Theorem xpord2indlem 8130

Description: Induction over the Cartesian product ordering. Note that the substitutions cover all possible cases of membership in the predecessor class. (Contributed by Scott Fenton, 22-Aug-2024.)

Hypotheses

Ref	Expression
xpord2.1	⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐴 × 𝐵) ∧ 𝑦 ∈ (𝐴 × 𝐵) ∧ (((1st ‘𝑥)𝑅(1st ‘𝑦) ∨ (1st ‘𝑥) = (1st ‘𝑦)) ∧ ((2nd ‘𝑥)𝑆(2nd ‘𝑦) ∨ (2nd ‘𝑥) = (2nd ‘𝑦)) ∧ 𝑥 ≠ 𝑦))}
xpord2indlem.1	⊢ 𝑅 Fr 𝐴
xpord2indlem.2	⊢ 𝑅 Po 𝐴
xpord2indlem.3	⊢ 𝑅 Se 𝐴
xpord2indlem.4	⊢ 𝑆 Fr 𝐵
xpord2indlem.5	⊢ 𝑆 Po 𝐵
xpord2indlem.6	⊢ 𝑆 Se 𝐵
xpord2indlem.7	⊢ (𝑎 = 𝑐 → (𝜑 ↔ 𝜓))
xpord2indlem.8	⊢ (𝑏 = 𝑑 → (𝜓 ↔ 𝜒))
xpord2indlem.9	⊢ (𝑎 = 𝑐 → (𝜃 ↔ 𝜒))
xpord2indlem.11	⊢ (𝑎 = 𝑋 → (𝜑 ↔ 𝜏))
xpord2indlem.12	⊢ (𝑏 = 𝑌 → (𝜏 ↔ 𝜂))
xpord2indlem.i	⊢ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → ((∀𝑐 ∈ Pred (𝑅, 𝐴, 𝑎)∀𝑑 ∈ Pred (𝑆, 𝐵, 𝑏)𝜒 ∧ ∀𝑐 ∈ Pred (𝑅, 𝐴, 𝑎)𝜓 ∧ ∀𝑑 ∈ Pred (𝑆, 𝐵, 𝑏)𝜃) → 𝜑))

Assertion

Ref	Expression
xpord2indlem	⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → 𝜂)

Distinct variable groups:   𝐴,𝑎,𝑏,𝑐,𝑑   𝜓,𝑎   𝜏,𝑎   𝑥,𝐴,𝑦   𝐵,𝑎,𝑏,𝑐,𝑑   𝜒,𝑏   𝜂,𝑏   𝑥,𝐵,𝑦   𝜑,𝑐   𝜃,𝑐   𝜓,𝑑   𝑅,𝑐,𝑑   𝑥,𝑅,𝑦   𝑆,𝑐,𝑑   𝑥,𝑆,𝑦   𝑇,𝑎,𝑏,𝑐,𝑑   𝑋,𝑎,𝑏   𝑌,𝑏

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑎,𝑏,𝑑)   𝜓(𝑥,𝑦,𝑏,𝑐)   𝜒(𝑥,𝑦,𝑎,𝑐,𝑑)   𝜃(𝑥,𝑦,𝑎,𝑏,𝑑)   𝜏(𝑥,𝑦,𝑏,𝑐,𝑑)   𝜂(𝑥,𝑦,𝑎,𝑐,𝑑)   𝑅(𝑎,𝑏)   𝑆(𝑎,𝑏)   𝑇(𝑥,𝑦)   𝑋(𝑥,𝑦,𝑐,𝑑)   𝑌(𝑥,𝑦,𝑎,𝑐,𝑑)

Proof of Theorem xpord2indlem

Step	Hyp	Ref	Expression
1		xpord2.1	. . . . 5 ⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐴 × 𝐵) ∧ 𝑦 ∈ (𝐴 × 𝐵) ∧ (((1st ‘𝑥)𝑅(1st ‘𝑦) ∨ (1st ‘𝑥) = (1st ‘𝑦)) ∧ ((2nd ‘𝑥)𝑆(2nd ‘𝑦) ∨ (2nd ‘𝑥) = (2nd ‘𝑦)) ∧ 𝑥 ≠ 𝑦))}
2		xpord2indlem.1	. . . . . 6 ⊢ 𝑅 Fr 𝐴
3	2	a1i 11	. . . . 5 ⊢ (⊤ → 𝑅 Fr 𝐴)
4		xpord2indlem.4	. . . . . 6 ⊢ 𝑆 Fr 𝐵
5	4	a1i 11	. . . . 5 ⊢ (⊤ → 𝑆 Fr 𝐵)
6	1, 3, 5	frxp2 8127	. . . 4 ⊢ (⊤ → 𝑇 Fr (𝐴 × 𝐵))
7		xpord2indlem.2	. . . . . 6 ⊢ 𝑅 Po 𝐴
8	7	a1i 11	. . . . 5 ⊢ (⊤ → 𝑅 Po 𝐴)
9		xpord2indlem.5	. . . . . 6 ⊢ 𝑆 Po 𝐵
10	9	a1i 11	. . . . 5 ⊢ (⊤ → 𝑆 Po 𝐵)
11	1, 8, 10	poxp2 8126	. . . 4 ⊢ (⊤ → 𝑇 Po (𝐴 × 𝐵))
12		xpord2indlem.3	. . . . . 6 ⊢ 𝑅 Se 𝐴
13	12	a1i 11	. . . . 5 ⊢ (⊤ → 𝑅 Se 𝐴)
14		xpord2indlem.6	. . . . . 6 ⊢ 𝑆 Se 𝐵
15	14	a1i 11	. . . . 5 ⊢ (⊤ → 𝑆 Se 𝐵)
16	1, 13, 15	sexp2 8129	. . . 4 ⊢ (⊤ → 𝑇 Se (𝐴 × 𝐵))
17	6, 11, 16	3jca 1129	. . 3 ⊢ (⊤ → (𝑇 Fr (𝐴 × 𝐵) ∧ 𝑇 Po (𝐴 × 𝐵) ∧ 𝑇 Se (𝐴 × 𝐵)))
18	17	mptru 1549	. 2 ⊢ (𝑇 Fr (𝐴 × 𝐵) ∧ 𝑇 Po (𝐴 × 𝐵) ∧ 𝑇 Se (𝐴 × 𝐵))
19	1	xpord2pred 8128	. . . . . . . . 9 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → Pred(𝑇, (𝐴 × 𝐵), ⟨𝑎, 𝑏⟩) = (((Pred(𝑅, 𝐴, 𝑎) ∪ {𝑎}) × (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏})) ∖ {⟨𝑎, 𝑏⟩}))
20	19	eleq2d 2820	. . . . . . . 8 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → (⟨𝑐, 𝑑⟩ ∈ Pred(𝑇, (𝐴 × 𝐵), ⟨𝑎, 𝑏⟩) ↔ ⟨𝑐, 𝑑⟩ ∈ (((Pred(𝑅, 𝐴, 𝑎) ∪ {𝑎}) × (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏})) ∖ {⟨𝑎, 𝑏⟩})))
21	20	imbi1d 342	. . . . . . 7 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → ((⟨𝑐, 𝑑⟩ ∈ Pred(𝑇, (𝐴 × 𝐵), ⟨𝑎, 𝑏⟩) → 𝜒) ↔ (⟨𝑐, 𝑑⟩ ∈ (((Pred(𝑅, 𝐴, 𝑎) ∪ {𝑎}) × (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏})) ∖ {⟨𝑎, 𝑏⟩}) → 𝜒)))
22		eldif 3958	. . . . . . . . . 10 ⊢ (⟨𝑐, 𝑑⟩ ∈ (((Pred(𝑅, 𝐴, 𝑎) ∪ {𝑎}) × (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏})) ∖ {⟨𝑎, 𝑏⟩}) ↔ (⟨𝑐, 𝑑⟩ ∈ ((Pred(𝑅, 𝐴, 𝑎) ∪ {𝑎}) × (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏})) ∧ ¬ ⟨𝑐, 𝑑⟩ ∈ {⟨𝑎, 𝑏⟩}))
23		opelxp 5712	. . . . . . . . . . 11 ⊢ (⟨𝑐, 𝑑⟩ ∈ ((Pred(𝑅, 𝐴, 𝑎) ∪ {𝑎}) × (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏})) ↔ (𝑐 ∈ (Pred(𝑅, 𝐴, 𝑎) ∪ {𝑎}) ∧ 𝑑 ∈ (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏})))
24		opex 5464	. . . . . . . . . . . . . 14 ⊢ ⟨𝑐, 𝑑⟩ ∈ V
25	24	elsn 4643	. . . . . . . . . . . . 13 ⊢ (⟨𝑐, 𝑑⟩ ∈ {⟨𝑎, 𝑏⟩} ↔ ⟨𝑐, 𝑑⟩ = ⟨𝑎, 𝑏⟩)
26	25	notbii 320	. . . . . . . . . . . 12 ⊢ (¬ ⟨𝑐, 𝑑⟩ ∈ {⟨𝑎, 𝑏⟩} ↔ ¬ ⟨𝑐, 𝑑⟩ = ⟨𝑎, 𝑏⟩)
27		df-ne 2942	. . . . . . . . . . . 12 ⊢ (⟨𝑐, 𝑑⟩ ≠ ⟨𝑎, 𝑏⟩ ↔ ¬ ⟨𝑐, 𝑑⟩ = ⟨𝑎, 𝑏⟩)
28		vex 3479	. . . . . . . . . . . . 13 ⊢ 𝑐 ∈ V
29		vex 3479	. . . . . . . . . . . . 13 ⊢ 𝑑 ∈ V
30	28, 29	opthne 5482	. . . . . . . . . . . 12 ⊢ (⟨𝑐, 𝑑⟩ ≠ ⟨𝑎, 𝑏⟩ ↔ (𝑐 ≠ 𝑎 ∨ 𝑑 ≠ 𝑏))
31	26, 27, 30	3bitr2i 299	. . . . . . . . . . 11 ⊢ (¬ ⟨𝑐, 𝑑⟩ ∈ {⟨𝑎, 𝑏⟩} ↔ (𝑐 ≠ 𝑎 ∨ 𝑑 ≠ 𝑏))
32	23, 31	anbi12i 628	. . . . . . . . . 10 ⊢ ((⟨𝑐, 𝑑⟩ ∈ ((Pred(𝑅, 𝐴, 𝑎) ∪ {𝑎}) × (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏})) ∧ ¬ ⟨𝑐, 𝑑⟩ ∈ {⟨𝑎, 𝑏⟩}) ↔ ((𝑐 ∈ (Pred(𝑅, 𝐴, 𝑎) ∪ {𝑎}) ∧ 𝑑 ∈ (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏})) ∧ (𝑐 ≠ 𝑎 ∨ 𝑑 ≠ 𝑏)))
33	22, 32	bitri 275	. . . . . . . . 9 ⊢ (⟨𝑐, 𝑑⟩ ∈ (((Pred(𝑅, 𝐴, 𝑎) ∪ {𝑎}) × (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏})) ∖ {⟨𝑎, 𝑏⟩}) ↔ ((𝑐 ∈ (Pred(𝑅, 𝐴, 𝑎) ∪ {𝑎}) ∧ 𝑑 ∈ (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏})) ∧ (𝑐 ≠ 𝑎 ∨ 𝑑 ≠ 𝑏)))
34	33	imbi1i 350	. . . . . . . 8 ⊢ ((⟨𝑐, 𝑑⟩ ∈ (((Pred(𝑅, 𝐴, 𝑎) ∪ {𝑎}) × (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏})) ∖ {⟨𝑎, 𝑏⟩}) → 𝜒) ↔ (((𝑐 ∈ (Pred(𝑅, 𝐴, 𝑎) ∪ {𝑎}) ∧ 𝑑 ∈ (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏})) ∧ (𝑐 ≠ 𝑎 ∨ 𝑑 ≠ 𝑏)) → 𝜒))
35		impexp 452	. . . . . . . 8 ⊢ ((((𝑐 ∈ (Pred(𝑅, 𝐴, 𝑎) ∪ {𝑎}) ∧ 𝑑 ∈ (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏})) ∧ (𝑐 ≠ 𝑎 ∨ 𝑑 ≠ 𝑏)) → 𝜒) ↔ ((𝑐 ∈ (Pred(𝑅, 𝐴, 𝑎) ∪ {𝑎}) ∧ 𝑑 ∈ (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏})) → ((𝑐 ≠ 𝑎 ∨ 𝑑 ≠ 𝑏) → 𝜒)))
36	34, 35	bitri 275	. . . . . . 7 ⊢ ((⟨𝑐, 𝑑⟩ ∈ (((Pred(𝑅, 𝐴, 𝑎) ∪ {𝑎}) × (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏})) ∖ {⟨𝑎, 𝑏⟩}) → 𝜒) ↔ ((𝑐 ∈ (Pred(𝑅, 𝐴, 𝑎) ∪ {𝑎}) ∧ 𝑑 ∈ (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏})) → ((𝑐 ≠ 𝑎 ∨ 𝑑 ≠ 𝑏) → 𝜒)))
37	21, 36	bitrdi 287	. . . . . 6 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → ((⟨𝑐, 𝑑⟩ ∈ Pred(𝑇, (𝐴 × 𝐵), ⟨𝑎, 𝑏⟩) → 𝜒) ↔ ((𝑐 ∈ (Pred(𝑅, 𝐴, 𝑎) ∪ {𝑎}) ∧ 𝑑 ∈ (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏})) → ((𝑐 ≠ 𝑎 ∨ 𝑑 ≠ 𝑏) → 𝜒))))
38	37	2albidv 1927	. . . . 5 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → (∀𝑐∀𝑑(⟨𝑐, 𝑑⟩ ∈ Pred(𝑇, (𝐴 × 𝐵), ⟨𝑎, 𝑏⟩) → 𝜒) ↔ ∀𝑐∀𝑑((𝑐 ∈ (Pred(𝑅, 𝐴, 𝑎) ∪ {𝑎}) ∧ 𝑑 ∈ (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏})) → ((𝑐 ≠ 𝑎 ∨ 𝑑 ≠ 𝑏) → 𝜒))))
39		r2al 3195	. . . . 5 ⊢ (∀𝑐 ∈ (Pred(𝑅, 𝐴, 𝑎) ∪ {𝑎})∀𝑑 ∈ (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏})((𝑐 ≠ 𝑎 ∨ 𝑑 ≠ 𝑏) → 𝜒) ↔ ∀𝑐∀𝑑((𝑐 ∈ (Pred(𝑅, 𝐴, 𝑎) ∪ {𝑎}) ∧ 𝑑 ∈ (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏})) → ((𝑐 ≠ 𝑎 ∨ 𝑑 ≠ 𝑏) → 𝜒)))
40	38, 39	bitr4di 289	. . . 4 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → (∀𝑐∀𝑑(⟨𝑐, 𝑑⟩ ∈ Pred(𝑇, (𝐴 × 𝐵), ⟨𝑎, 𝑏⟩) → 𝜒) ↔ ∀𝑐 ∈ (Pred(𝑅, 𝐴, 𝑎) ∪ {𝑎})∀𝑑 ∈ (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏})((𝑐 ≠ 𝑎 ∨ 𝑑 ≠ 𝑏) → 𝜒)))
41		ssun1 4172	. . . . . . . . 9 ⊢ Pred(𝑅, 𝐴, 𝑎) ⊆ (Pred(𝑅, 𝐴, 𝑎) ∪ {𝑎})
42		ssralv 4050	. . . . . . . . 9 ⊢ (Pred(𝑅, 𝐴, 𝑎) ⊆ (Pred(𝑅, 𝐴, 𝑎) ∪ {𝑎}) → (∀𝑐 ∈ (Pred(𝑅, 𝐴, 𝑎) ∪ {𝑎})∀𝑑 ∈ (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏})((𝑐 ≠ 𝑎 ∨ 𝑑 ≠ 𝑏) → 𝜒) → ∀𝑐 ∈ Pred (𝑅, 𝐴, 𝑎)∀𝑑 ∈ (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏})((𝑐 ≠ 𝑎 ∨ 𝑑 ≠ 𝑏) → 𝜒)))
43	41, 42	ax-mp 5	. . . . . . . 8 ⊢ (∀𝑐 ∈ (Pred(𝑅, 𝐴, 𝑎) ∪ {𝑎})∀𝑑 ∈ (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏})((𝑐 ≠ 𝑎 ∨ 𝑑 ≠ 𝑏) → 𝜒) → ∀𝑐 ∈ Pred (𝑅, 𝐴, 𝑎)∀𝑑 ∈ (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏})((𝑐 ≠ 𝑎 ∨ 𝑑 ≠ 𝑏) → 𝜒))
44		ssun1 4172	. . . . . . . . . 10 ⊢ Pred(𝑆, 𝐵, 𝑏) ⊆ (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏})
45		ssralv 4050	. . . . . . . . . 10 ⊢ (Pred(𝑆, 𝐵, 𝑏) ⊆ (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏}) → (∀𝑑 ∈ (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏})((𝑐 ≠ 𝑎 ∨ 𝑑 ≠ 𝑏) → 𝜒) → ∀𝑑 ∈ Pred (𝑆, 𝐵, 𝑏)((𝑐 ≠ 𝑎 ∨ 𝑑 ≠ 𝑏) → 𝜒)))
46	44, 45	ax-mp 5	. . . . . . . . 9 ⊢ (∀𝑑 ∈ (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏})((𝑐 ≠ 𝑎 ∨ 𝑑 ≠ 𝑏) → 𝜒) → ∀𝑑 ∈ Pred (𝑆, 𝐵, 𝑏)((𝑐 ≠ 𝑎 ∨ 𝑑 ≠ 𝑏) → 𝜒))
47	46	ralimi 3084	. . . . . . . 8 ⊢ (∀𝑐 ∈ Pred (𝑅, 𝐴, 𝑎)∀𝑑 ∈ (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏})((𝑐 ≠ 𝑎 ∨ 𝑑 ≠ 𝑏) → 𝜒) → ∀𝑐 ∈ Pred (𝑅, 𝐴, 𝑎)∀𝑑 ∈ Pred (𝑆, 𝐵, 𝑏)((𝑐 ≠ 𝑎 ∨ 𝑑 ≠ 𝑏) → 𝜒))
48	43, 47	syl 17	. . . . . . 7 ⊢ (∀𝑐 ∈ (Pred(𝑅, 𝐴, 𝑎) ∪ {𝑎})∀𝑑 ∈ (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏})((𝑐 ≠ 𝑎 ∨ 𝑑 ≠ 𝑏) → 𝜒) → ∀𝑐 ∈ Pred (𝑅, 𝐴, 𝑎)∀𝑑 ∈ Pred (𝑆, 𝐵, 𝑏)((𝑐 ≠ 𝑎 ∨ 𝑑 ≠ 𝑏) → 𝜒))
49		predpoirr 6332	. . . . . . . . . . . . . 14 ⊢ (𝑆 Po 𝐵 → ¬ 𝑏 ∈ Pred(𝑆, 𝐵, 𝑏))
50	9, 49	ax-mp 5	. . . . . . . . . . . . 13 ⊢ ¬ 𝑏 ∈ Pred(𝑆, 𝐵, 𝑏)
51		eleq1w 2817	. . . . . . . . . . . . 13 ⊢ (𝑑 = 𝑏 → (𝑑 ∈ Pred(𝑆, 𝐵, 𝑏) ↔ 𝑏 ∈ Pred(𝑆, 𝐵, 𝑏)))
52	50, 51	mtbiri 327	. . . . . . . . . . . 12 ⊢ (𝑑 = 𝑏 → ¬ 𝑑 ∈ Pred(𝑆, 𝐵, 𝑏))
53	52	necon2ai 2971	. . . . . . . . . . 11 ⊢ (𝑑 ∈ Pred(𝑆, 𝐵, 𝑏) → 𝑑 ≠ 𝑏)
54	53	olcd 873	. . . . . . . . . 10 ⊢ (𝑑 ∈ Pred(𝑆, 𝐵, 𝑏) → (𝑐 ≠ 𝑎 ∨ 𝑑 ≠ 𝑏))
55		pm2.27 42	. . . . . . . . . 10 ⊢ ((𝑐 ≠ 𝑎 ∨ 𝑑 ≠ 𝑏) → (((𝑐 ≠ 𝑎 ∨ 𝑑 ≠ 𝑏) → 𝜒) → 𝜒))
56	54, 55	syl 17	. . . . . . . . 9 ⊢ (𝑑 ∈ Pred(𝑆, 𝐵, 𝑏) → (((𝑐 ≠ 𝑎 ∨ 𝑑 ≠ 𝑏) → 𝜒) → 𝜒))
57	56	ralimia 3081	. . . . . . . 8 ⊢ (∀𝑑 ∈ Pred (𝑆, 𝐵, 𝑏)((𝑐 ≠ 𝑎 ∨ 𝑑 ≠ 𝑏) → 𝜒) → ∀𝑑 ∈ Pred (𝑆, 𝐵, 𝑏)𝜒)
58	57	ralimi 3084	. . . . . . 7 ⊢ (∀𝑐 ∈ Pred (𝑅, 𝐴, 𝑎)∀𝑑 ∈ Pred (𝑆, 𝐵, 𝑏)((𝑐 ≠ 𝑎 ∨ 𝑑 ≠ 𝑏) → 𝜒) → ∀𝑐 ∈ Pred (𝑅, 𝐴, 𝑎)∀𝑑 ∈ Pred (𝑆, 𝐵, 𝑏)𝜒)
59	48, 58	syl 17	. . . . . 6 ⊢ (∀𝑐 ∈ (Pred(𝑅, 𝐴, 𝑎) ∪ {𝑎})∀𝑑 ∈ (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏})((𝑐 ≠ 𝑎 ∨ 𝑑 ≠ 𝑏) → 𝜒) → ∀𝑐 ∈ Pred (𝑅, 𝐴, 𝑎)∀𝑑 ∈ Pred (𝑆, 𝐵, 𝑏)𝜒)
60		ssun2 4173	. . . . . . . . . . 11 ⊢ {𝑏} ⊆ (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏})
61		ssralv 4050	. . . . . . . . . . 11 ⊢ ({𝑏} ⊆ (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏}) → (∀𝑑 ∈ (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏})((𝑐 ≠ 𝑎 ∨ 𝑑 ≠ 𝑏) → 𝜒) → ∀𝑑 ∈ {𝑏} ((𝑐 ≠ 𝑎 ∨ 𝑑 ≠ 𝑏) → 𝜒)))
62	60, 61	ax-mp 5	. . . . . . . . . 10 ⊢ (∀𝑑 ∈ (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏})((𝑐 ≠ 𝑎 ∨ 𝑑 ≠ 𝑏) → 𝜒) → ∀𝑑 ∈ {𝑏} ((𝑐 ≠ 𝑎 ∨ 𝑑 ≠ 𝑏) → 𝜒))
63	62	ralimi 3084	. . . . . . . . 9 ⊢ (∀𝑐 ∈ Pred (𝑅, 𝐴, 𝑎)∀𝑑 ∈ (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏})((𝑐 ≠ 𝑎 ∨ 𝑑 ≠ 𝑏) → 𝜒) → ∀𝑐 ∈ Pred (𝑅, 𝐴, 𝑎)∀𝑑 ∈ {𝑏} ((𝑐 ≠ 𝑎 ∨ 𝑑 ≠ 𝑏) → 𝜒))
64	43, 63	syl 17	. . . . . . . 8 ⊢ (∀𝑐 ∈ (Pred(𝑅, 𝐴, 𝑎) ∪ {𝑎})∀𝑑 ∈ (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏})((𝑐 ≠ 𝑎 ∨ 𝑑 ≠ 𝑏) → 𝜒) → ∀𝑐 ∈ Pred (𝑅, 𝐴, 𝑎)∀𝑑 ∈ {𝑏} ((𝑐 ≠ 𝑎 ∨ 𝑑 ≠ 𝑏) → 𝜒))
65		vex 3479	. . . . . . . . . 10 ⊢ 𝑏 ∈ V
66		neeq1 3004	. . . . . . . . . . . 12 ⊢ (𝑑 = 𝑏 → (𝑑 ≠ 𝑏 ↔ 𝑏 ≠ 𝑏))
67	66	orbi2d 915	. . . . . . . . . . 11 ⊢ (𝑑 = 𝑏 → ((𝑐 ≠ 𝑎 ∨ 𝑑 ≠ 𝑏) ↔ (𝑐 ≠ 𝑎 ∨ 𝑏 ≠ 𝑏)))
68		xpord2indlem.8	. . . . . . . . . . . . 13 ⊢ (𝑏 = 𝑑 → (𝜓 ↔ 𝜒))
69	68	equcoms 2024	. . . . . . . . . . . 12 ⊢ (𝑑 = 𝑏 → (𝜓 ↔ 𝜒))
70	69	bicomd 222	. . . . . . . . . . 11 ⊢ (𝑑 = 𝑏 → (𝜒 ↔ 𝜓))
71	67, 70	imbi12d 345	. . . . . . . . . 10 ⊢ (𝑑 = 𝑏 → (((𝑐 ≠ 𝑎 ∨ 𝑑 ≠ 𝑏) → 𝜒) ↔ ((𝑐 ≠ 𝑎 ∨ 𝑏 ≠ 𝑏) → 𝜓)))
72	65, 71	ralsn 4685	. . . . . . . . 9 ⊢ (∀𝑑 ∈ {𝑏} ((𝑐 ≠ 𝑎 ∨ 𝑑 ≠ 𝑏) → 𝜒) ↔ ((𝑐 ≠ 𝑎 ∨ 𝑏 ≠ 𝑏) → 𝜓))
73	72	ralbii 3094	. . . . . . . 8 ⊢ (∀𝑐 ∈ Pred (𝑅, 𝐴, 𝑎)∀𝑑 ∈ {𝑏} ((𝑐 ≠ 𝑎 ∨ 𝑑 ≠ 𝑏) → 𝜒) ↔ ∀𝑐 ∈ Pred (𝑅, 𝐴, 𝑎)((𝑐 ≠ 𝑎 ∨ 𝑏 ≠ 𝑏) → 𝜓))
74	64, 73	sylib 217	. . . . . . 7 ⊢ (∀𝑐 ∈ (Pred(𝑅, 𝐴, 𝑎) ∪ {𝑎})∀𝑑 ∈ (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏})((𝑐 ≠ 𝑎 ∨ 𝑑 ≠ 𝑏) → 𝜒) → ∀𝑐 ∈ Pred (𝑅, 𝐴, 𝑎)((𝑐 ≠ 𝑎 ∨ 𝑏 ≠ 𝑏) → 𝜓))
75		predpoirr 6332	. . . . . . . . . . . . 13 ⊢ (𝑅 Po 𝐴 → ¬ 𝑎 ∈ Pred(𝑅, 𝐴, 𝑎))
76	7, 75	ax-mp 5	. . . . . . . . . . . 12 ⊢ ¬ 𝑎 ∈ Pred(𝑅, 𝐴, 𝑎)
77		eleq1w 2817	. . . . . . . . . . . 12 ⊢ (𝑐 = 𝑎 → (𝑐 ∈ Pred(𝑅, 𝐴, 𝑎) ↔ 𝑎 ∈ Pred(𝑅, 𝐴, 𝑎)))
78	76, 77	mtbiri 327	. . . . . . . . . . 11 ⊢ (𝑐 = 𝑎 → ¬ 𝑐 ∈ Pred(𝑅, 𝐴, 𝑎))
79	78	necon2ai 2971	. . . . . . . . . 10 ⊢ (𝑐 ∈ Pred(𝑅, 𝐴, 𝑎) → 𝑐 ≠ 𝑎)
80	79	orcd 872	. . . . . . . . 9 ⊢ (𝑐 ∈ Pred(𝑅, 𝐴, 𝑎) → (𝑐 ≠ 𝑎 ∨ 𝑏 ≠ 𝑏))
81		pm2.27 42	. . . . . . . . 9 ⊢ ((𝑐 ≠ 𝑎 ∨ 𝑏 ≠ 𝑏) → (((𝑐 ≠ 𝑎 ∨ 𝑏 ≠ 𝑏) → 𝜓) → 𝜓))
82	80, 81	syl 17	. . . . . . . 8 ⊢ (𝑐 ∈ Pred(𝑅, 𝐴, 𝑎) → (((𝑐 ≠ 𝑎 ∨ 𝑏 ≠ 𝑏) → 𝜓) → 𝜓))
83	82	ralimia 3081	. . . . . . 7 ⊢ (∀𝑐 ∈ Pred (𝑅, 𝐴, 𝑎)((𝑐 ≠ 𝑎 ∨ 𝑏 ≠ 𝑏) → 𝜓) → ∀𝑐 ∈ Pred (𝑅, 𝐴, 𝑎)𝜓)
84	74, 83	syl 17	. . . . . 6 ⊢ (∀𝑐 ∈ (Pred(𝑅, 𝐴, 𝑎) ∪ {𝑎})∀𝑑 ∈ (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏})((𝑐 ≠ 𝑎 ∨ 𝑑 ≠ 𝑏) → 𝜒) → ∀𝑐 ∈ Pred (𝑅, 𝐴, 𝑎)𝜓)
85		ssun2 4173	. . . . . . . . . 10 ⊢ {𝑎} ⊆ (Pred(𝑅, 𝐴, 𝑎) ∪ {𝑎})
86		ssralv 4050	. . . . . . . . . 10 ⊢ ({𝑎} ⊆ (Pred(𝑅, 𝐴, 𝑎) ∪ {𝑎}) → (∀𝑐 ∈ (Pred(𝑅, 𝐴, 𝑎) ∪ {𝑎})∀𝑑 ∈ (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏})((𝑐 ≠ 𝑎 ∨ 𝑑 ≠ 𝑏) → 𝜒) → ∀𝑐 ∈ {𝑎}∀𝑑 ∈ (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏})((𝑐 ≠ 𝑎 ∨ 𝑑 ≠ 𝑏) → 𝜒)))
87	85, 86	ax-mp 5	. . . . . . . . 9 ⊢ (∀𝑐 ∈ (Pred(𝑅, 𝐴, 𝑎) ∪ {𝑎})∀𝑑 ∈ (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏})((𝑐 ≠ 𝑎 ∨ 𝑑 ≠ 𝑏) → 𝜒) → ∀𝑐 ∈ {𝑎}∀𝑑 ∈ (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏})((𝑐 ≠ 𝑎 ∨ 𝑑 ≠ 𝑏) → 𝜒))
88	46	ralimi 3084	. . . . . . . . 9 ⊢ (∀𝑐 ∈ {𝑎}∀𝑑 ∈ (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏})((𝑐 ≠ 𝑎 ∨ 𝑑 ≠ 𝑏) → 𝜒) → ∀𝑐 ∈ {𝑎}∀𝑑 ∈ Pred (𝑆, 𝐵, 𝑏)((𝑐 ≠ 𝑎 ∨ 𝑑 ≠ 𝑏) → 𝜒))
89	87, 88	syl 17	. . . . . . . 8 ⊢ (∀𝑐 ∈ (Pred(𝑅, 𝐴, 𝑎) ∪ {𝑎})∀𝑑 ∈ (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏})((𝑐 ≠ 𝑎 ∨ 𝑑 ≠ 𝑏) → 𝜒) → ∀𝑐 ∈ {𝑎}∀𝑑 ∈ Pred (𝑆, 𝐵, 𝑏)((𝑐 ≠ 𝑎 ∨ 𝑑 ≠ 𝑏) → 𝜒))
90		vex 3479	. . . . . . . . 9 ⊢ 𝑎 ∈ V
91		neeq1 3004	. . . . . . . . . . . 12 ⊢ (𝑐 = 𝑎 → (𝑐 ≠ 𝑎 ↔ 𝑎 ≠ 𝑎))
92	91	orbi1d 916	. . . . . . . . . . 11 ⊢ (𝑐 = 𝑎 → ((𝑐 ≠ 𝑎 ∨ 𝑑 ≠ 𝑏) ↔ (𝑎 ≠ 𝑎 ∨ 𝑑 ≠ 𝑏)))
93		xpord2indlem.9	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑐 → (𝜃 ↔ 𝜒))
94	93	equcoms 2024	. . . . . . . . . . . 12 ⊢ (𝑐 = 𝑎 → (𝜃 ↔ 𝜒))
95	94	bicomd 222	. . . . . . . . . . 11 ⊢ (𝑐 = 𝑎 → (𝜒 ↔ 𝜃))
96	92, 95	imbi12d 345	. . . . . . . . . 10 ⊢ (𝑐 = 𝑎 → (((𝑐 ≠ 𝑎 ∨ 𝑑 ≠ 𝑏) → 𝜒) ↔ ((𝑎 ≠ 𝑎 ∨ 𝑑 ≠ 𝑏) → 𝜃)))
97	96	ralbidv 3178	. . . . . . . . 9 ⊢ (𝑐 = 𝑎 → (∀𝑑 ∈ Pred (𝑆, 𝐵, 𝑏)((𝑐 ≠ 𝑎 ∨ 𝑑 ≠ 𝑏) → 𝜒) ↔ ∀𝑑 ∈ Pred (𝑆, 𝐵, 𝑏)((𝑎 ≠ 𝑎 ∨ 𝑑 ≠ 𝑏) → 𝜃)))
98	90, 97	ralsn 4685	. . . . . . . 8 ⊢ (∀𝑐 ∈ {𝑎}∀𝑑 ∈ Pred (𝑆, 𝐵, 𝑏)((𝑐 ≠ 𝑎 ∨ 𝑑 ≠ 𝑏) → 𝜒) ↔ ∀𝑑 ∈ Pred (𝑆, 𝐵, 𝑏)((𝑎 ≠ 𝑎 ∨ 𝑑 ≠ 𝑏) → 𝜃))
99	89, 98	sylib 217	. . . . . . 7 ⊢ (∀𝑐 ∈ (Pred(𝑅, 𝐴, 𝑎) ∪ {𝑎})∀𝑑 ∈ (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏})((𝑐 ≠ 𝑎 ∨ 𝑑 ≠ 𝑏) → 𝜒) → ∀𝑑 ∈ Pred (𝑆, 𝐵, 𝑏)((𝑎 ≠ 𝑎 ∨ 𝑑 ≠ 𝑏) → 𝜃))
100	53	olcd 873	. . . . . . . . 9 ⊢ (𝑑 ∈ Pred(𝑆, 𝐵, 𝑏) → (𝑎 ≠ 𝑎 ∨ 𝑑 ≠ 𝑏))
101		pm2.27 42	. . . . . . . . 9 ⊢ ((𝑎 ≠ 𝑎 ∨ 𝑑 ≠ 𝑏) → (((𝑎 ≠ 𝑎 ∨ 𝑑 ≠ 𝑏) → 𝜃) → 𝜃))
102	100, 101	syl 17	. . . . . . . 8 ⊢ (𝑑 ∈ Pred(𝑆, 𝐵, 𝑏) → (((𝑎 ≠ 𝑎 ∨ 𝑑 ≠ 𝑏) → 𝜃) → 𝜃))
103	102	ralimia 3081	. . . . . . 7 ⊢ (∀𝑑 ∈ Pred (𝑆, 𝐵, 𝑏)((𝑎 ≠ 𝑎 ∨ 𝑑 ≠ 𝑏) → 𝜃) → ∀𝑑 ∈ Pred (𝑆, 𝐵, 𝑏)𝜃)
104	99, 103	syl 17	. . . . . 6 ⊢ (∀𝑐 ∈ (Pred(𝑅, 𝐴, 𝑎) ∪ {𝑎})∀𝑑 ∈ (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏})((𝑐 ≠ 𝑎 ∨ 𝑑 ≠ 𝑏) → 𝜒) → ∀𝑑 ∈ Pred (𝑆, 𝐵, 𝑏)𝜃)
105	59, 84, 104	3jca 1129	. . . . 5 ⊢ (∀𝑐 ∈ (Pred(𝑅, 𝐴, 𝑎) ∪ {𝑎})∀𝑑 ∈ (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏})((𝑐 ≠ 𝑎 ∨ 𝑑 ≠ 𝑏) → 𝜒) → (∀𝑐 ∈ Pred (𝑅, 𝐴, 𝑎)∀𝑑 ∈ Pred (𝑆, 𝐵, 𝑏)𝜒 ∧ ∀𝑐 ∈ Pred (𝑅, 𝐴, 𝑎)𝜓 ∧ ∀𝑑 ∈ Pred (𝑆, 𝐵, 𝑏)𝜃))
106		xpord2indlem.i	. . . . 5 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → ((∀𝑐 ∈ Pred (𝑅, 𝐴, 𝑎)∀𝑑 ∈ Pred (𝑆, 𝐵, 𝑏)𝜒 ∧ ∀𝑐 ∈ Pred (𝑅, 𝐴, 𝑎)𝜓 ∧ ∀𝑑 ∈ Pred (𝑆, 𝐵, 𝑏)𝜃) → 𝜑))
107	105, 106	syl5 34	. . . 4 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → (∀𝑐 ∈ (Pred(𝑅, 𝐴, 𝑎) ∪ {𝑎})∀𝑑 ∈ (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏})((𝑐 ≠ 𝑎 ∨ 𝑑 ≠ 𝑏) → 𝜒) → 𝜑))
108	40, 107	sylbid 239	. . 3 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → (∀𝑐∀𝑑(⟨𝑐, 𝑑⟩ ∈ Pred(𝑇, (𝐴 × 𝐵), ⟨𝑎, 𝑏⟩) → 𝜒) → 𝜑))
109		xpord2indlem.7	. . 3 ⊢ (𝑎 = 𝑐 → (𝜑 ↔ 𝜓))
110		xpord2indlem.11	. . 3 ⊢ (𝑎 = 𝑋 → (𝜑 ↔ 𝜏))
111		xpord2indlem.12	. . 3 ⊢ (𝑏 = 𝑌 → (𝜏 ↔ 𝜂))
112	108, 109, 68, 110, 111	frpoins3xpg 8123	. 2 ⊢ (((𝑇 Fr (𝐴 × 𝐵) ∧ 𝑇 Po (𝐴 × 𝐵) ∧ 𝑇 Se (𝐴 × 𝐵)) ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵)) → 𝜂)
113	18, 112	mpan 689	1 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → 𝜂)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 397   ∨ wo 846   ∧ w3a 1088  ∀wal 1540   = wceq 1542  ⊤wtru 1543   ∈ wcel 2107   ≠ wne 2941  ∀wral 3062   ∖ cdif 3945   ∪ cun 3946   ⊆ wss 3948  {csn 4628  ⟨cop 4634   class class class wbr 5148  {copab 5210   Po wpo 5586   Fr wfr 5628   Se wse 5629   × cxp 5674  Predcpred 6297  ‘cfv 6541  1^st c1st 7970  2^nd c2nd 7971

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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7722

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-po 5588  df-fr 5631  df-se 5632  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 6298  df-iota 6493  df-fun 6543  df-fv 6549  df-1st 7972  df-2nd 7973

This theorem is referenced by:  xpord2ind  8131