Theorem on3ind 33427

Description: Triple induction over ordinals. (Contributed by Scott Fenton, 4-Sep-2024.)

Hypotheses

Ref	Expression
on3ind.1	⊢ (𝑎 = 𝑑 → (𝜑 ↔ 𝜓))
on3ind.2	⊢ (𝑏 = 𝑒 → (𝜓 ↔ 𝜒))
on3ind.3	⊢ (𝑐 = 𝑓 → (𝜒 ↔ 𝜃))
on3ind.4	⊢ (𝑎 = 𝑑 → (𝜏 ↔ 𝜃))
on3ind.5	⊢ (𝑏 = 𝑒 → (𝜂 ↔ 𝜏))
on3ind.6	⊢ (𝑏 = 𝑒 → (𝜁 ↔ 𝜃))
on3ind.7	⊢ (𝑐 = 𝑓 → (𝜎 ↔ 𝜏))
on3ind.8	⊢ (𝑎 = 𝑋 → (𝜑 ↔ 𝜌))
on3ind.9	⊢ (𝑏 = 𝑌 → (𝜌 ↔ 𝜇))
on3ind.10	⊢ (𝑐 = 𝑍 → (𝜇 ↔ 𝜆))
on3ind.i	⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) → (((∀𝑑 ∈ 𝑎 ∀𝑒 ∈ 𝑏 ∀𝑓 ∈ 𝑐 𝜃 ∧ ∀𝑑 ∈ 𝑎 ∀𝑒 ∈ 𝑏 𝜒 ∧ ∀𝑑 ∈ 𝑎 ∀𝑓 ∈ 𝑐 𝜁) ∧ (∀𝑑 ∈ 𝑎 𝜓 ∧ ∀𝑒 ∈ 𝑏 ∀𝑓 ∈ 𝑐 𝜏 ∧ ∀𝑒 ∈ 𝑏 𝜎) ∧ ∀𝑓 ∈ 𝑐 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.

Step	Hyp	Ref	Expression
1		eqid 2758	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((On × On) × On) ∧ 𝑦 ∈ ((On × On) × On) ∧ ((((1st ‘(1st ‘𝑥)) E (1st ‘(1st ‘𝑦)) ∨ (1st ‘(1st ‘𝑥)) = (1st ‘(1st ‘𝑦))) ∧ ((2nd ‘(1st ‘𝑥)) E (2nd ‘(1st ‘𝑦)) ∨ (2nd ‘(1st ‘𝑥)) = (2nd ‘(1st ‘𝑦))) ∧ ((2nd ‘𝑥) E (2nd ‘𝑦) ∨ (2nd ‘𝑥) = (2nd ‘𝑦))) ∧ 𝑥 ≠ 𝑦))} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((On × On) × On) ∧ 𝑦 ∈ ((On × On) × On) ∧ ((((1st ‘(1st ‘𝑥)) E (1st ‘(1st ‘𝑦)) ∨ (1st ‘(1st ‘𝑥)) = (1st ‘(1st ‘𝑦))) ∧ ((2nd ‘(1st ‘𝑥)) E (2nd ‘(1st ‘𝑦)) ∨ (2nd ‘(1st ‘𝑥)) = (2nd ‘(1st ‘𝑦))) ∧ ((2nd ‘𝑥) E (2nd ‘𝑦) ∨ (2nd ‘𝑥) = (2nd ‘𝑦))) ∧ 𝑥 ≠ 𝑦))}
2		onfr 6213	. 2 ⊢ E Fr On
3		epweon 7502	. . 3 ⊢ E We On
4		weso 5519	. . 3 ⊢ ( E We On → E Or On)
5		sopo 5465	. . 3 ⊢ ( E Or On → E Po On)
6	3, 4, 5	mp2b 10	. 2 ⊢ E Po On
7		epse 5511	. 2 ⊢ E Se On
8		on3ind.1	. 2 ⊢ (𝑎 = 𝑑 → (𝜑 ↔ 𝜓))
9		on3ind.2	. 2 ⊢ (𝑏 = 𝑒 → (𝜓 ↔ 𝜒))
10		on3ind.3	. 2 ⊢ (𝑐 = 𝑓 → (𝜒 ↔ 𝜃))
11		on3ind.4	. 2 ⊢ (𝑎 = 𝑑 → (𝜏 ↔ 𝜃))
12		on3ind.5	. 2 ⊢ (𝑏 = 𝑒 → (𝜂 ↔ 𝜏))
13		on3ind.6	. 2 ⊢ (𝑏 = 𝑒 → (𝜁 ↔ 𝜃))
14		on3ind.7	. 2 ⊢ (𝑐 = 𝑓 → (𝜎 ↔ 𝜏))
15		on3ind.8	. 2 ⊢ (𝑎 = 𝑋 → (𝜑 ↔ 𝜌))
16		on3ind.9	. 2 ⊢ (𝑏 = 𝑌 → (𝜌 ↔ 𝜇))
17		on3ind.10	. 2 ⊢ (𝑐 = 𝑍 → (𝜇 ↔ 𝜆))
18		predon 7511	. . . . . . 7 ⊢ (𝑎 ∈ On → Pred( E , On, 𝑎) = 𝑎)
19	18	3ad2ant1 1130	. . . . . 6 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) → Pred( E , On, 𝑎) = 𝑎)
20		predon 7511	. . . . . . . 8 ⊢ (𝑏 ∈ On → Pred( E , On, 𝑏) = 𝑏)
21	20	3ad2ant2 1131	. . . . . . 7 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) → Pred( E , On, 𝑏) = 𝑏)
22		predon 7511	. . . . . . . . 9 ⊢ (𝑐 ∈ On → Pred( E , On, 𝑐) = 𝑐)
23	22	3ad2ant3 1132	. . . . . . . 8 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) → Pred( E , On, 𝑐) = 𝑐)
24	23	raleqdv 3329	. . . . . . 7 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) → (∀𝑓 ∈ Pred ( E , On, 𝑐)𝜃 ↔ ∀𝑓 ∈ 𝑐 𝜃))
25	21, 24	raleqbidv 3319	. . . . . 6 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) → (∀𝑒 ∈ Pred ( E , On, 𝑏)∀𝑓 ∈ Pred ( E , On, 𝑐)𝜃 ↔ ∀𝑒 ∈ 𝑏 ∀𝑓 ∈ 𝑐 𝜃))
26	19, 25	raleqbidv 3319	. . . . 5 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) → (∀𝑑 ∈ Pred ( E , On, 𝑎)∀𝑒 ∈ Pred ( E , On, 𝑏)∀𝑓 ∈ Pred ( E , On, 𝑐)𝜃 ↔ ∀𝑑 ∈ 𝑎 ∀𝑒 ∈ 𝑏 ∀𝑓 ∈ 𝑐 𝜃))
27	21	raleqdv 3329	. . . . . 6 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) → (∀𝑒 ∈ Pred ( E , On, 𝑏)𝜒 ↔ ∀𝑒 ∈ 𝑏 𝜒))
28	19, 27	raleqbidv 3319	. . . . 5 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) → (∀𝑑 ∈ Pred ( E , On, 𝑎)∀𝑒 ∈ Pred ( E , On, 𝑏)𝜒 ↔ ∀𝑑 ∈ 𝑎 ∀𝑒 ∈ 𝑏 𝜒))
29	23	raleqdv 3329	. . . . . 6 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) → (∀𝑓 ∈ Pred ( E , On, 𝑐)𝜁 ↔ ∀𝑓 ∈ 𝑐 𝜁))
30	19, 29	raleqbidv 3319	. . . . 5 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) → (∀𝑑 ∈ Pred ( E , On, 𝑎)∀𝑓 ∈ Pred ( E , On, 𝑐)𝜁 ↔ ∀𝑑 ∈ 𝑎 ∀𝑓 ∈ 𝑐 𝜁))
31	26, 28, 30	3anbi123d 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) ↔ (∀𝑑 ∈ 𝑎 ∀𝑒 ∈ 𝑏 ∀𝑓 ∈ 𝑐 𝜃 ∧ ∀𝑑 ∈ 𝑎 ∀𝑒 ∈ 𝑏 𝜒 ∧ ∀𝑑 ∈ 𝑎 ∀𝑓 ∈ 𝑐 𝜁)))
32	19	raleqdv 3329	. . . . 5 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) → (∀𝑑 ∈ Pred ( E , On, 𝑎)𝜓 ↔ ∀𝑑 ∈ 𝑎 𝜓))
33	23	raleqdv 3329	. . . . . 6 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) → (∀𝑓 ∈ Pred ( E , On, 𝑐)𝜏 ↔ ∀𝑓 ∈ 𝑐 𝜏))
34	21, 33	raleqbidv 3319	. . . . 5 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) → (∀𝑒 ∈ Pred ( E , On, 𝑏)∀𝑓 ∈ Pred ( E , On, 𝑐)𝜏 ↔ ∀𝑒 ∈ 𝑏 ∀𝑓 ∈ 𝑐 𝜏))
35	21	raleqdv 3329	. . . . 5 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) → (∀𝑒 ∈ Pred ( E , On, 𝑏)𝜎 ↔ ∀𝑒 ∈ 𝑏 𝜎))
36	32, 34, 35	3anbi123d 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) ↔ (∀𝑑 ∈ 𝑎 𝜓 ∧ ∀𝑒 ∈ 𝑏 ∀𝑓 ∈ 𝑐 𝜏 ∧ ∀𝑒 ∈ 𝑏 𝜎)))
37	23	raleqdv 3329	. . . 4 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) → (∀𝑓 ∈ Pred ( E , On, 𝑐)𝜂 ↔ ∀𝑓 ∈ 𝑐 𝜂))
38	31, 36, 37	3anbi123d 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) ↔ ((∀𝑑 ∈ 𝑎 ∀𝑒 ∈ 𝑏 ∀𝑓 ∈ 𝑐 𝜃 ∧ ∀𝑑 ∈ 𝑎 ∀𝑒 ∈ 𝑏 𝜒 ∧ ∀𝑑 ∈ 𝑎 ∀𝑓 ∈ 𝑐 𝜁) ∧ (∀𝑑 ∈ 𝑎 𝜓 ∧ ∀𝑒 ∈ 𝑏 ∀𝑓 ∈ 𝑐 𝜏 ∧ ∀𝑒 ∈ 𝑏 𝜎) ∧ ∀𝑓 ∈ 𝑐 𝜂)))
39		on3ind.i	. . 3 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) → (((∀𝑑 ∈ 𝑎 ∀𝑒 ∈ 𝑏 ∀𝑓 ∈ 𝑐 𝜃 ∧ ∀𝑑 ∈ 𝑎 ∀𝑒 ∈ 𝑏 𝜒 ∧ ∀𝑑 ∈ 𝑎 ∀𝑓 ∈ 𝑐 𝜁) ∧ (∀𝑑 ∈ 𝑎 𝜓 ∧ ∀𝑒 ∈ 𝑏 ∀𝑓 ∈ 𝑐 𝜏 ∧ ∀𝑒 ∈ 𝑏 𝜎) ∧ ∀𝑓 ∈ 𝑐 𝜂) → 𝜑))
40	38, 39	sylbid 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))
41	1, 2, 6, 7, 2, 6, 7, 2, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 40	xpord3ind 33368	1 ⊢ ((𝑋 ∈ On ∧ 𝑌 ∈ On ∧ 𝑍 ∈ On) → 𝜆)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   ≠ wne 2951  ∀wral 3070   class class class wbr 5036  {copab 5098   E cep 5438   Po wpo 5445   Or wor 5446   We wwe 5486   × cxp 5526  Predcpred 6130  Oncon0 6174  ‘cfv 6340  1^st c1st 7697  2^nd c2nd 7698

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5173  ax-nul 5180  ax-pow 5238  ax-pr 5302  ax-un 7465

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-pss 3879  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-iun 4888  df-br 5037  df-opab 5099  df-mpt 5117  df-tr 5143  df-id 5434  df-eprel 5439  df-po 5447  df-so 5448  df-fr 5487  df-se 5488  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-pred 6131  df-ord 6177  df-on 6178  df-iota 6299  df-fun 6342  df-fv 6348  df-1st 7699  df-2nd 7700

This theorem is referenced by: (None)