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Theorem on2ind 8653
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 6393 . 2 E Fr On
2 epweon 7746 . . 3 E We On
3 weso 5661 . . 3 ( E We On → E Or On)
4 sopo 5601 . . 3 ( E Or On → E Po On)
52, 3, 4mp2b 10 . 2 E Po On
6 epse 5653 . 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 7757 . . . . . 6 (𝑎 ∈ On → Pred( E , On, 𝑎) = 𝑎)
1312adantr 481 . . . . 5 ((𝑎 ∈ On ∧ 𝑏 ∈ On) → Pred( E , On, 𝑎) = 𝑎)
14 predon 7757 . . . . . . 7 (𝑏 ∈ On → Pred( E , On, 𝑏) = 𝑏)
1514adantl 482 . . . . . 6 ((𝑎 ∈ On ∧ 𝑏 ∈ On) → Pred( E , On, 𝑏) = 𝑏)
1615raleqdv 3325 . . . . 5 ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (∀𝑑 ∈ Pred ( E , On, 𝑏)𝜒 ↔ ∀𝑑𝑏 𝜒))
1713, 16raleqbidv 3342 . . . 4 ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (∀𝑐 ∈ Pred ( E , On, 𝑎)∀𝑑 ∈ Pred ( E , On, 𝑏)𝜒 ↔ ∀𝑐𝑎𝑑𝑏 𝜒))
1813raleqdv 3325 . . . 4 ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (∀𝑐 ∈ Pred ( E , On, 𝑎)𝜓 ↔ ∀𝑐𝑎 𝜓))
1915raleqdv 3325 . . . 4 ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (∀𝑑 ∈ Pred ( E , On, 𝑏)𝜃 ↔ ∀𝑑𝑏 𝜃))
2017, 18, 193anbi123d 1436 . . 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 8118 1 ((𝑋 ∈ On ∧ 𝑌 ∈ On) → 𝜂)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wral 3061   E cep 5573   Po wpo 5580   Or wor 5581   We wwe 5624  Predcpred 6289  Oncon0 6354
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5293  ax-nul 5300  ax-pow 5357  ax-pr 5421  ax-un 7709
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3775  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-pss 3964  df-nul 4320  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5568  df-eprel 5574  df-po 5582  df-so 5583  df-fr 5625  df-se 5626  df-we 5627  df-xp 5676  df-rel 5677  df-cnv 5678  df-co 5679  df-dm 5680  df-rn 5681  df-res 5682  df-ima 5683  df-pred 6290  df-ord 6357  df-on 6358  df-iota 6485  df-fun 6535  df-fv 6541  df-1st 7959  df-2nd 7960
This theorem is referenced by:  naddcllem  8660  naddcom  8666  naddsuc2  41978  naddgeoa  41980
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