Theorem no3inds 28351

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 2760	. . 3 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ ((L‘𝑦) ∪ (R‘𝑦))} = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ ((L‘𝑦) ∪ (R‘𝑦))}
2	1	lrrecfr 28336	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ ((L‘𝑦) ∪ (R‘𝑦))} Fr No
3	1	lrrecpo 28334	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ ((L‘𝑦) ∪ (R‘𝑦))} Po No
4	1	lrrecse 28335	. 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 28337	. . . . . . 7 ⊢ (𝑎 ∈ No → Pred({⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ ((L‘𝑦) ∪ (R‘𝑦))}, No, 𝑎) = ((L‘𝑎) ∪ (R‘𝑎)))
16	15	3ad2ant1 1151	. . . . . 6 ⊢ ((𝑎 ∈ No ∧ 𝑏 ∈ No ∧ 𝑐 ∈ No) → Pred({⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ ((L‘𝑦) ∪ (R‘𝑦))}, No, 𝑎) = ((L‘𝑎) ∪ (R‘𝑎)))
17	1	lrrecpred 28337	. . . . . . . 8 ⊢ (𝑏 ∈ No → Pred({⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ ((L‘𝑦) ∪ (R‘𝑦))}, No, 𝑏) = ((L‘𝑏) ∪ (R‘𝑏)))
18	17	3ad2ant2 1152	. . . . . . 7 ⊢ ((𝑎 ∈ No ∧ 𝑏 ∈ No ∧ 𝑐 ∈ No) → Pred({⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ ((L‘𝑦) ∪ (R‘𝑦))}, No, 𝑏) = ((L‘𝑏) ∪ (R‘𝑏)))
19	1	lrrecpred 28337	. . . . . . . . 9 ⊢ (𝑐 ∈ No → Pred({⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ ((L‘𝑦) ∪ (R‘𝑦))}, No, 𝑐) = ((L‘𝑐) ∪ (R‘𝑐)))
20	19	3ad2ant3 1153	. . . . . . . 8 ⊢ ((𝑎 ∈ No ∧ 𝑏 ∈ No ∧ 𝑐 ∈ No) → Pred({⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ ((L‘𝑦) ∪ (R‘𝑦))}, No, 𝑐) = ((L‘𝑐) ∪ (R‘𝑐)))
21	20	raleqdv 3319	. . . . . . 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 3334	. . . . . 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 3334	. . . . 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 3319	. . . . . 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 3334	. . . . 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 3319	. . . . . 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 3334	. . . . 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 1464	. . . 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 3319	. . . . 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 3319	. . . . . 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 3334	. . . . 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 3319	. . . . 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 1464	. . . 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 3319	. . . 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 1464	. . 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 243	. 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 8177	1 ⊢ ((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ 𝑍 ∈ No) → 𝜆)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144  ∀wral 3076   ∪ cun 4148  {copab 5166  Predcpred 6305  ‘cfv 6541  Nocsur 28004  Lcleft 28218  Rcright 28219

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7753

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-pss 4013  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4835  df-ot 4858  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-se 5604  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6306  df-ord 6367  df-on 6368  df-suc 6370  df-iota 6496  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-riota 7379  df-ov 7425  df-oprab 7426  df-mpo 7427  df-1st 8003  df-2nd 8004  df-frecs 8304  df-wrecs 8335  df-recs 8384  df-1o 8481  df-2o 8482  df-no 28007  df-lts 28008  df-bday 28009  df-slts 28151  df-cuts 28153  df-made 28220  df-old 28221  df-left 28223  df-right 28224

This theorem is used by:  leadds1  28382  addsass  28398  addsdi  28548  mulsass  28559