Theorem no3inds 27964

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:   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 no3inds

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

Step	Hyp	Ref	Expression
1		eqid 2737	. . 3 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))} = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))}
2	1	lrrecfr 27949	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))} Fr No
3	1	lrrecpo 27947	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))} Po No
4	1	lrrecse 27948	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))} Se No
5		no3inds.1	. 2 ⊢ (𝑎 = 𝑑 → (𝜑 ↔ 𝜓))
6		no3inds.2	. 2 ⊢ (𝑏 = 𝑒 → (𝜓 ↔ 𝜒))
7		no3inds.3	. 2 ⊢ (𝑐 = 𝑓 → (𝜒 ↔ 𝜃))
8		no3inds.4	. 2 ⊢ (𝑎 = 𝑑 → (𝜏 ↔ 𝜃))
9		no3inds.5	. 2 ⊢ (𝑏 = 𝑒 → (𝜂 ↔ 𝜏))
10		no3inds.6	. 2 ⊢ (𝑏 = 𝑒 → (𝜁 ↔ 𝜃))
11		no3inds.7	. 2 ⊢ (𝑐 = 𝑓 → (𝜎 ↔ 𝜏))
12		no3inds.8	. 2 ⊢ (𝑎 = 𝑋 → (𝜑 ↔ 𝜌))
13		no3inds.9	. 2 ⊢ (𝑏 = 𝑌 → (𝜌 ↔ 𝜇))
14		no3inds.10	. 2 ⊢ (𝑐 = 𝑍 → (𝜇 ↔ 𝜆))
15	1	lrrecpred 27950	. . . . . . 7 ⊢ (𝑎 ∈ No → Pred({⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))}, No , 𝑎) = (( L ‘𝑎) ∪ ( R ‘𝑎)))
16	15	3ad2ant1 1134	. . . . . 6 ⊢ ((𝑎 ∈ No ∧ 𝑏 ∈ No ∧ 𝑐 ∈ No ) → Pred({⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))}, No , 𝑎) = (( L ‘𝑎) ∪ ( R ‘𝑎)))
17	1	lrrecpred 27950	. . . . . . . 8 ⊢ (𝑏 ∈ No → Pred({⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))}, No , 𝑏) = (( L ‘𝑏) ∪ ( R ‘𝑏)))
18	17	3ad2ant2 1135	. . . . . . 7 ⊢ ((𝑎 ∈ No ∧ 𝑏 ∈ No ∧ 𝑐 ∈ No ) → Pred({⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))}, No , 𝑏) = (( L ‘𝑏) ∪ ( R ‘𝑏)))
19	1	lrrecpred 27950	. . . . . . . . 9 ⊢ (𝑐 ∈ No → Pred({⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))}, No , 𝑐) = (( L ‘𝑐) ∪ ( R ‘𝑐)))
20	19	3ad2ant3 1136	. . . . . . . 8 ⊢ ((𝑎 ∈ No ∧ 𝑏 ∈ No ∧ 𝑐 ∈ No ) → Pred({⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))}, No , 𝑐) = (( L ‘𝑐) ∪ ( R ‘𝑐)))
21	20	raleqdv 3296	. . . . . . 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))
22	18, 21	raleqbidv 3312	. . . . . 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))
23	16, 22	raleqbidv 3312	. . . . 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))
24	18	raleqdv 3296	. . . . . 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))
25	16, 24	raleqbidv 3312	. . . . 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))
26	20	raleqdv 3296	. . . . . 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))
27	16, 26	raleqbidv 3312	. . . . 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))
28	23, 25, 27	3anbi123d 1439	. . . 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)))
29	16	raleqdv 3296	. . . . 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))
30	20	raleqdv 3296	. . . . . 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))
31	18, 30	raleqbidv 3312	. . . . 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))
32	18	raleqdv 3296	. . . . 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))
33	29, 31, 32	3anbi123d 1439	. . . 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)))
34	20	raleqdv 3296	. . . 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))
35	28, 33, 34	3anbi123d 1439	. . 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)))
36		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))
37	35, 36	sylbid 240	. 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))
38	2, 3, 4, 2, 3, 4, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 37	xpord3ind 8099	1 ⊢ ((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ 𝑍 ∈ No ) → 𝜆)

Syntax hints:   → wi 4   ↔ wb 206   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3052   ∪ cun 3888  {copab 5148  Predcpred 6258  ‘cfv 6492   No csur 27617   L cleft 27831   R cright 27832

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-tp 4573  df-op 4575  df-ot 4577  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-se 5578  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-1st 7935  df-2nd 7936  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-1o 8398  df-2o 8399  df-no 27620  df-lts 27621  df-bday 27622  df-slts 27764  df-cuts 27766  df-made 27833  df-old 27834  df-left 27836  df-right 27837

This theorem is referenced by:  leadds1  27995  addsass  28011  addsdi  28161  mulsass  28172