝜑𝜃))
3533, 34sylan9r 512 . . . . . . . . . . . . 13 ((𝑥 = suc 𝑦𝑥 = 𝐵) → (𝐵 ∈ On → 𝜃))
3635exlimiv 1931 . . . . . . . . . . . 12 (∃𝑥(𝑥 = suc 𝑦𝑥 = 𝐵) → (𝐵 ∈ On → 𝜃))
3732, 36sylbi 220 . . . . . . . . . . 11 (suc 𝑦 = 𝐵 → (𝐵 ∈ On → 𝜃))
3837eqcoms 2809 . . . . . . . . . 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 485 . . . . . 6 ((𝑦 ∈ On ∧ 𝐵 ∈ On) → ((𝐵 ⊆ suc 𝑦𝐵 = suc 𝑦) → ((𝐵𝑦𝜒) → (𝐵 ⊆ suc 𝑦𝜃))))
43 df-ne 2991 . . . . . . . . 9 (𝐵 ≠ suc 𝑦 ↔ ¬ 𝐵 = suc 𝑦)
4443anbi2i 625 . . . . . . . 8 ((𝐵 ⊆ suc 𝑦𝐵 ≠ suc 𝑦) ↔ (𝐵 ⊆ suc 𝑦 ∧ ¬ 𝐵 = suc 𝑦))
45 annim 407 . . . . . . . 8 ((𝐵 ⊆ suc 𝑦 ∧ ¬ 𝐵 = suc 𝑦) ↔ ¬ (𝐵 ⊆ suc 𝑦𝐵 = suc 𝑦))
4644, 45bitri 278 . . . . . . 7 ((𝐵 ⊆ suc 𝑦𝐵 ≠ suc 𝑦) ↔ ¬ (𝐵 ⊆ suc 𝑦𝐵 = suc 𝑦))
47 onsssuc 6250 . . . . . . . . . 10 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵𝑦𝐵 ∈ suc 𝑦))
48 suceloni 7512 . . . . . . . . . . 11 (𝑦 ∈ On → suc 𝑦 ∈ On)
49 onelpss 6203 . . . . . . . . . . 11 ((𝐵 ∈ On ∧ suc 𝑦 ∈ On) → (𝐵 ∈ suc 𝑦 ↔ (𝐵 ⊆ suc 𝑦𝐵 ≠ suc 𝑦)))
5048, 49sylan2 595 . . . . . . . . . 10 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 ∈ suc 𝑦 ↔ (𝐵 ⊆ suc 𝑦𝐵 ≠ suc 𝑦)))
5147, 50bitrd 282 . . . . . . . . 9 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵𝑦 ↔ (𝐵 ⊆ suc 𝑦𝐵 ≠ suc 𝑦)))
5251ancoms 462 . . . . . . . 8 ((𝑦 ∈ On ∧ 𝐵 ∈ On) → (𝐵𝑦 ↔ (𝐵 ⊆ suc 𝑦𝐵 ≠ suc 𝑦)))
53 tfindsg.6 . . . . . . . . . . . 12 (((𝑦 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐵𝑦) → (𝜒𝜃))
5453ex 416 . . . . . . . . . . 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 263 . . . . . . 7 ((𝑦 ∈ On ∧ 𝐵 ∈ On) → ((𝐵 ⊆ suc 𝑦𝐵 ≠ suc 𝑦) → ((𝐵𝑦𝜒) → (𝐵 ⊆ suc 𝑦𝜃))))
5946, 58syl5bir 246 . . . . . 6 ((𝑦 ∈ On ∧ 𝐵 ∈ On) → (¬ (𝐵 ⊆ suc 𝑦𝐵 = suc 𝑦) → ((𝐵𝑦𝜒) → (𝐵 ⊆ suc 𝑦𝜃))))
6042, 59pm2.61d 182 . . . . 5 ((𝑦 ∈ On ∧ 𝐵 ∈ On) → ((𝐵𝑦𝜒) → (𝐵 ⊆ suc 𝑦𝜃)))
6160ex 416 . . . 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 3148 . . . . . . . 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 423 . . . . 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 7558 . 2 (𝐴 ∈ On → (𝐵 ∈ On → (𝐵𝐴𝜏)))
7271imp31 421 1 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐵𝐴) → 𝜏)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538  ∃wex 1781   ∈ wcel 2112   ≠ wne 2990  ∀wral 3109   ⊆ wss 3884  ∅c0 4246  Oncon0 6163  Lim wlim 6164  suc csuc 6165 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298  ax-un 7445 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-tr 5140  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169 This theorem is referenced by:  tfindsg2  7560  oaordi  8159  infensuc  8683  r1ordg  9195
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