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Theorem on2ind 8690
Description: Double induction over ordinal numbers. (Contributed by Scott Fenton, 26-Aug-2024.)
Hypotheses
Ref Expression
on2ind.1 (𝑎 = 𝑐 → (𝜑𝜓))
on2ind.2 (𝑏 = 𝑑 → (𝜓𝜒))
on2ind.3 (𝑎 = 𝑐 → (𝜃𝜒))
on2ind.4 (𝑎 = 𝑋 → (𝜑𝜏))
on2ind.5 (𝑏 = 𝑌 → (𝜏𝜂))
on2ind.i ((𝑎 ∈ On ∧ 𝑏 ∈ On) → ((∀𝑐𝑎𝑑𝑏 𝜒 ∧ ∀𝑐𝑎 𝜓 ∧ ∀𝑑𝑏 𝜃) → 𝜑))
Assertion
Ref Expression
on2ind ((𝑋 ∈ On ∧ 𝑌 ∈ On) → 𝜂)
Distinct variable groups:   𝑎,𝑏,𝑐,𝑑   𝜓,𝑎   𝜏,𝑎   𝑏,𝑐   𝜒,𝑏   𝑏,𝑑   𝜂,𝑏   𝑐,𝑑   𝜑,𝑐   𝜃,𝑐   𝜓,𝑑   𝑋,𝑎,𝑏   𝑌,𝑏
Allowed substitution hints:   𝜑(𝑎,𝑏,𝑑)   𝜓(𝑏,𝑐)   𝜒(𝑎,𝑐,𝑑)   𝜃(𝑎,𝑏,𝑑)   𝜏(𝑏,𝑐,𝑑)   𝜂(𝑎,𝑐,𝑑)   𝑋(𝑐,𝑑)   𝑌(𝑎,𝑐,𝑑)

Proof of Theorem on2ind
StepHypRef Expression
1 onfr 6408 . 2 E Fr On
2 epweon 7777 . . 3 E We On
3 weso 5669 . . 3 ( E We On → E Or On)
4 sopo 5609 . . 3 ( E Or On → E Po On)
52, 3, 4mp2b 10 . 2 E Po On
6 epse 5661 . 2 E Se On
7 on2ind.1 . 2 (𝑎 = 𝑐 → (𝜑𝜓))
8 on2ind.2 . 2 (𝑏 = 𝑑 → (𝜓𝜒))
9 on2ind.3 . 2 (𝑎 = 𝑐 → (𝜃𝜒))
10 on2ind.4 . 2 (𝑎 = 𝑋 → (𝜑𝜏))
11 on2ind.5 . 2 (𝑏 = 𝑌 → (𝜏𝜂))
12 predon 7788 . . . . . 6 (𝑎 ∈ On → Pred( E , On, 𝑎) = 𝑎)
1312adantr 480 . . . . 5 ((𝑎 ∈ On ∧ 𝑏 ∈ On) → Pred( E , On, 𝑎) = 𝑎)
14 predon 7788 . . . . . . 7 (𝑏 ∈ On → Pred( E , On, 𝑏) = 𝑏)
1514adantl 481 . . . . . 6 ((𝑎 ∈ On ∧ 𝑏 ∈ On) → Pred( E , On, 𝑏) = 𝑏)
1615raleqdv 3322 . . . . 5 ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (∀𝑑 ∈ Pred ( E , On, 𝑏)𝜒 ↔ ∀𝑑𝑏 𝜒))
1713, 16raleqbidv 3339 . . . 4 ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (∀𝑐 ∈ Pred ( E , On, 𝑎)∀𝑑 ∈ Pred ( E , On, 𝑏)𝜒 ↔ ∀𝑐𝑎𝑑𝑏 𝜒))
1813raleqdv 3322 . . . 4 ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (∀𝑐 ∈ Pred ( E , On, 𝑎)𝜓 ↔ ∀𝑐𝑎 𝜓))
1915raleqdv 3322 . . . 4 ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (∀𝑑 ∈ Pred ( E , On, 𝑏)𝜃 ↔ ∀𝑑𝑏 𝜃))
2017, 18, 193anbi123d 1433 . . 3 ((𝑎 ∈ On ∧ 𝑏 ∈ On) → ((∀𝑐 ∈ Pred ( E , On, 𝑎)∀𝑑 ∈ Pred ( E , On, 𝑏)𝜒 ∧ ∀𝑐 ∈ Pred ( E , On, 𝑎)𝜓 ∧ ∀𝑑 ∈ Pred ( E , On, 𝑏)𝜃) ↔ (∀𝑐𝑎𝑑𝑏 𝜒 ∧ ∀𝑐𝑎 𝜓 ∧ ∀𝑑𝑏 𝜃)))
21 on2ind.i . . 3 ((𝑎 ∈ On ∧ 𝑏 ∈ On) → ((∀𝑐𝑎𝑑𝑏 𝜒 ∧ ∀𝑐𝑎 𝜓 ∧ ∀𝑑𝑏 𝜃) → 𝜑))
2220, 21sylbid 239 . 2 ((𝑎 ∈ On ∧ 𝑏 ∈ On) → ((∀𝑐 ∈ Pred ( E , On, 𝑎)∀𝑑 ∈ Pred ( E , On, 𝑏)𝜒 ∧ ∀𝑐 ∈ Pred ( E , On, 𝑎)𝜓 ∧ ∀𝑑 ∈ Pred ( E , On, 𝑏)𝜃) → 𝜑))
231, 5, 6, 1, 5, 6, 7, 8, 9, 10, 11, 22xpord2ind 8153 1 ((𝑋 ∈ On ∧ 𝑌 ∈ On) → 𝜂)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  w3a 1085   = wceq 1534  wcel 2099  wral 3058   E cep 5581   Po wpo 5588   Or wor 5589   We wwe 5632  Predcpred 6304  Oncon0 6369
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-se 5634  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6305  df-ord 6372  df-on 6373  df-iota 6500  df-fun 6550  df-fv 6556  df-1st 7993  df-2nd 7994
This theorem is referenced by:  naddcllem  8697  naddcom  8703  naddsuc2  42822  naddgeoa  42824
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