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Theorem on3ind 8691
Description: Triple induction over ordinals. (Contributed by Scott Fenton, 4-Sep-2024.)
Hypotheses
Ref Expression
on3ind.1 (𝑎 = 𝑑 → (𝜑𝜓))
on3ind.2 (𝑏 = 𝑒 → (𝜓𝜒))
on3ind.3 (𝑐 = 𝑓 → (𝜒𝜃))
on3ind.4 (𝑎 = 𝑑 → (𝜏𝜃))
on3ind.5 (𝑏 = 𝑒 → (𝜂𝜏))
on3ind.6 (𝑏 = 𝑒 → (𝜁𝜃))
on3ind.7 (𝑐 = 𝑓 → (𝜎𝜏))
on3ind.8 (𝑎 = 𝑋 → (𝜑𝜌))
on3ind.9 (𝑏 = 𝑌 → (𝜌𝜇))
on3ind.10 (𝑐 = 𝑍 → (𝜇𝜆))
on3ind.i ((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) → (((∀𝑑𝑎𝑒𝑏𝑓𝑐 𝜃 ∧ ∀𝑑𝑎𝑒𝑏 𝜒 ∧ ∀𝑑𝑎𝑓𝑐 𝜁) ∧ (∀𝑑𝑎 𝜓 ∧ ∀𝑒𝑏𝑓𝑐 𝜏 ∧ ∀𝑒𝑏 𝜎) ∧ ∀𝑓𝑐 𝜂) → 𝜑))
Assertion
Ref Expression
on3ind ((𝑋 ∈ On ∧ 𝑌 ∈ On ∧ 𝑍 ∈ On) → 𝜆)
Distinct variable groups:   𝑋,𝑎,𝑏,𝑐,𝑑,𝑒,𝑓   𝑌,𝑎,𝑏,𝑐,𝑑,𝑒,𝑓   𝑍,𝑎,𝑏,𝑐,𝑑,𝑒,𝑓   𝜓,𝑎   𝜌,𝑎   𝜃,𝑎,𝑏,𝑐   𝜒,𝑏,𝑓   𝜇,𝑏   𝜆,𝑐   𝜑,𝑑   𝜏,𝑑   𝜂,𝑒   𝜓,𝑒   𝜁,𝑒   𝜎,𝑓
Allowed substitution hints:   𝜑(𝑒,𝑓,𝑎,𝑏,𝑐)   𝜓(𝑓,𝑏,𝑐,𝑑)   𝜒(𝑒,𝑎,𝑐,𝑑)   𝜃(𝑒,𝑓,𝑑)   𝜏(𝑒,𝑓,𝑎,𝑏,𝑐)   𝜂(𝑓,𝑎,𝑏,𝑐,𝑑)   𝜁(𝑓,𝑎,𝑏,𝑐,𝑑)   𝜎(𝑒,𝑎,𝑏,𝑐,𝑑)   𝜌(𝑒,𝑓,𝑏,𝑐,𝑑)   𝜇(𝑒,𝑓,𝑎,𝑐,𝑑)   𝜆(𝑒,𝑓,𝑎,𝑏,𝑑)

Proof of Theorem on3ind
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 on3ind.1 . 2 (𝑎 = 𝑑 → (𝜑𝜓))
8 on3ind.2 . 2 (𝑏 = 𝑒 → (𝜓𝜒))
9 on3ind.3 . 2 (𝑐 = 𝑓 → (𝜒𝜃))
10 on3ind.4 . 2 (𝑎 = 𝑑 → (𝜏𝜃))
11 on3ind.5 . 2 (𝑏 = 𝑒 → (𝜂𝜏))
12 on3ind.6 . 2 (𝑏 = 𝑒 → (𝜁𝜃))
13 on3ind.7 . 2 (𝑐 = 𝑓 → (𝜎𝜏))
14 on3ind.8 . 2 (𝑎 = 𝑋 → (𝜑𝜌))
15 on3ind.9 . 2 (𝑏 = 𝑌 → (𝜌𝜇))
16 on3ind.10 . 2 (𝑐 = 𝑍 → (𝜇𝜆))
17 predon 7788 . . . . . . 7 (𝑎 ∈ On → Pred( E , On, 𝑎) = 𝑎)
18173ad2ant1 1131 . . . . . 6 ((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) → Pred( E , On, 𝑎) = 𝑎)
19 predon 7788 . . . . . . . 8 (𝑏 ∈ On → Pred( E , On, 𝑏) = 𝑏)
20193ad2ant2 1132 . . . . . . 7 ((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) → Pred( E , On, 𝑏) = 𝑏)
21 predon 7788 . . . . . . . . 9 (𝑐 ∈ On → Pred( E , On, 𝑐) = 𝑐)
22213ad2ant3 1133 . . . . . . . 8 ((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) → Pred( E , On, 𝑐) = 𝑐)
2322raleqdv 3322 . . . . . . 7 ((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) → (∀𝑓 ∈ Pred ( E , On, 𝑐)𝜃 ↔ ∀𝑓𝑐 𝜃))
2420, 23raleqbidv 3339 . . . . . 6 ((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) → (∀𝑒 ∈ Pred ( E , On, 𝑏)∀𝑓 ∈ Pred ( E , On, 𝑐)𝜃 ↔ ∀𝑒𝑏𝑓𝑐 𝜃))
2518, 24raleqbidv 3339 . . . . 5 ((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) → (∀𝑑 ∈ Pred ( E , On, 𝑎)∀𝑒 ∈ Pred ( E , On, 𝑏)∀𝑓 ∈ Pred ( E , On, 𝑐)𝜃 ↔ ∀𝑑𝑎𝑒𝑏𝑓𝑐 𝜃))
2620raleqdv 3322 . . . . . 6 ((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) → (∀𝑒 ∈ Pred ( E , On, 𝑏)𝜒 ↔ ∀𝑒𝑏 𝜒))
2718, 26raleqbidv 3339 . . . . 5 ((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) → (∀𝑑 ∈ Pred ( E , On, 𝑎)∀𝑒 ∈ Pred ( E , On, 𝑏)𝜒 ↔ ∀𝑑𝑎𝑒𝑏 𝜒))
2822raleqdv 3322 . . . . . 6 ((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) → (∀𝑓 ∈ Pred ( E , On, 𝑐)𝜁 ↔ ∀𝑓𝑐 𝜁))
2918, 28raleqbidv 3339 . . . . 5 ((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) → (∀𝑑 ∈ Pred ( E , On, 𝑎)∀𝑓 ∈ Pred ( E , On, 𝑐)𝜁 ↔ ∀𝑑𝑎𝑓𝑐 𝜁))
3025, 27, 293anbi123d 1433 . . . 4 ((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) → ((∀𝑑 ∈ Pred ( E , On, 𝑎)∀𝑒 ∈ Pred ( E , On, 𝑏)∀𝑓 ∈ Pred ( E , On, 𝑐)𝜃 ∧ ∀𝑑 ∈ Pred ( E , On, 𝑎)∀𝑒 ∈ Pred ( E , On, 𝑏)𝜒 ∧ ∀𝑑 ∈ Pred ( E , On, 𝑎)∀𝑓 ∈ Pred ( E , On, 𝑐)𝜁) ↔ (∀𝑑𝑎𝑒𝑏𝑓𝑐 𝜃 ∧ ∀𝑑𝑎𝑒𝑏 𝜒 ∧ ∀𝑑𝑎𝑓𝑐 𝜁)))
3118raleqdv 3322 . . . . 5 ((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) → (∀𝑑 ∈ Pred ( E , On, 𝑎)𝜓 ↔ ∀𝑑𝑎 𝜓))
3222raleqdv 3322 . . . . . 6 ((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) → (∀𝑓 ∈ Pred ( E , On, 𝑐)𝜏 ↔ ∀𝑓𝑐 𝜏))
3320, 32raleqbidv 3339 . . . . 5 ((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) → (∀𝑒 ∈ Pred ( E , On, 𝑏)∀𝑓 ∈ Pred ( E , On, 𝑐)𝜏 ↔ ∀𝑒𝑏𝑓𝑐 𝜏))
3420raleqdv 3322 . . . . 5 ((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) → (∀𝑒 ∈ Pred ( E , On, 𝑏)𝜎 ↔ ∀𝑒𝑏 𝜎))
3531, 33, 343anbi123d 1433 . . . 4 ((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) → ((∀𝑑 ∈ Pred ( E , On, 𝑎)𝜓 ∧ ∀𝑒 ∈ Pred ( E , On, 𝑏)∀𝑓 ∈ Pred ( E , On, 𝑐)𝜏 ∧ ∀𝑒 ∈ Pred ( E , On, 𝑏)𝜎) ↔ (∀𝑑𝑎 𝜓 ∧ ∀𝑒𝑏𝑓𝑐 𝜏 ∧ ∀𝑒𝑏 𝜎)))
3622raleqdv 3322 . . . 4 ((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) → (∀𝑓 ∈ Pred ( E , On, 𝑐)𝜂 ↔ ∀𝑓𝑐 𝜂))
3730, 35, 363anbi123d 1433 . . 3 ((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) → (((∀𝑑 ∈ Pred ( E , On, 𝑎)∀𝑒 ∈ Pred ( E , On, 𝑏)∀𝑓 ∈ Pred ( E , On, 𝑐)𝜃 ∧ ∀𝑑 ∈ Pred ( E , On, 𝑎)∀𝑒 ∈ Pred ( E , On, 𝑏)𝜒 ∧ ∀𝑑 ∈ Pred ( E , On, 𝑎)∀𝑓 ∈ Pred ( E , On, 𝑐)𝜁) ∧ (∀𝑑 ∈ Pred ( E , On, 𝑎)𝜓 ∧ ∀𝑒 ∈ Pred ( E , On, 𝑏)∀𝑓 ∈ Pred ( E , On, 𝑐)𝜏 ∧ ∀𝑒 ∈ Pred ( E , On, 𝑏)𝜎) ∧ ∀𝑓 ∈ Pred ( E , On, 𝑐)𝜂) ↔ ((∀𝑑𝑎𝑒𝑏𝑓𝑐 𝜃 ∧ ∀𝑑𝑎𝑒𝑏 𝜒 ∧ ∀𝑑𝑎𝑓𝑐 𝜁) ∧ (∀𝑑𝑎 𝜓 ∧ ∀𝑒𝑏𝑓𝑐 𝜏 ∧ ∀𝑒𝑏 𝜎) ∧ ∀𝑓𝑐 𝜂)))
38 on3ind.i . . 3 ((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) → (((∀𝑑𝑎𝑒𝑏𝑓𝑐 𝜃 ∧ ∀𝑑𝑎𝑒𝑏 𝜒 ∧ ∀𝑑𝑎𝑓𝑐 𝜁) ∧ (∀𝑑𝑎 𝜓 ∧ ∀𝑒𝑏𝑓𝑐 𝜏 ∧ ∀𝑒𝑏 𝜎) ∧ ∀𝑓𝑐 𝜂) → 𝜑))
3937, 38sylbid 239 . 2 ((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) → (((∀𝑑 ∈ Pred ( E , On, 𝑎)∀𝑒 ∈ Pred ( E , On, 𝑏)∀𝑓 ∈ Pred ( E , On, 𝑐)𝜃 ∧ ∀𝑑 ∈ Pred ( E , On, 𝑎)∀𝑒 ∈ Pred ( E , On, 𝑏)𝜒 ∧ ∀𝑑 ∈ Pred ( E , On, 𝑎)∀𝑓 ∈ Pred ( E , On, 𝑐)𝜁) ∧ (∀𝑑 ∈ Pred ( E , On, 𝑎)𝜓 ∧ ∀𝑒 ∈ Pred ( E , On, 𝑏)∀𝑓 ∈ Pred ( E , On, 𝑐)𝜏 ∧ ∀𝑒 ∈ Pred ( E , On, 𝑏)𝜎) ∧ ∀𝑓 ∈ Pred ( E , On, 𝑐)𝜂) → 𝜑))
401, 5, 6, 1, 5, 6, 1, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 39xpord3ind 8161 1 ((𝑋 ∈ On ∧ 𝑌 ∈ On ∧ 𝑍 ∈ On) → 𝜆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  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-ot 4638  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:  naddass  8717
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