Theorem no3inds 34115

Description: Triple induction over surreal numbers. (Contributed by Scott Fenton, 9-Oct-2024.)

Hypotheses

Ref	Expression
no3inds.1	⊢ (𝑎 = 𝑑 → (𝜑 ↔ 𝜓))
no3inds.2	⊢ (𝑏 = 𝑒 → (𝜓 ↔ 𝜒))
no3inds.3	⊢ (𝑐 = 𝑓 → (𝜒 ↔ 𝜃))
no3inds.4	⊢ (𝑎 = 𝑑 → (𝜏 ↔ 𝜃))
no3inds.5	⊢ (𝑏 = 𝑒 → (𝜂 ↔ 𝜏))
no3inds.6	⊢ (𝑏 = 𝑒 → (𝜁 ↔ 𝜃))
no3inds.7	⊢ (𝑐 = 𝑓 → (𝜎 ↔ 𝜏))
no3inds.8	⊢ (𝑎 = 𝑋 → (𝜑 ↔ 𝜌))
no3inds.9	⊢ (𝑏 = 𝑌 → (𝜌 ↔ 𝜇))
no3inds.10	⊢ (𝑐 = 𝑍 → (𝜇 ↔ 𝜆))
no3inds.i	⊢ ((𝑎 ∈ No ∧ 𝑏 ∈ No ∧ 𝑐 ∈ No ) → (((∀𝑑 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))∀𝑒 ∈ (( L ‘𝑏) ∪ ( R ‘𝑏))∀𝑓 ∈ (( L ‘𝑐) ∪ ( R ‘𝑐))𝜃 ∧ ∀𝑑 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))∀𝑒 ∈ (( L ‘𝑏) ∪ ( R ‘𝑏))𝜒 ∧ ∀𝑑 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))∀𝑓 ∈ (( L ‘𝑐) ∪ ( R ‘𝑐))𝜁) ∧ (∀𝑑 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))𝜓 ∧ ∀𝑒 ∈ (( L ‘𝑏) ∪ ( R ‘𝑏))∀𝑓 ∈ (( L ‘𝑐) ∪ ( R ‘𝑐))𝜏 ∧ ∀𝑒 ∈ (( L ‘𝑏) ∪ ( R ‘𝑏))𝜎) ∧ ∀𝑓 ∈ (( L ‘𝑐) ∪ ( R ‘𝑐))𝜂) → 𝜑))

Assertion

Ref	Expression
no3inds	⊢ ((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ 𝑍 ∈ No ) → 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 no3inds

Dummy variables 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2738	. 2 ⊢ {⟨𝑧, 𝑤⟩ ∣ (𝑧 ∈ (( No × No ) × No ) ∧ 𝑤 ∈ (( No × No ) × No ) ∧ ((((1st ‘(1st ‘𝑧)){⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))} (1st ‘(1st ‘𝑤)) ∨ (1st ‘(1st ‘𝑧)) = (1st ‘(1st ‘𝑤))) ∧ ((2nd ‘(1st ‘𝑧)){⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))} (2nd ‘(1st ‘𝑤)) ∨ (2nd ‘(1st ‘𝑧)) = (2nd ‘(1st ‘𝑤))) ∧ ((2nd ‘𝑧){⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))} (2nd ‘𝑤) ∨ (2nd ‘𝑧) = (2nd ‘𝑤))) ∧ 𝑧 ≠ 𝑤))} = {⟨𝑧, 𝑤⟩ ∣ (𝑧 ∈ (( No × No ) × No ) ∧ 𝑤 ∈ (( No × No ) × No ) ∧ ((((1st ‘(1st ‘𝑧)){⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))} (1st ‘(1st ‘𝑤)) ∨ (1st ‘(1st ‘𝑧)) = (1st ‘(1st ‘𝑤))) ∧ ((2nd ‘(1st ‘𝑧)){⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))} (2nd ‘(1st ‘𝑤)) ∨ (2nd ‘(1st ‘𝑧)) = (2nd ‘(1st ‘𝑤))) ∧ ((2nd ‘𝑧){⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))} (2nd ‘𝑤) ∨ (2nd ‘𝑧) = (2nd ‘𝑤))) ∧ 𝑧 ≠ 𝑤))}
2		eqid 2738	. . 3 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))} = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))}
3	2	lrrecfr 34100	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))} Fr No
4	2	lrrecpo 34098	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))} Po No
5	2	lrrecse 34099	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))} Se No
6		no3inds.1	. 2 ⊢ (𝑎 = 𝑑 → (𝜑 ↔ 𝜓))
7		no3inds.2	. 2 ⊢ (𝑏 = 𝑒 → (𝜓 ↔ 𝜒))
8		no3inds.3	. 2 ⊢ (𝑐 = 𝑓 → (𝜒 ↔ 𝜃))
9		no3inds.4	. 2 ⊢ (𝑎 = 𝑑 → (𝜏 ↔ 𝜃))
10		no3inds.5	. 2 ⊢ (𝑏 = 𝑒 → (𝜂 ↔ 𝜏))
11		no3inds.6	. 2 ⊢ (𝑏 = 𝑒 → (𝜁 ↔ 𝜃))
12		no3inds.7	. 2 ⊢ (𝑐 = 𝑓 → (𝜎 ↔ 𝜏))
13		no3inds.8	. 2 ⊢ (𝑎 = 𝑋 → (𝜑 ↔ 𝜌))
14		no3inds.9	. 2 ⊢ (𝑏 = 𝑌 → (𝜌 ↔ 𝜇))
15		no3inds.10	. 2 ⊢ (𝑐 = 𝑍 → (𝜇 ↔ 𝜆))
16	2	lrrecpred 34101	. . . . . . 7 ⊢ (𝑎 ∈ No → Pred({⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))}, No , 𝑎) = (( L ‘𝑎) ∪ ( R ‘𝑎)))
17	16	3ad2ant1 1132	. . . . . 6 ⊢ ((𝑎 ∈ No ∧ 𝑏 ∈ No ∧ 𝑐 ∈ No ) → Pred({⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))}, No , 𝑎) = (( L ‘𝑎) ∪ ( R ‘𝑎)))
18	2	lrrecpred 34101	. . . . . . . 8 ⊢ (𝑏 ∈ No → Pred({⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))}, No , 𝑏) = (( L ‘𝑏) ∪ ( R ‘𝑏)))
19	18	3ad2ant2 1133	. . . . . . 7 ⊢ ((𝑎 ∈ No ∧ 𝑏 ∈ No ∧ 𝑐 ∈ No ) → Pred({⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))}, No , 𝑏) = (( L ‘𝑏) ∪ ( R ‘𝑏)))
20	2	lrrecpred 34101	. . . . . . . . 9 ⊢ (𝑐 ∈ No → Pred({⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))}, No , 𝑐) = (( L ‘𝑐) ∪ ( R ‘𝑐)))
21	20	3ad2ant3 1134	. . . . . . . 8 ⊢ ((𝑎 ∈ No ∧ 𝑏 ∈ No ∧ 𝑐 ∈ No ) → Pred({⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))}, No , 𝑐) = (( L ‘𝑐) ∪ ( R ‘𝑐)))
22	21	raleqdv 3348	. . . . . . 7 ⊢ ((𝑎 ∈ No ∧ 𝑏 ∈ No ∧ 𝑐 ∈ No ) → (∀𝑓 ∈ Pred ({⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))}, No , 𝑐)𝜃 ↔ ∀𝑓 ∈ (( L ‘𝑐) ∪ ( R ‘𝑐))𝜃))
23	19, 22	raleqbidv 3336	. . . . . 6 ⊢ ((𝑎 ∈ No ∧ 𝑏 ∈ No ∧ 𝑐 ∈ No ) → (∀𝑒 ∈ Pred ({⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))}, No , 𝑏)∀𝑓 ∈ Pred ({⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))}, No , 𝑐)𝜃 ↔ ∀𝑒 ∈ (( L ‘𝑏) ∪ ( R ‘𝑏))∀𝑓 ∈ (( L ‘𝑐) ∪ ( R ‘𝑐))𝜃))
24	17, 23	raleqbidv 3336	. . . . 5 ⊢ ((𝑎 ∈ No ∧ 𝑏 ∈ No ∧ 𝑐 ∈ No ) → (∀𝑑 ∈ Pred ({⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))}, No , 𝑎)∀𝑒 ∈ Pred ({⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))}, No , 𝑏)∀𝑓 ∈ Pred ({⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))}, No , 𝑐)𝜃 ↔ ∀𝑑 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))∀𝑒 ∈ (( L ‘𝑏) ∪ ( R ‘𝑏))∀𝑓 ∈ (( L ‘𝑐) ∪ ( R ‘𝑐))𝜃))
25	19	raleqdv 3348	. . . . . 6 ⊢ ((𝑎 ∈ No ∧ 𝑏 ∈ No ∧ 𝑐 ∈ No ) → (∀𝑒 ∈ Pred ({⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))}, No , 𝑏)𝜒 ↔ ∀𝑒 ∈ (( L ‘𝑏) ∪ ( R ‘𝑏))𝜒))
26	17, 25	raleqbidv 3336	. . . . 5 ⊢ ((𝑎 ∈ No ∧ 𝑏 ∈ No ∧ 𝑐 ∈ No ) → (∀𝑑 ∈ Pred ({⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))}, No , 𝑎)∀𝑒 ∈ Pred ({⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))}, No , 𝑏)𝜒 ↔ ∀𝑑 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))∀𝑒 ∈ (( L ‘𝑏) ∪ ( R ‘𝑏))𝜒))
27	21	raleqdv 3348	. . . . . 6 ⊢ ((𝑎 ∈ No ∧ 𝑏 ∈ No ∧ 𝑐 ∈ No ) → (∀𝑓 ∈ Pred ({⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))}, No , 𝑐)𝜁 ↔ ∀𝑓 ∈ (( L ‘𝑐) ∪ ( R ‘𝑐))𝜁))
28	17, 27	raleqbidv 3336	. . . . 5 ⊢ ((𝑎 ∈ No ∧ 𝑏 ∈ No ∧ 𝑐 ∈ No ) → (∀𝑑 ∈ Pred ({⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))}, No , 𝑎)∀𝑓 ∈ Pred ({⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))}, No , 𝑐)𝜁 ↔ ∀𝑑 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))∀𝑓 ∈ (( L ‘𝑐) ∪ ( R ‘𝑐))𝜁))
29	24, 26, 28	3anbi123d 1435	. . . 4 ⊢ ((𝑎 ∈ No ∧ 𝑏 ∈ No ∧ 𝑐 ∈ No ) → ((∀𝑑 ∈ Pred ({⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))}, No , 𝑎)∀𝑒 ∈ Pred ({⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))}, No , 𝑏)∀𝑓 ∈ Pred ({⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))}, No , 𝑐)𝜃 ∧ ∀𝑑 ∈ Pred ({⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))}, No , 𝑎)∀𝑒 ∈ Pred ({⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))}, No , 𝑏)𝜒 ∧ ∀𝑑 ∈ Pred ({⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))}, No , 𝑎)∀𝑓 ∈ Pred ({⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))}, No , 𝑐)𝜁) ↔ (∀𝑑 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))∀𝑒 ∈ (( L ‘𝑏) ∪ ( R ‘𝑏))∀𝑓 ∈ (( L ‘𝑐) ∪ ( R ‘𝑐))𝜃 ∧ ∀𝑑 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))∀𝑒 ∈ (( L ‘𝑏) ∪ ( R ‘𝑏))𝜒 ∧ ∀𝑑 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))∀𝑓 ∈ (( L ‘𝑐) ∪ ( R ‘𝑐))𝜁)))
30	17	raleqdv 3348	. . . . 5 ⊢ ((𝑎 ∈ No ∧ 𝑏 ∈ No ∧ 𝑐 ∈ No ) → (∀𝑑 ∈ Pred ({⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))}, No , 𝑎)𝜓 ↔ ∀𝑑 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))𝜓))
31	21	raleqdv 3348	. . . . . 6 ⊢ ((𝑎 ∈ No ∧ 𝑏 ∈ No ∧ 𝑐 ∈ No ) → (∀𝑓 ∈ Pred ({⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))}, No , 𝑐)𝜏 ↔ ∀𝑓 ∈ (( L ‘𝑐) ∪ ( R ‘𝑐))𝜏))
32	19, 31	raleqbidv 3336	. . . . 5 ⊢ ((𝑎 ∈ No ∧ 𝑏 ∈ No ∧ 𝑐 ∈ No ) → (∀𝑒 ∈ Pred ({⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))}, No , 𝑏)∀𝑓 ∈ Pred ({⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))}, No , 𝑐)𝜏 ↔ ∀𝑒 ∈ (( L ‘𝑏) ∪ ( R ‘𝑏))∀𝑓 ∈ (( L ‘𝑐) ∪ ( R ‘𝑐))𝜏))
33	19	raleqdv 3348	. . . . 5 ⊢ ((𝑎 ∈ No ∧ 𝑏 ∈ No ∧ 𝑐 ∈ No ) → (∀𝑒 ∈ Pred ({⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))}, No , 𝑏)𝜎 ↔ ∀𝑒 ∈ (( L ‘𝑏) ∪ ( R ‘𝑏))𝜎))
34	30, 32, 33	3anbi123d 1435	. . . 4 ⊢ ((𝑎 ∈ No ∧ 𝑏 ∈ No ∧ 𝑐 ∈ No ) → ((∀𝑑 ∈ Pred ({⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))}, No , 𝑎)𝜓 ∧ ∀𝑒 ∈ Pred ({⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))}, No , 𝑏)∀𝑓 ∈ Pred ({⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))}, No , 𝑐)𝜏 ∧ ∀𝑒 ∈ Pred ({⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))}, No , 𝑏)𝜎) ↔ (∀𝑑 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))𝜓 ∧ ∀𝑒 ∈ (( L ‘𝑏) ∪ ( R ‘𝑏))∀𝑓 ∈ (( L ‘𝑐) ∪ ( R ‘𝑐))𝜏 ∧ ∀𝑒 ∈ (( L ‘𝑏) ∪ ( R ‘𝑏))𝜎)))
35	21	raleqdv 3348	. . . 4 ⊢ ((𝑎 ∈ No ∧ 𝑏 ∈ No ∧ 𝑐 ∈ No ) → (∀𝑓 ∈ Pred ({⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))}, No , 𝑐)𝜂 ↔ ∀𝑓 ∈ (( L ‘𝑐) ∪ ( R ‘𝑐))𝜂))
36	29, 34, 35	3anbi123d 1435	. . 3 ⊢ ((𝑎 ∈ No ∧ 𝑏 ∈ No ∧ 𝑐 ∈ No ) → (((∀𝑑 ∈ Pred ({⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))}, No , 𝑎)∀𝑒 ∈ Pred ({⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))}, No , 𝑏)∀𝑓 ∈ Pred ({⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))}, No , 𝑐)𝜃 ∧ ∀𝑑 ∈ Pred ({⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))}, No , 𝑎)∀𝑒 ∈ Pred ({⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))}, No , 𝑏)𝜒 ∧ ∀𝑑 ∈ Pred ({⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))}, No , 𝑎)∀𝑓 ∈ Pred ({⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))}, No , 𝑐)𝜁) ∧ (∀𝑑 ∈ Pred ({⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))}, No , 𝑎)𝜓 ∧ ∀𝑒 ∈ Pred ({⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))}, No , 𝑏)∀𝑓 ∈ Pred ({⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))}, No , 𝑐)𝜏 ∧ ∀𝑒 ∈ Pred ({⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))}, No , 𝑏)𝜎) ∧ ∀𝑓 ∈ Pred ({⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))}, No , 𝑐)𝜂) ↔ ((∀𝑑 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))∀𝑒 ∈ (( L ‘𝑏) ∪ ( R ‘𝑏))∀𝑓 ∈ (( L ‘𝑐) ∪ ( R ‘𝑐))𝜃 ∧ ∀𝑑 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))∀𝑒 ∈ (( L ‘𝑏) ∪ ( R ‘𝑏))𝜒 ∧ ∀𝑑 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))∀𝑓 ∈ (( L ‘𝑐) ∪ ( R ‘𝑐))𝜁) ∧ (∀𝑑 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))𝜓 ∧ ∀𝑒 ∈ (( L ‘𝑏) ∪ ( R ‘𝑏))∀𝑓 ∈ (( L ‘𝑐) ∪ ( R ‘𝑐))𝜏 ∧ ∀𝑒 ∈ (( L ‘𝑏) ∪ ( R ‘𝑏))𝜎) ∧ ∀𝑓 ∈ (( L ‘𝑐) ∪ ( R ‘𝑐))𝜂)))
37		no3inds.i	. . 3 ⊢ ((𝑎 ∈ No ∧ 𝑏 ∈ No ∧ 𝑐 ∈ No ) → (((∀𝑑 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))∀𝑒 ∈ (( L ‘𝑏) ∪ ( R ‘𝑏))∀𝑓 ∈ (( L ‘𝑐) ∪ ( R ‘𝑐))𝜃 ∧ ∀𝑑 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))∀𝑒 ∈ (( L ‘𝑏) ∪ ( R ‘𝑏))𝜒 ∧ ∀𝑑 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))∀𝑓 ∈ (( L ‘𝑐) ∪ ( R ‘𝑐))𝜁) ∧ (∀𝑑 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))𝜓 ∧ ∀𝑒 ∈ (( L ‘𝑏) ∪ ( R ‘𝑏))∀𝑓 ∈ (( L ‘𝑐) ∪ ( R ‘𝑐))𝜏 ∧ ∀𝑒 ∈ (( L ‘𝑏) ∪ ( R ‘𝑏))𝜎) ∧ ∀𝑓 ∈ (( L ‘𝑐) ∪ ( R ‘𝑐))𝜂) → 𝜑))
38	36, 37	sylbid 239	. 2 ⊢ ((𝑎 ∈ No ∧ 𝑏 ∈ No ∧ 𝑐 ∈ No ) → (((∀𝑑 ∈ Pred ({⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))}, No , 𝑎)∀𝑒 ∈ Pred ({⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))}, No , 𝑏)∀𝑓 ∈ Pred ({⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))}, No , 𝑐)𝜃 ∧ ∀𝑑 ∈ Pred ({⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))}, No , 𝑎)∀𝑒 ∈ Pred ({⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))}, No , 𝑏)𝜒 ∧ ∀𝑑 ∈ Pred ({⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))}, No , 𝑎)∀𝑓 ∈ Pred ({⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))}, No , 𝑐)𝜁) ∧ (∀𝑑 ∈ Pred ({⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))}, No , 𝑎)𝜓 ∧ ∀𝑒 ∈ Pred ({⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))}, No , 𝑏)∀𝑓 ∈ Pred ({⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))}, No , 𝑐)𝜏 ∧ ∀𝑒 ∈ Pred ({⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))}, No , 𝑏)𝜎) ∧ ∀𝑓 ∈ Pred ({⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))}, No , 𝑐)𝜂) → 𝜑))
39	1, 3, 4, 5, 3, 4, 5, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 38	xpord3ind 33800	1 ⊢ ((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ 𝑍 ∈ No ) → 𝜆)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 844   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106   ≠ wne 2943  ∀wral 3064   ∪ cun 3885   class class class wbr 5074  {copab 5136   × cxp 5587  Predcpred 6201  ‘cfv 6433  1^st c1st 7829  2^nd c2nd 7830   No csur 33843   L cleft 34029   R cright 34030

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  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 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-se 5545  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-1o 8297  df-2o 8298  df-no 33846  df-slt 33847  df-bday 33848  df-sslt 33976  df-scut 33978  df-made 34031  df-old 34032  df-left 34034  df-right 34035

This theorem is referenced by: (None)