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Theorem tfindsg 7045
 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 3619 . . . . . . 7 (𝑥 = ∅ → (𝐵𝑥𝐵 ⊆ ∅))
21adantl 482 . . . . . 6 ((𝐵 = ∅ ∧ 𝑥 = ∅) → (𝐵𝑥𝐵 ⊆ ∅))
3 eqeq2 2631 . . . . . . . 8 (𝐵 = ∅ → (𝑥 = 𝐵𝑥 = ∅))
4 tfindsg.1 . . . . . . . 8 (𝑥 = 𝐵 → (𝜑𝜓))
53, 4syl6bir 244 . . . . . . 7 (𝐵 = ∅ → (𝑥 = ∅ → (𝜑𝜓)))
65imp 445 . . . . . 6 ((𝐵 = ∅ ∧ 𝑥 = ∅) → (𝜑𝜓))
72, 6imbi12d 334 . . . . 5 ((𝐵 = ∅ ∧ 𝑥 = ∅) → ((𝐵𝑥𝜑) ↔ (𝐵 ⊆ ∅ → 𝜓)))
81imbi1d 331 . . . . . 6 (𝑥 = ∅ → ((𝐵𝑥𝜑) ↔ (𝐵 ⊆ ∅ → 𝜑)))
9 ss0 3965 . . . . . . . . 9 (𝐵 ⊆ ∅ → 𝐵 = ∅)
109con3i 150 . . . . . . . 8 𝐵 = ∅ → ¬ 𝐵 ⊆ ∅)
1110pm2.21d 118 . . . . . . 7 𝐵 = ∅ → (𝐵 ⊆ ∅ → (𝜑𝜓)))
1211pm5.74d 262 . . . . . 6 𝐵 = ∅ → ((𝐵 ⊆ ∅ → 𝜑) ↔ (𝐵 ⊆ ∅ → 𝜓)))
138, 12sylan9bbr 736 . . . . 5 ((¬ 𝐵 = ∅ ∧ 𝑥 = ∅) → ((𝐵𝑥𝜑) ↔ (𝐵 ⊆ ∅ → 𝜓)))
147, 13pm2.61ian 830 . . . 4 (𝑥 = ∅ → ((𝐵𝑥𝜑) ↔ (𝐵 ⊆ ∅ → 𝜓)))
1514imbi2d 330 . . 3 (𝑥 = ∅ → ((𝐵 ∈ On → (𝐵𝑥𝜑)) ↔ (𝐵 ∈ On → (𝐵 ⊆ ∅ → 𝜓))))
16 sseq2 3619 . . . . 5 (𝑥 = 𝑦 → (𝐵𝑥𝐵𝑦))
17 tfindsg.2 . . . . 5 (𝑥 = 𝑦 → (𝜑𝜒))
1816, 17imbi12d 334 . . . 4 (𝑥 = 𝑦 → ((𝐵𝑥𝜑) ↔ (𝐵𝑦𝜒)))
1918imbi2d 330 . . 3 (𝑥 = 𝑦 → ((𝐵 ∈ On → (𝐵𝑥𝜑)) ↔ (𝐵 ∈ On → (𝐵𝑦𝜒))))
20 sseq2 3619 . . . . 5 (𝑥 = suc 𝑦 → (𝐵𝑥𝐵 ⊆ suc 𝑦))
21 tfindsg.3 . . . . 5 (𝑥 = suc 𝑦 → (𝜑𝜃))
2220, 21imbi12d 334 . . . 4 (𝑥 = suc 𝑦 → ((𝐵𝑥𝜑) ↔ (𝐵 ⊆ suc 𝑦𝜃)))
2322imbi2d 330 . . 3 (𝑥 = suc 𝑦 → ((𝐵 ∈ On → (𝐵𝑥𝜑)) ↔ (𝐵 ∈ On → (𝐵 ⊆ suc 𝑦𝜃))))
24 sseq2 3619 . . . . 5 (𝑥 = 𝐴 → (𝐵𝑥𝐵𝐴))
25 tfindsg.4 . . . . 5 (𝑥 = 𝐴 → (𝜑𝜏))
2624, 25imbi12d 334 . . . 4 (𝑥 = 𝐴 → ((𝐵𝑥𝜑) ↔ (𝐵𝐴𝜏)))
2726imbi2d 330 . . 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 3198 . . . . . . . . . . . . . 14 𝑦 ∈ V
3130sucex 6996 . . . . . . . . . . . . 13 suc 𝑦 ∈ V
3231eqvinc 3324 . . . . . . . . . . . 12 (suc 𝑦 = 𝐵 ↔ ∃𝑥(𝑥 = suc 𝑦𝑥 = 𝐵))
3328, 4syl5ibr 236 . . . . . . . . . . . . . 14 (𝑥 = 𝐵 → (𝐵 ∈ On → 𝜑))
3421biimpd 219 . . . . . . . . . . . . . 14 (𝑥 = suc 𝑦 → (𝜑𝜃))
3533, 34sylan9r 689 . . . . . . . . . . . . 13 ((𝑥 = suc 𝑦𝑥 = 𝐵) → (𝐵 ∈ On → 𝜃))
3635exlimiv 1856 . . . . . . . . . . . 12 (∃𝑥(𝑥 = suc 𝑦𝑥 = 𝐵) → (𝐵 ∈ On → 𝜃))
3732, 36sylbi 207 . . . . . . . . . . 11 (suc 𝑦 = 𝐵 → (𝐵 ∈ On → 𝜃))
3837eqcoms 2628 . . . . . . . . . 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 482 . . . . . 6 ((𝑦 ∈ On ∧ 𝐵 ∈ On) → ((𝐵 ⊆ suc 𝑦𝐵 = suc 𝑦) → ((𝐵𝑦𝜒) → (𝐵 ⊆ suc 𝑦𝜃))))
43 df-ne 2792 . . . . . . . . 9 (𝐵 ≠ suc 𝑦 ↔ ¬ 𝐵 = suc 𝑦)
4443anbi2i 729 . . . . . . . 8 ((𝐵 ⊆ suc 𝑦𝐵 ≠ suc 𝑦) ↔ (𝐵 ⊆ suc 𝑦 ∧ ¬ 𝐵 = suc 𝑦))
45 annim 441 . . . . . . . 8 ((𝐵 ⊆ suc 𝑦 ∧ ¬ 𝐵 = suc 𝑦) ↔ ¬ (𝐵 ⊆ suc 𝑦𝐵 = suc 𝑦))
4644, 45bitri 264 . . . . . . 7 ((𝐵 ⊆ suc 𝑦𝐵 ≠ suc 𝑦) ↔ ¬ (𝐵 ⊆ suc 𝑦𝐵 = suc 𝑦))
47 onsssuc 5801 . . . . . . . . . 10 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵𝑦𝐵 ∈ suc 𝑦))
48 suceloni 6998 . . . . . . . . . . 11 (𝑦 ∈ On → suc 𝑦 ∈ On)
49 onelpss 5752 . . . . . . . . . . 11 ((𝐵 ∈ On ∧ suc 𝑦 ∈ On) → (𝐵 ∈ suc 𝑦 ↔ (𝐵 ⊆ suc 𝑦𝐵 ≠ suc 𝑦)))
5048, 49sylan2 491 . . . . . . . . . 10 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 ∈ suc 𝑦 ↔ (𝐵 ⊆ suc 𝑦𝐵 ≠ suc 𝑦)))
5147, 50bitrd 268 . . . . . . . . 9 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵𝑦 ↔ (𝐵 ⊆ suc 𝑦𝐵 ≠ suc 𝑦)))
5251ancoms 469 . . . . . . . 8 ((𝑦 ∈ On ∧ 𝐵 ∈ On) → (𝐵𝑦 ↔ (𝐵 ⊆ suc 𝑦𝐵 ≠ suc 𝑦)))
53 tfindsg.6 . . . . . . . . . . . 12 (((𝑦 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐵𝑦) → (𝜒𝜃))
5453ex 450 . . . . . . . . . . 11 ((𝑦 ∈ On ∧ 𝐵 ∈ On) → (𝐵𝑦 → (𝜒𝜃)))
55 ax-1 6 . . . . . . . . . . 11 (𝜃 → (𝐵 ⊆ suc 𝑦𝜃))
5654, 55syl8 76 . . . . . . . . . 10 ((𝑦 ∈ On ∧ 𝐵 ∈ On) → (𝐵𝑦 → (𝜒 → (𝐵 ⊆ suc 𝑦𝜃))))
5756a2d 29 . . . . . . . . 9 ((𝑦 ∈ On ∧ 𝐵 ∈ On) → ((𝐵𝑦𝜒) → (𝐵𝑦 → (𝐵 ⊆ suc 𝑦𝜃))))
5857com23 86 . . . . . . . 8 ((𝑦 ∈ On ∧ 𝐵 ∈ On) → (𝐵𝑦 → ((𝐵𝑦𝜒) → (𝐵 ⊆ suc 𝑦𝜃))))
5952, 58sylbird 250 . . . . . . 7 ((𝑦 ∈ On ∧ 𝐵 ∈ On) → ((𝐵 ⊆ suc 𝑦𝐵 ≠ suc 𝑦) → ((𝐵𝑦𝜒) → (𝐵 ⊆ suc 𝑦𝜃))))
6046, 59syl5bir 233 . . . . . 6 ((𝑦 ∈ On ∧ 𝐵 ∈ On) → (¬ (𝐵 ⊆ suc 𝑦𝐵 = suc 𝑦) → ((𝐵𝑦𝜒) → (𝐵 ⊆ suc 𝑦𝜃))))
6142, 60pm2.61d 170 . . . . 5 ((𝑦 ∈ On ∧ 𝐵 ∈ On) → ((𝐵𝑦𝜒) → (𝐵 ⊆ suc 𝑦𝜃)))
6261ex 450 . . . 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 42 . . . . . . . . 9 (𝐵 ∈ On → ((𝐵 ∈ On → (𝐵𝑦𝜒)) → (𝐵𝑦𝜒)))
6564ralimdv 2960 . . . . . . . 8 (𝐵 ∈ On → (∀𝑦𝑥 (𝐵 ∈ On → (𝐵𝑦𝜒)) → ∀𝑦𝑥 (𝐵𝑦𝜒)))
6665ad2antlr 762 . . . . . . 7 (((Lim 𝑥𝐵 ∈ On) ∧ 𝐵𝑥) → (∀𝑦𝑥 (𝐵 ∈ On → (𝐵𝑦𝜒)) → ∀𝑦𝑥 (𝐵𝑦𝜒)))
67 tfindsg.7 . . . . . . 7 (((Lim 𝑥𝐵 ∈ On) ∧ 𝐵𝑥) → (∀𝑦𝑥 (𝐵𝑦𝜒) → 𝜑))
6866, 67syld 47 . . . . . 6 (((Lim 𝑥𝐵 ∈ On) ∧ 𝐵𝑥) → (∀𝑦𝑥 (𝐵 ∈ On → (𝐵𝑦𝜒)) → 𝜑))
6968exp31 629 . . . . 5 (Lim 𝑥 → (𝐵 ∈ On → (𝐵𝑥 → (∀𝑦𝑥 (𝐵 ∈ On → (𝐵𝑦𝜒)) → 𝜑))))
7069com3l 89 . . . 4 (𝐵 ∈ On → (𝐵𝑥 → (Lim 𝑥 → (∀𝑦𝑥 (𝐵 ∈ On → (𝐵𝑦𝜒)) → 𝜑))))
7170com4t 93 . . 3 (Lim 𝑥 → (∀𝑦𝑥 (𝐵 ∈ On → (𝐵𝑦𝜒)) → (𝐵 ∈ On → (𝐵𝑥𝜑))))
7215, 19, 23, 27, 29, 63, 71tfinds 7044 . 2 (𝐴 ∈ On → (𝐵 ∈ On → (𝐵𝐴𝜏)))
7372imp31 448 1 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐵𝐴) → 𝜏)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1481  ∃wex 1702   ∈ wcel 1988   ≠ wne 2791  ∀wral 2909   ⊆ wss 3567  ∅c0 3907  Oncon0 5711  Lim wlim 5712  suc csuc 5713 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897  ax-un 6934 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-tr 4744  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717 This theorem is referenced by:  tfindsg2  7046  oaordi  7611  infensuc  8123  r1ordg  8626
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