Theorem on3ind 8710

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 6423	. 2 ⊢ E Fr On
2		epweon 7816	. . 3 ⊢ E We On
3		weso 5647	. . 3 ⊢ ( E We On → E Or On)
4		sopo 5583	. . 3 ⊢ ( E Or On → E Po On)
5	2, 3, 4	mp2b 10	. 2 ⊢ E Po On
6		epse 5638	. 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 7827	. . . . . . 7 ⊢ (𝑎 ∈ On → Pred( E , On, 𝑎) = 𝑎)
18	17	3ad2ant1 1151	. . . . . 6 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) → Pred( E , On, 𝑎) = 𝑎)
19		predon 7827	. . . . . . . 8 ⊢ (𝑏 ∈ On → Pred( E , On, 𝑏) = 𝑏)
20	19	3ad2ant2 1152	. . . . . . 7 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) → Pred( E , On, 𝑏) = 𝑏)
21		predon 7827	. . . . . . . . 9 ⊢ (𝑐 ∈ On → Pred( E , On, 𝑐) = 𝑐)
22	21	3ad2ant3 1153	. . . . . . . 8 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) → Pred( E , On, 𝑐) = 𝑐)
23	22	raleqdv 3323	. . . . . . 7 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) → (∀𝑓 ∈ Pred ( E , On, 𝑐)𝜃 ↔ ∀𝑓 ∈ 𝑐 𝜃))
24	20, 23	raleqbidv 3338	. . . . . 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 3338	. . . . 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 3323	. . . . . 6 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) → (∀𝑒 ∈ Pred ( E , On, 𝑏)𝜒 ↔ ∀𝑒 ∈ 𝑏 𝜒))
27	18, 26	raleqbidv 3338	. . . . 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 3323	. . . . . 6 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) → (∀𝑓 ∈ Pred ( E , On, 𝑐)𝜁 ↔ ∀𝑓 ∈ 𝑐 𝜁))
29	18, 28	raleqbidv 3338	. . . . 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 1465	. . . 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 3323	. . . . 5 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) → (∀𝑑 ∈ Pred ( E , On, 𝑎)𝜓 ↔ ∀𝑑 ∈ 𝑎 𝜓))
32	22	raleqdv 3323	. . . . . 6 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) → (∀𝑓 ∈ Pred ( E , On, 𝑐)𝜏 ↔ ∀𝑓 ∈ 𝑐 𝜏))
33	20, 32	raleqbidv 3338	. . . . 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 3323	. . . . 5 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) → (∀𝑒 ∈ Pred ( E , On, 𝑏)𝜎 ↔ ∀𝑒 ∈ 𝑏 𝜎))
35	31, 33, 34	3anbi123d 1465	. . . 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 3323	. . . 4 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) → (∀𝑓 ∈ Pred ( E , On, 𝑐)𝜂 ↔ ∀𝑓 ∈ 𝑐 𝜂))
37	30, 35, 36	3anbi123d 1465	. . 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 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))
40	1, 5, 6, 1, 5, 6, 1, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 39	xpord3ind 8203	1 ⊢ ((𝑋 ∈ On ∧ 𝑌 ∈ On ∧ 𝑍 ∈ On) → 𝜆)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  ∀wral 3080   E cep 5555   Po wpo 5562   Or wor 5563   We wwe 5608  Predcpred 6324  Oncon0 6383

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7778

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-ot 4862  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5551  df-eprel 5556  df-po 5564  df-so 5565  df-fr 5609  df-se 5610  df-we 5611  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-pred 6325  df-ord 6386  df-on 6387  df-iota 6515  df-fun 6563  df-fv 6569  df-1st 8028  df-2nd 8029

This theorem is used by:  naddass  8737  nadddi  37164