Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  on3ind Structured version   Visualization version   GIF version

Theorem on3ind 33462
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:   𝑎,𝑏,𝑐,𝑑,𝑒,𝑓   𝜓,𝑎   𝜌,𝑎   𝜃,𝑎   𝑏,𝑐   𝜒,𝑏   𝑏,𝑑,𝑒,𝑓   𝜇,𝑏   𝜃,𝑏   𝑐,𝑑,𝑒,𝑓   𝜆,𝑐   𝜃,𝑐   𝜒,𝑓   𝑒,𝑑,𝑓   𝜑,𝑑   𝜏,𝑑   𝜂,𝑒   𝑒,𝑓   𝜓,𝑒   𝜁,𝑒   𝜎,𝑓   𝑋,𝑎,𝑏,𝑐   𝑌,𝑏,𝑐   𝑍,𝑐
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)   𝑋(𝑒,𝑓,𝑑)   𝑌(𝑒,𝑓,𝑎,𝑑)   𝑍(𝑒,𝑓,𝑎,𝑏,𝑑)

Proof of Theorem on3ind
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2738 . 2 {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((On × On) × On) ∧ 𝑦 ∈ ((On × On) × On) ∧ ((((1st ‘(1st𝑥)) E (1st ‘(1st𝑦)) ∨ (1st ‘(1st𝑥)) = (1st ‘(1st𝑦))) ∧ ((2nd ‘(1st𝑥)) E (2nd ‘(1st𝑦)) ∨ (2nd ‘(1st𝑥)) = (2nd ‘(1st𝑦))) ∧ ((2nd𝑥) E (2nd𝑦) ∨ (2nd𝑥) = (2nd𝑦))) ∧ 𝑥𝑦))} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((On × On) × On) ∧ 𝑦 ∈ ((On × On) × On) ∧ ((((1st ‘(1st𝑥)) E (1st ‘(1st𝑦)) ∨ (1st ‘(1st𝑥)) = (1st ‘(1st𝑦))) ∧ ((2nd ‘(1st𝑥)) E (2nd ‘(1st𝑦)) ∨ (2nd ‘(1st𝑥)) = (2nd ‘(1st𝑦))) ∧ ((2nd𝑥) E (2nd𝑦) ∨ (2nd𝑥) = (2nd𝑦))) ∧ 𝑥𝑦))}
2 onfr 6205 . 2 E Fr On
3 epweon 7510 . . 3 E We On
4 weso 5510 . . 3 ( E We On → E Or On)
5 sopo 5456 . . 3 ( E Or On → E Po On)
63, 4, 5mp2b 10 . 2 E Po On
7 epse 5502 . 2 E Se On
8 on3ind.1 . 2 (𝑎 = 𝑑 → (𝜑𝜓))
9 on3ind.2 . 2 (𝑏 = 𝑒 → (𝜓𝜒))
10 on3ind.3 . 2 (𝑐 = 𝑓 → (𝜒𝜃))
11 on3ind.4 . 2 (𝑎 = 𝑑 → (𝜏𝜃))
12 on3ind.5 . 2 (𝑏 = 𝑒 → (𝜂𝜏))
13 on3ind.6 . 2 (𝑏 = 𝑒 → (𝜁𝜃))
14 on3ind.7 . 2 (𝑐 = 𝑓 → (𝜎𝜏))
15 on3ind.8 . 2 (𝑎 = 𝑋 → (𝜑𝜌))
16 on3ind.9 . 2 (𝑏 = 𝑌 → (𝜌𝜇))
17 on3ind.10 . 2 (𝑐 = 𝑍 → (𝜇𝜆))
18 predon 7519 . . . . . . 7 (𝑎 ∈ On → Pred( E , On, 𝑎) = 𝑎)
19183ad2ant1 1134 . . . . . 6 ((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) → Pred( E , On, 𝑎) = 𝑎)
20 predon 7519 . . . . . . . 8 (𝑏 ∈ On → Pred( E , On, 𝑏) = 𝑏)
21203ad2ant2 1135 . . . . . . 7 ((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) → Pred( E , On, 𝑏) = 𝑏)
22 predon 7519 . . . . . . . . 9 (𝑐 ∈ On → Pred( E , On, 𝑐) = 𝑐)
23223ad2ant3 1136 . . . . . . . 8 ((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) → Pred( E , On, 𝑐) = 𝑐)
2423raleqdv 3315 . . . . . . 7 ((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) → (∀𝑓 ∈ Pred ( E , On, 𝑐)𝜃 ↔ ∀𝑓𝑐 𝜃))
2521, 24raleqbidv 3303 . . . . . 6 ((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) → (∀𝑒 ∈ Pred ( E , On, 𝑏)∀𝑓 ∈ Pred ( E , On, 𝑐)𝜃 ↔ ∀𝑒𝑏𝑓𝑐 𝜃))
2619, 25raleqbidv 3303 . . . . 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))
2721raleqdv 3315 . . . . . 6 ((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) → (∀𝑒 ∈ Pred ( E , On, 𝑏)𝜒 ↔ ∀𝑒𝑏 𝜒))
2819, 27raleqbidv 3303 . . . . 5 ((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) → (∀𝑑 ∈ Pred ( E , On, 𝑎)∀𝑒 ∈ Pred ( E , On, 𝑏)𝜒 ↔ ∀𝑑𝑎𝑒𝑏 𝜒))
2923raleqdv 3315 . . . . . 6 ((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) → (∀𝑓 ∈ Pred ( E , On, 𝑐)𝜁 ↔ ∀𝑓𝑐 𝜁))
3019, 29raleqbidv 3303 . . . . 5 ((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) → (∀𝑑 ∈ Pred ( E , On, 𝑎)∀𝑓 ∈ Pred ( E , On, 𝑐)𝜁 ↔ ∀𝑑𝑎𝑓𝑐 𝜁))
3126, 28, 303anbi123d 1437 . . . 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)))
3219raleqdv 3315 . . . . 5 ((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) → (∀𝑑 ∈ Pred ( E , On, 𝑎)𝜓 ↔ ∀𝑑𝑎 𝜓))
3323raleqdv 3315 . . . . . 6 ((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) → (∀𝑓 ∈ Pred ( E , On, 𝑐)𝜏 ↔ ∀𝑓𝑐 𝜏))
3421, 33raleqbidv 3303 . . . . 5 ((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) → (∀𝑒 ∈ Pred ( E , On, 𝑏)∀𝑓 ∈ Pred ( E , On, 𝑐)𝜏 ↔ ∀𝑒𝑏𝑓𝑐 𝜏))
3521raleqdv 3315 . . . . 5 ((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) → (∀𝑒 ∈ Pred ( E , On, 𝑏)𝜎 ↔ ∀𝑒𝑏 𝜎))
3632, 34, 353anbi123d 1437 . . . 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)))
3723raleqdv 3315 . . . 4 ((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) → (∀𝑓 ∈ Pred ( E , On, 𝑐)𝜂 ↔ ∀𝑓𝑐 𝜂))
3831, 36, 373anbi123d 1437 . . 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)))
39 on3ind.i . . 3 ((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) → (((∀𝑑𝑎𝑒𝑏𝑓𝑐 𝜃 ∧ ∀𝑑𝑎𝑒𝑏 𝜒 ∧ ∀𝑑𝑎𝑓𝑐 𝜁) ∧ (∀𝑑𝑎 𝜓 ∧ ∀𝑒𝑏𝑓𝑐 𝜏 ∧ ∀𝑒𝑏 𝜎) ∧ ∀𝑓𝑐 𝜂) → 𝜑))
4038, 39sylbid 243 . 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))
411, 2, 6, 7, 2, 6, 7, 2, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 40xpord3ind 33403 1 ((𝑋 ∈ On ∧ 𝑌 ∈ On ∧ 𝑍 ∈ On) → 𝜆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wo 846  w3a 1088   = wceq 1542  wcel 2113  wne 2934  wral 3053   class class class wbr 5027  {copab 5089   E cep 5429   Po wpo 5436   Or wor 5437   We wwe 5477   × cxp 5517  Predcpred 6122  Oncon0 6166  cfv 6333  1st c1st 7705  2nd c2nd 7706
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-sep 5164  ax-nul 5171  ax-pow 5229  ax-pr 5293  ax-un 7473
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3399  df-sbc 3680  df-csb 3789  df-dif 3844  df-un 3846  df-in 3848  df-ss 3858  df-pss 3860  df-nul 4210  df-if 4412  df-pw 4487  df-sn 4514  df-pr 4516  df-tp 4518  df-op 4520  df-uni 4794  df-iun 4880  df-br 5028  df-opab 5090  df-mpt 5108  df-tr 5134  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-se 5479  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6123  df-ord 6169  df-on 6170  df-iota 6291  df-fun 6335  df-fv 6341  df-1st 7707  df-2nd 7708
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator