Theorem on3ind 8668

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

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)

Proof of Theorem on3ind

Step	Hyp	Ref	Expression
1		onfr 6403	. 2 ⊢ E Fr On
2		epweon 7761	. . 3 ⊢ E We On
3		weso 5667	. . 3 ⊢ ( E We On → E Or On)
4		sopo 5607	. . 3 ⊢ ( E Or On → E Po On)
5	2, 3, 4	mp2b 10	. 2 ⊢ E Po On
6		epse 5659	. 2 ⊢ E Se On
7		on3ind.1	. 2 ⊢ (𝑎 = 𝑑 → (𝜑 ↔ 𝜓))
8		on3ind.2	. 2 ⊢ (𝑏 = 𝑒 → (𝜓 ↔ 𝜒))
9		on3ind.3	. 2 ⊢ (𝑐 = 𝑓 → (𝜒 ↔ 𝜃))
10		on3ind.4	. 2 ⊢ (𝑎 = 𝑑 → (𝜏 ↔ 𝜃))
11		on3ind.5	. 2 ⊢ (𝑏 = 𝑒 → (𝜂 ↔ 𝜏))
12		on3ind.6	. 2 ⊢ (𝑏 = 𝑒 → (𝜁 ↔ 𝜃))
13		on3ind.7	. 2 ⊢ (𝑐 = 𝑓 → (𝜎 ↔ 𝜏))
14		on3ind.8	. 2 ⊢ (𝑎 = 𝑋 → (𝜑 ↔ 𝜌))
15		on3ind.9	. 2 ⊢ (𝑏 = 𝑌 → (𝜌 ↔ 𝜇))
16		on3ind.10	. 2 ⊢ (𝑐 = 𝑍 → (𝜇 ↔ 𝜆))
17		predon 7772	. . . . . . 7 ⊢ (𝑎 ∈ On → Pred( E , On, 𝑎) = 𝑎)
18	17	3ad2ant1 1133	. . . . . 6 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) → Pred( E , On, 𝑎) = 𝑎)
19		predon 7772	. . . . . . . 8 ⊢ (𝑏 ∈ On → Pred( E , On, 𝑏) = 𝑏)
20	19	3ad2ant2 1134	. . . . . . 7 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) → Pred( E , On, 𝑏) = 𝑏)
21		predon 7772	. . . . . . . . 9 ⊢ (𝑐 ∈ On → Pred( E , On, 𝑐) = 𝑐)
22	21	3ad2ant3 1135	. . . . . . . 8 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) → Pred( E , On, 𝑐) = 𝑐)
23	22	raleqdv 3325	. . . . . . 7 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) → (∀𝑓 ∈ Pred ( E , On, 𝑐)𝜃 ↔ ∀𝑓 ∈ 𝑐 𝜃))
24	20, 23	raleqbidv 3342	. . . . . 6 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) → (∀𝑒 ∈ Pred ( E , On, 𝑏)∀𝑓 ∈ Pred ( E , On, 𝑐)𝜃 ↔ ∀𝑒 ∈ 𝑏 ∀𝑓 ∈ 𝑐 𝜃))
25	18, 24	raleqbidv 3342	. . . . 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))
26	20	raleqdv 3325	. . . . . 6 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) → (∀𝑒 ∈ Pred ( E , On, 𝑏)𝜒 ↔ ∀𝑒 ∈ 𝑏 𝜒))
27	18, 26	raleqbidv 3342	. . . . 5 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) → (∀𝑑 ∈ Pred ( E , On, 𝑎)∀𝑒 ∈ Pred ( E , On, 𝑏)𝜒 ↔ ∀𝑑 ∈ 𝑎 ∀𝑒 ∈ 𝑏 𝜒))
28	22	raleqdv 3325	. . . . . 6 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) → (∀𝑓 ∈ Pred ( E , On, 𝑐)𝜁 ↔ ∀𝑓 ∈ 𝑐 𝜁))
29	18, 28	raleqbidv 3342	. . . . 5 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) → (∀𝑑 ∈ Pred ( E , On, 𝑎)∀𝑓 ∈ Pred ( E , On, 𝑐)𝜁 ↔ ∀𝑑 ∈ 𝑎 ∀𝑓 ∈ 𝑐 𝜁))
30	25, 27, 29	3anbi123d 1436	. . . 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)))
31	18	raleqdv 3325	. . . . 5 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) → (∀𝑑 ∈ Pred ( E , On, 𝑎)𝜓 ↔ ∀𝑑 ∈ 𝑎 𝜓))
32	22	raleqdv 3325	. . . . . 6 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) → (∀𝑓 ∈ Pred ( E , On, 𝑐)𝜏 ↔ ∀𝑓 ∈ 𝑐 𝜏))
33	20, 32	raleqbidv 3342	. . . . 5 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) → (∀𝑒 ∈ Pred ( E , On, 𝑏)∀𝑓 ∈ Pred ( E , On, 𝑐)𝜏 ↔ ∀𝑒 ∈ 𝑏 ∀𝑓 ∈ 𝑐 𝜏))
34	20	raleqdv 3325	. . . . 5 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) → (∀𝑒 ∈ Pred ( E , On, 𝑏)𝜎 ↔ ∀𝑒 ∈ 𝑏 𝜎))
35	31, 33, 34	3anbi123d 1436	. . . 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)))
36	22	raleqdv 3325	. . . 4 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) → (∀𝑓 ∈ Pred ( E , On, 𝑐)𝜂 ↔ ∀𝑓 ∈ 𝑐 𝜂))
37	30, 35, 36	3anbi123d 1436	. . 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)))
38		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))
39	37, 38	sylbid 239	. 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))
40	1, 5, 6, 1, 5, 6, 1, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 39	xpord3ind 8141	1 ⊢ ((𝑋 ∈ On ∧ 𝑌 ∈ On ∧ 𝑍 ∈ On) → 𝜆)

Syntax hints:   → wi 4   ↔ wb 205   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  ∀wral 3061   E cep 5579   Po wpo 5586   Or wor 5587   We wwe 5630  Predcpred 6299  Oncon0 6364

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 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724

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 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-ot 4637  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-se 5632  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-iota 6495  df-fun 6545  df-fv 6551  df-1st 7974  df-2nd 7975

This theorem is referenced by:  naddass  8694