Theorem on2ind 8594

Description: Double induction over ordinal numbers. (Contributed by Scott Fenton, 26-Aug-2024.)

Hypotheses

Ref	Expression
on2ind.1	⊢ (𝑎 = 𝑐 → (𝜑 ↔ 𝜓))
on2ind.2	⊢ (𝑏 = 𝑑 → (𝜓 ↔ 𝜒))
on2ind.3	⊢ (𝑎 = 𝑐 → (𝜃 ↔ 𝜒))
on2ind.4	⊢ (𝑎 = 𝑋 → (𝜑 ↔ 𝜏))
on2ind.5	⊢ (𝑏 = 𝑌 → (𝜏 ↔ 𝜂))
on2ind.i	⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → ((∀𝑐 ∈ 𝑎 ∀𝑑 ∈ 𝑏 𝜒 ∧ ∀𝑐 ∈ 𝑎 𝜓 ∧ ∀𝑑 ∈ 𝑏 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

Step	Hyp	Ref	Expression
1		onfr 6350	. 2 ⊢ E Fr On
2		epweon 7715	. . 3 ⊢ E We On
3		weso 5614	. . 3 ⊢ ( E We On → E Or On)
4		sopo 5550	. . 3 ⊢ ( E Or On → E Po On)
5	2, 3, 4	mp2b 10	. 2 ⊢ E Po On
6		epse 5605	. 2 ⊢ E Se On
7		on2ind.1	. 2 ⊢ (𝑎 = 𝑐 → (𝜑 ↔ 𝜓))
8		on2ind.2	. 2 ⊢ (𝑏 = 𝑑 → (𝜓 ↔ 𝜒))
9		on2ind.3	. 2 ⊢ (𝑎 = 𝑐 → (𝜃 ↔ 𝜒))
10		on2ind.4	. 2 ⊢ (𝑎 = 𝑋 → (𝜑 ↔ 𝜏))
11		on2ind.5	. 2 ⊢ (𝑏 = 𝑌 → (𝜏 ↔ 𝜂))
12		predon 7726	. . . . . 6 ⊢ (𝑎 ∈ On → Pred( E , On, 𝑎) = 𝑎)
13	12	adantr 480	. . . . 5 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → Pred( E , On, 𝑎) = 𝑎)
14		predon 7726	. . . . . . 7 ⊢ (𝑏 ∈ On → Pred( E , On, 𝑏) = 𝑏)
15	14	adantl 481	. . . . . 6 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → Pred( E , On, 𝑏) = 𝑏)
16	15	raleqdv 3290	. . . . 5 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (∀𝑑 ∈ Pred ( E , On, 𝑏)𝜒 ↔ ∀𝑑 ∈ 𝑏 𝜒))
17	13, 16	raleqbidv 3310	. . . 4 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (∀𝑐 ∈ Pred ( E , On, 𝑎)∀𝑑 ∈ Pred ( E , On, 𝑏)𝜒 ↔ ∀𝑐 ∈ 𝑎 ∀𝑑 ∈ 𝑏 𝜒))
18	13	raleqdv 3290	. . . 4 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (∀𝑐 ∈ Pred ( E , On, 𝑎)𝜓 ↔ ∀𝑐 ∈ 𝑎 𝜓))
19	15	raleqdv 3290	. . . 4 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (∀𝑑 ∈ Pred ( E , On, 𝑏)𝜃 ↔ ∀𝑑 ∈ 𝑏 𝜃))
20	17, 18, 19	3anbi123d 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) ↔ (∀𝑐 ∈ 𝑎 ∀𝑑 ∈ 𝑏 𝜒 ∧ ∀𝑐 ∈ 𝑎 𝜓 ∧ ∀𝑑 ∈ 𝑏 𝜃)))
21		on2ind.i	. . 3 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → ((∀𝑐 ∈ 𝑎 ∀𝑑 ∈ 𝑏 𝜒 ∧ ∀𝑐 ∈ 𝑎 𝜓 ∧ ∀𝑑 ∈ 𝑏 𝜃) → 𝜑))
22	20, 21	sylbid 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))
23	1, 5, 6, 1, 5, 6, 7, 8, 9, 10, 11, 22	xpord2ind 8088	1 ⊢ ((𝑋 ∈ On ∧ 𝑌 ∈ On) → 𝜂)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3044   E cep 5522   Po wpo 5529   Or wor 5530   We wwe 5575  Predcpred 6252  Oncon0 6311

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-se 5577  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6253  df-ord 6314  df-on 6315  df-iota 6442  df-fun 6488  df-fv 6494  df-1st 7931  df-2nd 7932

This theorem is referenced by:  naddcllem  8601  naddcom  8607  naddsuc2  8626  naddgeoa  43367