MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  on2ind Structured version   Visualization version   GIF version

Theorem on2ind 8707
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 6423 . 2 E Fr On
2 epweon 7795 . . 3 E We On
3 weso 5676 . . 3 ( E We On → E Or On)
4 sopo 5611 . . 3 ( E Or On → E Po On)
52, 3, 4mp2b 10 . 2 E Po On
6 epse 5667 . 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 7806 . . . . . 6 (𝑎 ∈ On → Pred( E , On, 𝑎) = 𝑎)
1312adantr 480 . . . . 5 ((𝑎 ∈ On ∧ 𝑏 ∈ On) → Pred( E , On, 𝑎) = 𝑎)
14 predon 7806 . . . . . . 7 (𝑏 ∈ On → Pred( E , On, 𝑏) = 𝑏)
1514adantl 481 . . . . . 6 ((𝑎 ∈ On ∧ 𝑏 ∈ On) → Pred( E , On, 𝑏) = 𝑏)
1615raleqdv 3326 . . . . 5 ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (∀𝑑 ∈ Pred ( E , On, 𝑏)𝜒 ↔ ∀𝑑𝑏 𝜒))
1713, 16raleqbidv 3346 . . . 4 ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (∀𝑐 ∈ Pred ( E , On, 𝑎)∀𝑑 ∈ Pred ( E , On, 𝑏)𝜒 ↔ ∀𝑐𝑎𝑑𝑏 𝜒))
1813raleqdv 3326 . . . 4 ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (∀𝑐 ∈ Pred ( E , On, 𝑎)𝜓 ↔ ∀𝑐𝑎 𝜓))
1915raleqdv 3326 . . . 4 ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (∀𝑑 ∈ Pred ( E , On, 𝑏)𝜃 ↔ ∀𝑑𝑏 𝜃))
2017, 18, 193anbi123d 1438 . . 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 240 . 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 8173 1 ((𝑋 ∈ On ∧ 𝑌 ∈ On) → 𝜂)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  wral 3061   E cep 5583   Po wpo 5590   Or wor 5591   We wwe 5636  Predcpred 6320  Oncon0 6384
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-iota 6514  df-fun 6563  df-fv 6569  df-1st 8014  df-2nd 8015
This theorem is referenced by:  naddcllem  8714  naddcom  8720  naddsuc2  8739  naddgeoa  43407
  Copyright terms: Public domain W3C validator