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Theorem frxp2 8185

Description: Another way of giving a well-founded order to a Cartesian product of two classes. (Contributed by Scott Fenton, 19-Aug-2024.)

**Hypotheses**
Ref	Expression
xpord2.1	⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐴 × 𝐵) ∧ 𝑦 ∈ (𝐴 × 𝐵) ∧ (((1^st ‘𝑥)𝑅(1^st ‘𝑦) ∨ (1^st ‘𝑥) = (1^st ‘𝑦)) ∧ ((2^nd ‘𝑥)𝑆(2^nd ‘𝑦) ∨ (2^nd ‘𝑥) = (2^nd ‘𝑦)) ∧ 𝑥 ≠ 𝑦))}
frxp2.1	⊢ (𝜑 → 𝑅 Fr 𝐴)
frxp2.2	⊢ (𝜑 → 𝑆 Fr 𝐵)

**Assertion**
Ref	Expression
frxp2	⊢ (𝜑 → 𝑇 Fr (𝐴 × 𝐵))

Distinct variable groups: 𝑥,𝐴,𝑦 𝑥,𝐵,𝑦 𝑥,𝑅,𝑦 𝑥,𝑆,𝑦 Allowed substitution hints: 𝜑(𝑥,𝑦) 𝑇(𝑥,𝑦)

Proof of Theorem frxp2 Dummy variables 𝑎 𝑏 𝑐 𝑑 𝑒 𝑓 𝑝 𝑞 𝑠 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dmss 5927	. . . . . . . 8 ⊢ (𝑠 ⊆ (𝐴 × 𝐵) → dom 𝑠 ⊆ dom (𝐴 × 𝐵))
2	1	ad2antrl 727	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑠 ⊆ (𝐴 × 𝐵) ∧ 𝑠 ≠ ∅)) → dom 𝑠 ⊆ dom (𝐴 × 𝐵))
3		dmxpss 6202	. . . . . . 7 ⊢ dom (𝐴 × 𝐵) ⊆ 𝐴
4	2, 3	sstrdi 4021	. . . . . 6 ⊢ ((𝜑 ∧ (𝑠 ⊆ (𝐴 × 𝐵) ∧ 𝑠 ≠ ∅)) → dom 𝑠 ⊆ 𝐴)
5		simprr 772	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑠 ⊆ (𝐴 × 𝐵) ∧ 𝑠 ≠ ∅)) → 𝑠 ≠ ∅)
6		relxp 5718	. . . . . . . . . . 11 ⊢ Rel (𝐴 × 𝐵)
7		relss 5805	. . . . . . . . . . 11 ⊢ (𝑠 ⊆ (𝐴 × 𝐵) → (Rel (𝐴 × 𝐵) → Rel 𝑠))
8	6, 7	mpi 20	. . . . . . . . . 10 ⊢ (𝑠 ⊆ (𝐴 × 𝐵) → Rel 𝑠)
9	8	ad2antrl 727	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑠 ⊆ (𝐴 × 𝐵) ∧ 𝑠 ≠ ∅)) → Rel 𝑠)
10		reldm0 5952	. . . . . . . . 9 ⊢ (Rel 𝑠 → (𝑠 = ∅ ↔ dom 𝑠 = ∅))
11	9, 10	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑠 ⊆ (𝐴 × 𝐵) ∧ 𝑠 ≠ ∅)) → (𝑠 = ∅ ↔ dom 𝑠 = ∅))
12	11	necon3bid 2991	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑠 ⊆ (𝐴 × 𝐵) ∧ 𝑠 ≠ ∅)) → (𝑠 ≠ ∅ ↔ dom 𝑠 ≠ ∅))
13	5, 12	mpbid 232	. . . . . 6 ⊢ ((𝜑 ∧ (𝑠 ⊆ (𝐴 × 𝐵) ∧ 𝑠 ≠ ∅)) → dom 𝑠 ≠ ∅)
14		frxp2.1	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 Fr 𝐴)
15	14	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑠 ⊆ (𝐴 × 𝐵) ∧ 𝑠 ≠ ∅)) → 𝑅 Fr 𝐴)
16		df-fr 5652	. . . . . . . 8 ⊢ (𝑅 Fr 𝐴 ↔ ∀𝑐((𝑐 ⊆ 𝐴 ∧ 𝑐 ≠ ∅) → ∃𝑎 ∈ 𝑐 ∀𝑏 ∈ 𝑐 ¬ 𝑏𝑅𝑎))
17	15, 16	sylib 218	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑠 ⊆ (𝐴 × 𝐵) ∧ 𝑠 ≠ ∅)) → ∀𝑐((𝑐 ⊆ 𝐴 ∧ 𝑐 ≠ ∅) → ∃𝑎 ∈ 𝑐 ∀𝑏 ∈ 𝑐 ¬ 𝑏𝑅𝑎))
18		vex 3492	. . . . . . . . 9 ⊢ 𝑠 ∈ V
19	18	dmex 7949	. . . . . . . 8 ⊢ dom 𝑠 ∈ V
20		sseq1 4034	. . . . . . . . . 10 ⊢ (𝑐 = dom 𝑠 → (𝑐 ⊆ 𝐴 ↔ dom 𝑠 ⊆ 𝐴))
21		neeq1 3009	. . . . . . . . . 10 ⊢ (𝑐 = dom 𝑠 → (𝑐 ≠ ∅ ↔ dom 𝑠 ≠ ∅))
22	20, 21	anbi12d 631	. . . . . . . . 9 ⊢ (𝑐 = dom 𝑠 → ((𝑐 ⊆ 𝐴 ∧ 𝑐 ≠ ∅) ↔ (dom 𝑠 ⊆ 𝐴 ∧ dom 𝑠 ≠ ∅)))
23		raleq 3331	. . . . . . . . . 10 ⊢ (𝑐 = dom 𝑠 → (∀𝑏 ∈ 𝑐 ¬ 𝑏𝑅𝑎 ↔ ∀𝑏 ∈ dom 𝑠 ¬ 𝑏𝑅𝑎))
24	23	rexeqbi1dv 3347	. . . . . . . . 9 ⊢ (𝑐 = dom 𝑠 → (∃𝑎 ∈ 𝑐 ∀𝑏 ∈ 𝑐 ¬ 𝑏𝑅𝑎 ↔ ∃𝑎 ∈ dom 𝑠∀𝑏 ∈ dom 𝑠 ¬ 𝑏𝑅𝑎))
25	22, 24	imbi12d 344	. . . . . . . 8 ⊢ (𝑐 = dom 𝑠 → (((𝑐 ⊆ 𝐴 ∧ 𝑐 ≠ ∅) → ∃𝑎 ∈ 𝑐 ∀𝑏 ∈ 𝑐 ¬ 𝑏𝑅𝑎) ↔ ((dom 𝑠 ⊆ 𝐴 ∧ dom 𝑠 ≠ ∅) → ∃𝑎 ∈ dom 𝑠∀𝑏 ∈ dom 𝑠 ¬ 𝑏𝑅𝑎)))
26	19, 25	spcv 3618	. . . . . . 7 ⊢ (∀𝑐((𝑐 ⊆ 𝐴 ∧ 𝑐 ≠ ∅) → ∃𝑎 ∈ 𝑐 ∀𝑏 ∈ 𝑐 ¬ 𝑏𝑅𝑎) → ((dom 𝑠 ⊆ 𝐴 ∧ dom 𝑠 ≠ ∅) → ∃𝑎 ∈ dom 𝑠∀𝑏 ∈ dom 𝑠 ¬ 𝑏𝑅𝑎))
27	17, 26	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ (𝑠 ⊆ (𝐴 × 𝐵) ∧ 𝑠 ≠ ∅)) → ((dom 𝑠 ⊆ 𝐴 ∧ dom 𝑠 ≠ ∅) → ∃𝑎 ∈ dom 𝑠∀𝑏 ∈ dom 𝑠 ¬ 𝑏𝑅𝑎))
28	4, 13, 27	mp2and 698	. . . . 5 ⊢ ((𝜑 ∧ (𝑠 ⊆ (𝐴 × 𝐵) ∧ 𝑠 ≠ ∅)) → ∃𝑎 ∈ dom 𝑠∀𝑏 ∈ dom 𝑠 ¬ 𝑏𝑅𝑎)
29		imassrn 6100	. . . . . . . 8 ⊢ (𝑠 “ {𝑎}) ⊆ ran 𝑠
30		rnss 5964	. . . . . . . . . . 11 ⊢ (𝑠 ⊆ (𝐴 × 𝐵) → ran 𝑠 ⊆ ran (𝐴 × 𝐵))
31	30	ad2antrl 727	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑠 ⊆ (𝐴 × 𝐵) ∧ 𝑠 ≠ ∅)) → ran 𝑠 ⊆ ran (𝐴 × 𝐵))
32	31	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑠 ⊆ (𝐴 × 𝐵) ∧ 𝑠 ≠ ∅)) ∧ (𝑎 ∈ dom 𝑠 ∧ ∀𝑏 ∈ dom 𝑠 ¬ 𝑏𝑅𝑎)) → ran 𝑠 ⊆ ran (𝐴 × 𝐵))
33		rnxpss 6203	. . . . . . . . 9 ⊢ ran (𝐴 × 𝐵) ⊆ 𝐵
34	32, 33	sstrdi 4021	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑠 ⊆ (𝐴 × 𝐵) ∧ 𝑠 ≠ ∅)) ∧ (𝑎 ∈ dom 𝑠 ∧ ∀𝑏 ∈ dom 𝑠 ¬ 𝑏𝑅𝑎)) → ran 𝑠 ⊆ 𝐵)
35	29, 34	sstrid 4020	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑠 ⊆ (𝐴 × 𝐵) ∧ 𝑠 ≠ ∅)) ∧ (𝑎 ∈ dom 𝑠 ∧ ∀𝑏 ∈ dom 𝑠 ¬ 𝑏𝑅𝑎)) → (𝑠 “ {𝑎}) ⊆ 𝐵)
36		simprl 770	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑠 ⊆ (𝐴 × 𝐵) ∧ 𝑠 ≠ ∅)) ∧ (𝑎 ∈ dom 𝑠 ∧ ∀𝑏 ∈ dom 𝑠 ¬ 𝑏𝑅𝑎)) → 𝑎 ∈ dom 𝑠)
37		imadisj 6109	. . . . . . . . . 10 ⊢ ((𝑠 “ {𝑎}) = ∅ ↔ (dom 𝑠 ∩ {𝑎}) = ∅)
38		disjsn 4736	. . . . . . . . . 10 ⊢ ((dom 𝑠 ∩ {𝑎}) = ∅ ↔ ¬ 𝑎 ∈ dom 𝑠)
39	37, 38	bitri 275	. . . . . . . . 9 ⊢ ((𝑠 “ {𝑎}) = ∅ ↔ ¬ 𝑎 ∈ dom 𝑠)
40	39	necon2abii 2997	. . . . . . . 8 ⊢ (𝑎 ∈ dom 𝑠 ↔ (𝑠 “ {𝑎}) ≠ ∅)
41	36, 40	sylib 218	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑠 ⊆ (𝐴 × 𝐵) ∧ 𝑠 ≠ ∅)) ∧ (𝑎 ∈ dom 𝑠 ∧ ∀𝑏 ∈ dom 𝑠 ¬ 𝑏𝑅𝑎)) → (𝑠 “ {𝑎}) ≠ ∅)
42		frxp2.2	. . . . . . . . . 10 ⊢ (𝜑 → 𝑆 Fr 𝐵)
43		df-fr 5652	. . . . . . . . . 10 ⊢ (𝑆 Fr 𝐵 ↔ ∀𝑒((𝑒 ⊆ 𝐵 ∧ 𝑒 ≠ ∅) → ∃𝑐 ∈ 𝑒 ∀𝑑 ∈ 𝑒 ¬ 𝑑𝑆𝑐))
44	42, 43	sylib 218	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑒((𝑒 ⊆ 𝐵 ∧ 𝑒 ≠ ∅) → ∃𝑐 ∈ 𝑒 ∀𝑑 ∈ 𝑒 ¬ 𝑑𝑆𝑐))
45	44	ad2antrr 725	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑠 ⊆ (𝐴 × 𝐵) ∧ 𝑠 ≠ ∅)) ∧ (𝑎 ∈ dom 𝑠 ∧ ∀𝑏 ∈ dom 𝑠 ¬ 𝑏𝑅𝑎)) → ∀𝑒((𝑒 ⊆ 𝐵 ∧ 𝑒 ≠ ∅) → ∃𝑐 ∈ 𝑒 ∀𝑑 ∈ 𝑒 ¬ 𝑑𝑆𝑐))
46	18	imaex 7954	. . . . . . . . 9 ⊢ (𝑠 “ {𝑎}) ∈ V
47		sseq1 4034	. . . . . . . . . . 11 ⊢ (𝑒 = (𝑠 “ {𝑎}) → (𝑒 ⊆ 𝐵 ↔ (𝑠 “ {𝑎}) ⊆ 𝐵))
48		neeq1 3009	. . . . . . . . . . 11 ⊢ (𝑒 = (𝑠 “ {𝑎}) → (𝑒 ≠ ∅ ↔ (𝑠 “ {𝑎}) ≠ ∅))
49	47, 48	anbi12d 631	. . . . . . . . . 10 ⊢ (𝑒 = (𝑠 “ {𝑎}) → ((𝑒 ⊆ 𝐵 ∧ 𝑒 ≠ ∅) ↔ ((𝑠 “ {𝑎}) ⊆ 𝐵 ∧ (𝑠 “ {𝑎}) ≠ ∅)))
50		raleq 3331	. . . . . . . . . . 11 ⊢ (𝑒 = (𝑠 “ {𝑎}) → (∀𝑑 ∈ 𝑒 ¬ 𝑑𝑆𝑐 ↔ ∀𝑑 ∈ (𝑠 “ {𝑎}) ¬ 𝑑𝑆𝑐))
51	50	rexeqbi1dv 3347	. . . . . . . . . 10 ⊢ (𝑒 = (𝑠 “ {𝑎}) → (∃𝑐 ∈ 𝑒 ∀𝑑 ∈ 𝑒 ¬ 𝑑𝑆𝑐 ↔ ∃𝑐 ∈ (𝑠 “ {𝑎})∀𝑑 ∈ (𝑠 “ {𝑎}) ¬ 𝑑𝑆𝑐))
52	49, 51	imbi12d 344	. . . . . . . . 9 ⊢ (𝑒 = (𝑠 “ {𝑎}) → (((𝑒 ⊆ 𝐵 ∧ 𝑒 ≠ ∅) → ∃𝑐 ∈ 𝑒 ∀𝑑 ∈ 𝑒 ¬ 𝑑𝑆𝑐) ↔ (((𝑠 “ {𝑎}) ⊆ 𝐵 ∧ (𝑠 “ {𝑎}) ≠ ∅) → ∃𝑐 ∈ (𝑠 “ {𝑎})∀𝑑 ∈ (𝑠 “ {𝑎}) ¬ 𝑑𝑆𝑐)))
53	46, 52	spcv 3618	. . . . . . . 8 ⊢ (∀𝑒((𝑒 ⊆ 𝐵 ∧ 𝑒 ≠ ∅) → ∃𝑐 ∈ 𝑒 ∀𝑑 ∈ 𝑒 ¬ 𝑑𝑆𝑐) → (((𝑠 “ {𝑎}) ⊆ 𝐵 ∧ (𝑠 “ {𝑎}) ≠ ∅) → ∃𝑐 ∈ (𝑠 “ {𝑎})∀𝑑 ∈ (𝑠 “ {𝑎}) ¬ 𝑑𝑆𝑐))
54	45, 53	syl 17	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑠 ⊆ (𝐴 × 𝐵) ∧ 𝑠 ≠ ∅)) ∧ (𝑎 ∈ dom 𝑠 ∧ ∀𝑏 ∈ dom 𝑠 ¬ 𝑏𝑅𝑎)) → (((𝑠 “ {𝑎}) ⊆ 𝐵 ∧ (𝑠 “ {𝑎}) ≠ ∅) → ∃𝑐 ∈ (𝑠 “ {𝑎})∀𝑑 ∈ (𝑠 “ {𝑎}) ¬ 𝑑𝑆𝑐))
55	35, 41, 54	mp2and 698	. . . . . 6 ⊢ (((𝜑 ∧ (𝑠 ⊆ (𝐴 × 𝐵) ∧ 𝑠 ≠ ∅)) ∧ (𝑎 ∈ dom 𝑠 ∧ ∀𝑏 ∈ dom 𝑠 ¬ 𝑏𝑅𝑎)) → ∃𝑐 ∈ (𝑠 “ {𝑎})∀𝑑 ∈ (𝑠 “ {𝑎}) ¬ 𝑑𝑆𝑐)
56		breq2 5170	. . . . . . . . 9 ⊢ (𝑝 = ⟨𝑎, 𝑐⟩ → (𝑞𝑇𝑝 ↔ 𝑞𝑇⟨𝑎, 𝑐⟩))
57	56	notbid 318	. . . . . . . 8 ⊢ (𝑝 = ⟨𝑎, 𝑐⟩ → (¬ 𝑞𝑇𝑝 ↔ ¬ 𝑞𝑇⟨𝑎, 𝑐⟩))
58	57	ralbidv 3184	. . . . . . 7 ⊢ (𝑝 = ⟨𝑎, 𝑐⟩ → (∀𝑞 ∈ 𝑠 ¬ 𝑞𝑇𝑝 ↔ ∀𝑞 ∈ 𝑠 ¬ 𝑞𝑇⟨𝑎, 𝑐⟩))
59		simprl 770	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑠 ⊆ (𝐴 × 𝐵) ∧ 𝑠 ≠ ∅)) ∧ (𝑎 ∈ dom 𝑠 ∧ ∀𝑏 ∈ dom 𝑠 ¬ 𝑏𝑅𝑎)) ∧ (𝑐 ∈ (𝑠 “ {𝑎}) ∧ ∀𝑑 ∈ (𝑠 “ {𝑎}) ¬ 𝑑𝑆𝑐)) → 𝑐 ∈ (𝑠 “ {𝑎}))
60		vex 3492	. . . . . . . . 9 ⊢ 𝑎 ∈ V
61		vex 3492	. . . . . . . . 9 ⊢ 𝑐 ∈ V
62	60, 61	elimasn 6119	. . . . . . . 8 ⊢ (𝑐 ∈ (𝑠 “ {𝑎}) ↔ ⟨𝑎, 𝑐⟩ ∈ 𝑠)
63	59, 62	sylib 218	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑠 ⊆ (𝐴 × 𝐵) ∧ 𝑠 ≠ ∅)) ∧ (𝑎 ∈ dom 𝑠 ∧ ∀𝑏 ∈ dom 𝑠 ¬ 𝑏𝑅𝑎)) ∧ (𝑐 ∈ (𝑠 “ {𝑎}) ∧ ∀𝑑 ∈ (𝑠 “ {𝑎}) ¬ 𝑑𝑆𝑐)) → ⟨𝑎, 𝑐⟩ ∈ 𝑠)
64	9	ad2antrr 725	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑠 ⊆ (𝐴 × 𝐵) ∧ 𝑠 ≠ ∅)) ∧ (𝑎 ∈ dom 𝑠 ∧ ∀𝑏 ∈ dom 𝑠 ¬ 𝑏𝑅𝑎)) ∧ (𝑐 ∈ (𝑠 “ {𝑎}) ∧ ∀𝑑 ∈ (𝑠 “ {𝑎}) ¬ 𝑑𝑆𝑐)) → Rel 𝑠)
65		elrel 5822	. . . . . . . . . 10 ⊢ ((Rel 𝑠 ∧ 𝑞 ∈ 𝑠) → ∃𝑒∃𝑓 𝑞 = ⟨𝑒, 𝑓⟩)
66	64, 65	sylan 579	. . . . . . . . 9 ⊢ (((((𝜑 ∧ (𝑠 ⊆ (𝐴 × 𝐵) ∧ 𝑠 ≠ ∅)) ∧ (𝑎 ∈ dom 𝑠 ∧ ∀𝑏 ∈ dom 𝑠 ¬ 𝑏𝑅𝑎)) ∧ (𝑐 ∈ (𝑠 “ {𝑎}) ∧ ∀𝑑 ∈ (𝑠 “ {𝑎}) ¬ 𝑑𝑆𝑐)) ∧ 𝑞 ∈ 𝑠) → ∃𝑒∃𝑓 𝑞 = ⟨𝑒, 𝑓⟩)
67		breq1 5169	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑑 = 𝑓 → (𝑑𝑆𝑐 ↔ 𝑓𝑆𝑐))
68	67	notbid 318	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑑 = 𝑓 → (¬ 𝑑𝑆𝑐 ↔ ¬ 𝑓𝑆𝑐))
69		simplrr 777	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ (𝑠 ⊆ (𝐴 × 𝐵) ∧ 𝑠 ≠ ∅)) ∧ (𝑎 ∈ dom 𝑠 ∧ ∀𝑏 ∈ dom 𝑠 ¬ 𝑏𝑅𝑎)) ∧ (𝑐 ∈ (𝑠 “ {𝑎}) ∧ ∀𝑑 ∈ (𝑠 “ {𝑎}) ¬ 𝑑𝑆𝑐)) ∧ ⟨𝑎, 𝑓⟩ ∈ 𝑠) → ∀𝑑 ∈ (𝑠 “ {𝑎}) ¬ 𝑑𝑆𝑐)
70		vex 3492	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑓 ∈ V
71	60, 70	elimasn 6119	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓 ∈ (𝑠 “ {𝑎}) ↔ ⟨𝑎, 𝑓⟩ ∈ 𝑠)
72	71	biimpri 228	. . . . . . . . . . . . . . . . . . 19 ⊢ (⟨𝑎, 𝑓⟩ ∈ 𝑠 → 𝑓 ∈ (𝑠 “ {𝑎}))
73	72	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ (𝑠 ⊆ (𝐴 × 𝐵) ∧ 𝑠 ≠ ∅)) ∧ (𝑎 ∈ dom 𝑠 ∧ ∀𝑏 ∈ dom 𝑠 ¬ 𝑏𝑅𝑎)) ∧ (𝑐 ∈ (𝑠 “ {𝑎}) ∧ ∀𝑑 ∈ (𝑠 “ {𝑎}) ¬ 𝑑𝑆𝑐)) ∧ ⟨𝑎, 𝑓⟩ ∈ 𝑠) → 𝑓 ∈ (𝑠 “ {𝑎}))
74	68, 69, 73	rspcdva 3636	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ (𝑠 ⊆ (𝐴 × 𝐵) ∧ 𝑠 ≠ ∅)) ∧ (𝑎 ∈ dom 𝑠 ∧ ∀𝑏 ∈ dom 𝑠 ¬ 𝑏𝑅𝑎)) ∧ (𝑐 ∈ (𝑠 “ {𝑎}) ∧ ∀𝑑 ∈ (𝑠 “ {𝑎}) ¬ 𝑑𝑆𝑐)) ∧ ⟨𝑎, 𝑓⟩ ∈ 𝑠) → ¬ 𝑓𝑆𝑐)
75	74	intnanrd 489	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ (𝑠 ⊆ (𝐴 × 𝐵) ∧ 𝑠 ≠ ∅)) ∧ (𝑎 ∈ dom 𝑠 ∧ ∀𝑏 ∈ dom 𝑠 ¬ 𝑏𝑅𝑎)) ∧ (𝑐 ∈ (𝑠 “ {𝑎}) ∧ ∀𝑑 ∈ (𝑠 “ {𝑎}) ¬ 𝑑𝑆𝑐)) ∧ ⟨𝑎, 𝑓⟩ ∈ 𝑠) → ¬ (𝑓𝑆𝑐 ∧ 𝑓 ≠ 𝑐))
76		opeq1 4897	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑒 = 𝑎 → ⟨𝑒, 𝑓⟩ = ⟨𝑎, 𝑓⟩)
77	76	eleq1d 2829	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑒 = 𝑎 → (⟨𝑒, 𝑓⟩ ∈ 𝑠 ↔ ⟨𝑎, 𝑓⟩ ∈ 𝑠))
78	77	anbi2d 629	. . . . . . . . . . . . . . . . 17 ⊢ (𝑒 = 𝑎 → (((((𝜑 ∧ (𝑠 ⊆ (𝐴 × 𝐵) ∧ 𝑠 ≠ ∅)) ∧ (𝑎 ∈ dom 𝑠 ∧ ∀𝑏 ∈ dom 𝑠 ¬ 𝑏𝑅𝑎)) ∧ (𝑐 ∈ (𝑠 “ {𝑎}) ∧ ∀𝑑 ∈ (𝑠 “ {𝑎}) ¬ 𝑑𝑆𝑐)) ∧ ⟨𝑒, 𝑓⟩ ∈ 𝑠) ↔ ((((𝜑 ∧ (𝑠 ⊆ (𝐴 × 𝐵) ∧ 𝑠 ≠ ∅)) ∧ (𝑎 ∈ dom 𝑠 ∧ ∀𝑏 ∈ dom 𝑠 ¬ 𝑏𝑅𝑎)) ∧ (𝑐 ∈ (𝑠 “ {𝑎}) ∧ ∀𝑑 ∈ (𝑠 “ {𝑎}) ¬ 𝑑𝑆𝑐)) ∧ ⟨𝑎, 𝑓⟩ ∈ 𝑠)))
79		3anass 1095	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑒𝑅𝑎 ∨ 𝑒 = 𝑎) ∧ (𝑓𝑆𝑐 ∨ 𝑓 = 𝑐) ∧ (𝑒 ≠ 𝑎 ∨ 𝑓 ≠ 𝑐)) ↔ ((𝑒𝑅𝑎 ∨ 𝑒 = 𝑎) ∧ ((𝑓𝑆𝑐 ∨ 𝑓 = 𝑐) ∧ (𝑒 ≠ 𝑎 ∨ 𝑓 ≠ 𝑐))))
80		olc 867	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑒 = 𝑎 → (𝑒𝑅𝑎 ∨ 𝑒 = 𝑎))
81	80	biantrurd 532	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑒 = 𝑎 → (((𝑓𝑆𝑐 ∨ 𝑓 = 𝑐) ∧ (𝑒 ≠ 𝑎 ∨ 𝑓 ≠ 𝑐)) ↔ ((𝑒𝑅𝑎 ∨ 𝑒 = 𝑎) ∧ ((𝑓𝑆𝑐 ∨ 𝑓 = 𝑐) ∧ (𝑒 ≠ 𝑎 ∨ 𝑓 ≠ 𝑐)))))
82		neeq1 3009	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑒 = 𝑎 → (𝑒 ≠ 𝑎 ↔ 𝑎 ≠ 𝑎))
83	82	orbi1d 915	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑒 = 𝑎 → ((𝑒 ≠ 𝑎 ∨ 𝑓 ≠ 𝑐) ↔ (𝑎 ≠ 𝑎 ∨ 𝑓 ≠ 𝑐)))
84		neirr 2955	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ¬ 𝑎 ≠ 𝑎
85	84	biorfi 937	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑓 ≠ 𝑐 ↔ (𝑎 ≠ 𝑎 ∨ 𝑓 ≠ 𝑐))
86	83, 85	bitr4di 289	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑒 = 𝑎 → ((𝑒 ≠ 𝑎 ∨ 𝑓 ≠ 𝑐) ↔ 𝑓 ≠ 𝑐))
87	86	anbi2d 629	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑒 = 𝑎 → (((𝑓𝑆𝑐 ∨ 𝑓 = 𝑐) ∧ (𝑒 ≠ 𝑎 ∨ 𝑓 ≠ 𝑐)) ↔ ((𝑓𝑆𝑐 ∨ 𝑓 = 𝑐) ∧ 𝑓 ≠ 𝑐)))
88		andir 1009	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑓𝑆𝑐 ∨ 𝑓 = 𝑐) ∧ 𝑓 ≠ 𝑐) ↔ ((𝑓𝑆𝑐 ∧ 𝑓 ≠ 𝑐) ∨ (𝑓 = 𝑐 ∧ 𝑓 ≠ 𝑐)))
89		nonconne 2958	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ¬ (𝑓 = 𝑐 ∧ 𝑓 ≠ 𝑐)
90	89	biorfri 938	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑓𝑆𝑐 ∧ 𝑓 ≠ 𝑐) ↔ ((𝑓𝑆𝑐 ∧ 𝑓 ≠ 𝑐) ∨ (𝑓 = 𝑐 ∧ 𝑓 ≠ 𝑐)))
91	88, 90	bitr4i 278	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑓𝑆𝑐 ∨ 𝑓 = 𝑐) ∧ 𝑓 ≠ 𝑐) ↔ (𝑓𝑆𝑐 ∧ 𝑓 ≠ 𝑐))
92	87, 91	bitrdi 287	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑒 = 𝑎 → (((𝑓𝑆𝑐 ∨ 𝑓 = 𝑐) ∧ (𝑒 ≠ 𝑎 ∨ 𝑓 ≠ 𝑐)) ↔ (𝑓𝑆𝑐 ∧ 𝑓 ≠ 𝑐)))
93	81, 92	bitr3d 281	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑒 = 𝑎 → (((𝑒𝑅𝑎 ∨ 𝑒 = 𝑎) ∧ ((𝑓𝑆𝑐 ∨ 𝑓 = 𝑐) ∧ (𝑒 ≠ 𝑎 ∨ 𝑓 ≠ 𝑐))) ↔ (𝑓𝑆𝑐 ∧ 𝑓 ≠ 𝑐)))
94	79, 93	bitrid 283	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑒 = 𝑎 → (((𝑒𝑅𝑎 ∨ 𝑒 = 𝑎) ∧ (𝑓𝑆𝑐 ∨ 𝑓 = 𝑐) ∧ (𝑒 ≠ 𝑎 ∨ 𝑓 ≠ 𝑐)) ↔ (𝑓𝑆𝑐 ∧ 𝑓 ≠ 𝑐)))
95	94	notbid 318	. . . . . . . . . . . . . . . . 17 ⊢ (𝑒 = 𝑎 → (¬ ((𝑒𝑅𝑎 ∨ 𝑒 = 𝑎) ∧ (𝑓𝑆𝑐 ∨ 𝑓 = 𝑐) ∧ (𝑒 ≠ 𝑎 ∨ 𝑓 ≠ 𝑐)) ↔ ¬ (𝑓𝑆𝑐 ∧ 𝑓 ≠ 𝑐)))
96	78, 95	imbi12d 344	. . . . . . . . . . . . . . . 16 ⊢ (𝑒 = 𝑎 → ((((((𝜑 ∧ (𝑠 ⊆ (𝐴 × 𝐵) ∧ 𝑠 ≠ ∅)) ∧ (𝑎 ∈ dom 𝑠 ∧ ∀𝑏 ∈ dom 𝑠 ¬ 𝑏𝑅𝑎)) ∧ (𝑐 ∈ (𝑠 “ {𝑎}) ∧ ∀𝑑 ∈ (𝑠 “ {𝑎}) ¬ 𝑑𝑆𝑐)) ∧ ⟨𝑒, 𝑓⟩ ∈ 𝑠) → ¬ ((𝑒𝑅𝑎 ∨ 𝑒 = 𝑎) ∧ (𝑓𝑆𝑐 ∨ 𝑓 = 𝑐) ∧ (𝑒 ≠ 𝑎 ∨ 𝑓 ≠ 𝑐))) ↔ (((((𝜑 ∧ (𝑠 ⊆ (𝐴 × 𝐵) ∧ 𝑠 ≠ ∅)) ∧ (𝑎 ∈ dom 𝑠 ∧ ∀𝑏 ∈ dom 𝑠 ¬ 𝑏𝑅𝑎)) ∧ (𝑐 ∈ (𝑠 “ {𝑎}) ∧ ∀𝑑 ∈ (𝑠 “ {𝑎}) ¬ 𝑑𝑆𝑐)) ∧ ⟨𝑎, 𝑓⟩ ∈ 𝑠) → ¬ (𝑓𝑆𝑐 ∧ 𝑓 ≠ 𝑐))))
97	75, 96	mpbiri 258	. . . . . . . . . . . . . . 15 ⊢ (𝑒 = 𝑎 → (((((𝜑 ∧ (𝑠 ⊆ (𝐴 × 𝐵) ∧ 𝑠 ≠ ∅)) ∧ (𝑎 ∈ dom 𝑠 ∧ ∀𝑏 ∈ dom 𝑠 ¬ 𝑏𝑅𝑎)) ∧ (𝑐 ∈ (𝑠 “ {𝑎}) ∧ ∀𝑑 ∈ (𝑠 “ {𝑎}) ¬ 𝑑𝑆𝑐)) ∧ ⟨𝑒, 𝑓⟩ ∈ 𝑠) → ¬ ((𝑒𝑅𝑎 ∨ 𝑒 = 𝑎) ∧ (𝑓𝑆𝑐 ∨ 𝑓 = 𝑐) ∧ (𝑒 ≠ 𝑎 ∨ 𝑓 ≠ 𝑐))))
98	97	impcom 407	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ (𝑠 ⊆ (𝐴 × 𝐵) ∧ 𝑠 ≠ ∅)) ∧ (𝑎 ∈ dom 𝑠 ∧ ∀𝑏 ∈ dom 𝑠 ¬ 𝑏𝑅𝑎)) ∧ (𝑐 ∈ (𝑠 “ {𝑎}) ∧ ∀𝑑 ∈ (𝑠 “ {𝑎}) ¬ 𝑑𝑆𝑐)) ∧ ⟨𝑒, 𝑓⟩ ∈ 𝑠) ∧ 𝑒 = 𝑎) → ¬ ((𝑒𝑅𝑎 ∨ 𝑒 = 𝑎) ∧ (𝑓𝑆𝑐 ∨ 𝑓 = 𝑐) ∧ (𝑒 ≠ 𝑎 ∨ 𝑓 ≠ 𝑐)))
99		breq1 5169	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 = 𝑒 → (𝑏𝑅𝑎 ↔ 𝑒𝑅𝑎))
100	99	notbid 318	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 = 𝑒 → (¬ 𝑏𝑅𝑎 ↔ ¬ 𝑒𝑅𝑎))
101		simplrr 777	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ (𝑠 ⊆ (𝐴 × 𝐵) ∧ 𝑠 ≠ ∅)) ∧ (𝑎 ∈ dom 𝑠 ∧ ∀𝑏 ∈ dom 𝑠 ¬ 𝑏𝑅𝑎)) ∧ (𝑐 ∈ (𝑠 “ {𝑎}) ∧ ∀𝑑 ∈ (𝑠 “ {𝑎}) ¬ 𝑑𝑆𝑐)) → ∀𝑏 ∈ dom 𝑠 ¬ 𝑏𝑅𝑎)
102	101	ad2antrr 725	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝜑 ∧ (𝑠 ⊆ (𝐴 × 𝐵) ∧ 𝑠 ≠ ∅)) ∧ (𝑎 ∈ dom 𝑠 ∧ ∀𝑏 ∈ dom 𝑠 ¬ 𝑏𝑅𝑎)) ∧ (𝑐 ∈ (𝑠 “ {𝑎}) ∧ ∀𝑑 ∈ (𝑠 “ {𝑎}) ¬ 𝑑𝑆𝑐)) ∧ ⟨𝑒, 𝑓⟩ ∈ 𝑠) ∧ 𝑒 ≠ 𝑎) → ∀𝑏 ∈ dom 𝑠 ¬ 𝑏𝑅𝑎)
103		vex 3492	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑒 ∈ V
104	103, 70	opeldm 5932	. . . . . . . . . . . . . . . . . . 19 ⊢ (⟨𝑒, 𝑓⟩ ∈ 𝑠 → 𝑒 ∈ dom 𝑠)
105	104	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ (𝑠 ⊆ (𝐴 × 𝐵) ∧ 𝑠 ≠ ∅)) ∧ (𝑎 ∈ dom 𝑠 ∧ ∀𝑏 ∈ dom 𝑠 ¬ 𝑏𝑅𝑎)) ∧ (𝑐 ∈ (𝑠 “ {𝑎}) ∧ ∀𝑑 ∈ (𝑠 “ {𝑎}) ¬ 𝑑𝑆𝑐)) ∧ ⟨𝑒, 𝑓⟩ ∈ 𝑠) → 𝑒 ∈ dom 𝑠)
106	105	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝜑 ∧ (𝑠 ⊆ (𝐴 × 𝐵) ∧ 𝑠 ≠ ∅)) ∧ (𝑎 ∈ dom 𝑠 ∧ ∀𝑏 ∈ dom 𝑠 ¬ 𝑏𝑅𝑎)) ∧ (𝑐 ∈ (𝑠 “ {𝑎}) ∧ ∀𝑑 ∈ (𝑠 “ {𝑎}) ¬ 𝑑𝑆𝑐)) ∧ ⟨𝑒, 𝑓⟩ ∈ 𝑠) ∧ 𝑒 ≠ 𝑎) → 𝑒 ∈ dom 𝑠)
107	100, 102, 106	rspcdva 3636	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ (𝑠 ⊆ (𝐴 × 𝐵) ∧ 𝑠 ≠ ∅)) ∧ (𝑎 ∈ dom 𝑠 ∧ ∀𝑏 ∈ dom 𝑠 ¬ 𝑏𝑅𝑎)) ∧ (𝑐 ∈ (𝑠 “ {𝑎}) ∧ ∀𝑑 ∈ (𝑠 “ {𝑎}) ¬ 𝑑𝑆𝑐)) ∧ ⟨𝑒, 𝑓⟩ ∈ 𝑠) ∧ 𝑒 ≠ 𝑎) → ¬ 𝑒𝑅𝑎)
108		simpr 484	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝜑 ∧ (𝑠 ⊆ (𝐴 × 𝐵) ∧ 𝑠 ≠ ∅)) ∧ (𝑎 ∈ dom 𝑠 ∧ ∀𝑏 ∈ dom 𝑠 ¬ 𝑏𝑅𝑎)) ∧ (𝑐 ∈ (𝑠 “ {𝑎}) ∧ ∀𝑑 ∈ (𝑠 “ {𝑎}) ¬ 𝑑𝑆𝑐)) ∧ ⟨𝑒, 𝑓⟩ ∈ 𝑠) ∧ 𝑒 ≠ 𝑎) → 𝑒 ≠ 𝑎)
109	108	neneqd 2951	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ (𝑠 ⊆ (𝐴 × 𝐵) ∧ 𝑠 ≠ ∅)) ∧ (𝑎 ∈ dom 𝑠 ∧ ∀𝑏 ∈ dom 𝑠 ¬ 𝑏𝑅𝑎)) ∧ (𝑐 ∈ (𝑠 “ {𝑎}) ∧ ∀𝑑 ∈ (𝑠 “ {𝑎}) ¬ 𝑑𝑆𝑐)) ∧ ⟨𝑒, 𝑓⟩ ∈ 𝑠) ∧ 𝑒 ≠ 𝑎) → ¬ 𝑒 = 𝑎)
110		ioran 984	. . . . . . . . . . . . . . . 16 ⊢ (¬ (𝑒𝑅𝑎 ∨ 𝑒 = 𝑎) ↔ (¬ 𝑒𝑅𝑎 ∧ ¬ 𝑒 = 𝑎))
111	107, 109, 110	sylanbrc 582	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ (𝑠 ⊆ (𝐴 × 𝐵) ∧ 𝑠 ≠ ∅)) ∧ (𝑎 ∈ dom 𝑠 ∧ ∀𝑏 ∈ dom 𝑠 ¬ 𝑏𝑅𝑎)) ∧ (𝑐 ∈ (𝑠 “ {𝑎}) ∧ ∀𝑑 ∈ (𝑠 “ {𝑎}) ¬ 𝑑𝑆𝑐)) ∧ ⟨𝑒, 𝑓⟩ ∈ 𝑠) ∧ 𝑒 ≠ 𝑎) → ¬ (𝑒𝑅𝑎 ∨ 𝑒 = 𝑎))
112	111	intn3an1d 1479	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ (𝑠 ⊆ (𝐴 × 𝐵) ∧ 𝑠 ≠ ∅)) ∧ (𝑎 ∈ dom 𝑠 ∧ ∀𝑏 ∈ dom 𝑠 ¬ 𝑏𝑅𝑎)) ∧ (𝑐 ∈ (𝑠 “ {𝑎}) ∧ ∀𝑑 ∈ (𝑠 “ {𝑎}) ¬ 𝑑𝑆𝑐)) ∧ ⟨𝑒, 𝑓⟩ ∈ 𝑠) ∧ 𝑒 ≠ 𝑎) → ¬ ((𝑒𝑅𝑎 ∨ 𝑒 = 𝑎) ∧ (𝑓𝑆𝑐 ∨ 𝑓 = 𝑐) ∧ (𝑒 ≠ 𝑎 ∨ 𝑓 ≠ 𝑐)))
113	98, 112	pm2.61dane 3035	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ (𝑠 ⊆ (𝐴 × 𝐵) ∧ 𝑠 ≠ ∅)) ∧ (𝑎 ∈ dom 𝑠 ∧ ∀𝑏 ∈ dom 𝑠 ¬ 𝑏𝑅𝑎)) ∧ (𝑐 ∈ (𝑠 “ {𝑎}) ∧ ∀𝑑 ∈ (𝑠 “ {𝑎}) ¬ 𝑑𝑆𝑐)) ∧ ⟨𝑒, 𝑓⟩ ∈ 𝑠) → ¬ ((𝑒𝑅𝑎 ∨ 𝑒 = 𝑎) ∧ (𝑓𝑆𝑐 ∨ 𝑓 = 𝑐) ∧ (𝑒 ≠ 𝑎 ∨ 𝑓 ≠ 𝑐)))
114	113	intn3an3d 1481	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ (𝑠 ⊆ (𝐴 × 𝐵) ∧ 𝑠 ≠ ∅)) ∧ (𝑎 ∈ dom 𝑠 ∧ ∀𝑏 ∈ dom 𝑠 ¬ 𝑏𝑅𝑎)) ∧ (𝑐 ∈ (𝑠 “ {𝑎}) ∧ ∀𝑑 ∈ (𝑠 “ {𝑎}) ¬ 𝑑𝑆𝑐)) ∧ ⟨𝑒, 𝑓⟩ ∈ 𝑠) → ¬ ((𝑒 ∈ 𝐴 ∧ 𝑓 ∈ 𝐵) ∧ (𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐵) ∧ ((𝑒𝑅𝑎 ∨ 𝑒 = 𝑎) ∧ (𝑓𝑆𝑐 ∨ 𝑓 = 𝑐) ∧ (𝑒 ≠ 𝑎 ∨ 𝑓 ≠ 𝑐))))
115		eleq1 2832	. . . . . . . . . . . . . 14 ⊢ (𝑞 = ⟨𝑒, 𝑓⟩ → (𝑞 ∈ 𝑠 ↔ ⟨𝑒, 𝑓⟩ ∈ 𝑠))
116	115	anbi2d 629	. . . . . . . . . . . . 13 ⊢ (𝑞 = ⟨𝑒, 𝑓⟩ → (((((𝜑 ∧ (𝑠 ⊆ (𝐴 × 𝐵) ∧ 𝑠 ≠ ∅)) ∧ (𝑎 ∈ dom 𝑠 ∧ ∀𝑏 ∈ dom 𝑠 ¬ 𝑏𝑅𝑎)) ∧ (𝑐 ∈ (𝑠 “ {𝑎}) ∧ ∀𝑑 ∈ (𝑠 “ {𝑎}) ¬ 𝑑𝑆𝑐)) ∧ 𝑞 ∈ 𝑠) ↔ ((((𝜑 ∧ (𝑠 ⊆ (𝐴 × 𝐵) ∧ 𝑠 ≠ ∅)) ∧ (𝑎 ∈ dom 𝑠 ∧ ∀𝑏 ∈ dom 𝑠 ¬ 𝑏𝑅𝑎)) ∧ (𝑐 ∈ (𝑠 “ {𝑎}) ∧ ∀𝑑 ∈ (𝑠 “ {𝑎}) ¬ 𝑑𝑆𝑐)) ∧ ⟨𝑒, 𝑓⟩ ∈ 𝑠)))
117		breq1 5169	. . . . . . . . . . . . . . 15 ⊢ (𝑞 = ⟨𝑒, 𝑓⟩ → (𝑞𝑇⟨𝑎, 𝑐⟩ ↔ ⟨𝑒, 𝑓⟩𝑇⟨𝑎, 𝑐⟩))
118		xpord2.1	. . . . . . . . . . . . . . . 16 ⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐴 × 𝐵) ∧ 𝑦 ∈ (𝐴 × 𝐵) ∧ (((1^st ‘𝑥)𝑅(1^st ‘𝑦) ∨ (1^st ‘𝑥) = (1^st ‘𝑦)) ∧ ((2^nd ‘𝑥)𝑆(2^nd ‘𝑦) ∨ (2^nd ‘𝑥) = (2^nd ‘𝑦)) ∧ 𝑥 ≠ 𝑦))}
119	118	xpord2lem 8183	. . . . . . . . . . . . . . 15 ⊢ (⟨𝑒, 𝑓⟩𝑇⟨𝑎, 𝑐⟩ ↔ ((𝑒 ∈ 𝐴 ∧ 𝑓 ∈ 𝐵) ∧ (𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐵) ∧ ((𝑒𝑅𝑎 ∨ 𝑒 = 𝑎) ∧ (𝑓𝑆𝑐 ∨ 𝑓 = 𝑐) ∧ (𝑒 ≠ 𝑎 ∨ 𝑓 ≠ 𝑐))))
120	117, 119	bitrdi 287	. . . . . . . . . . . . . 14 ⊢ (𝑞 = ⟨𝑒, 𝑓⟩ → (𝑞𝑇⟨𝑎, 𝑐⟩ ↔ ((𝑒 ∈ 𝐴 ∧ 𝑓 ∈ 𝐵) ∧ (𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐵) ∧ ((𝑒𝑅𝑎 ∨ 𝑒 = 𝑎) ∧ (𝑓𝑆𝑐 ∨ 𝑓 = 𝑐) ∧ (𝑒 ≠ 𝑎 ∨ 𝑓 ≠ 𝑐)))))
121	120	notbid 318	. . . . . . . . . . . . 13 ⊢ (𝑞 = ⟨𝑒, 𝑓⟩ → (¬ 𝑞𝑇⟨𝑎, 𝑐⟩ ↔ ¬ ((𝑒 ∈ 𝐴 ∧ 𝑓 ∈ 𝐵) ∧ (𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐵) ∧ ((𝑒𝑅𝑎 ∨ 𝑒 = 𝑎) ∧ (𝑓𝑆𝑐 ∨ 𝑓 = 𝑐) ∧ (𝑒 ≠ 𝑎 ∨ 𝑓 ≠ 𝑐)))))
122	116, 121	imbi12d 344	. . . . . . . . . . . 12 ⊢ (𝑞 = ⟨𝑒, 𝑓⟩ → ((((((𝜑 ∧ (𝑠 ⊆ (𝐴 × 𝐵) ∧ 𝑠 ≠ ∅)) ∧ (𝑎 ∈ dom 𝑠 ∧ ∀𝑏 ∈ dom 𝑠 ¬ 𝑏𝑅𝑎)) ∧ (𝑐 ∈ (𝑠 “ {𝑎}) ∧ ∀𝑑 ∈ (𝑠 “ {𝑎}) ¬ 𝑑𝑆𝑐)) ∧ 𝑞 ∈ 𝑠) → ¬ 𝑞𝑇⟨𝑎, 𝑐⟩) ↔ (((((𝜑 ∧ (𝑠 ⊆ (𝐴 × 𝐵) ∧ 𝑠 ≠ ∅)) ∧ (𝑎 ∈ dom 𝑠 ∧ ∀𝑏 ∈ dom 𝑠 ¬ 𝑏𝑅𝑎)) ∧ (𝑐 ∈ (𝑠 “ {𝑎}) ∧ ∀𝑑 ∈ (𝑠 “ {𝑎}) ¬ 𝑑𝑆𝑐)) ∧ ⟨𝑒, 𝑓⟩ ∈ 𝑠) → ¬ ((𝑒 ∈ 𝐴 ∧ 𝑓 ∈ 𝐵) ∧ (𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐵) ∧ ((𝑒𝑅𝑎 ∨ 𝑒 = 𝑎) ∧ (𝑓𝑆𝑐 ∨ 𝑓 = 𝑐) ∧ (𝑒 ≠ 𝑎 ∨ 𝑓 ≠ 𝑐))))))
123	114, 122	mpbiri 258	. . . . . . . . . . 11 ⊢ (𝑞 = ⟨𝑒, 𝑓⟩ → (((((𝜑 ∧ (𝑠 ⊆ (𝐴 × 𝐵) ∧ 𝑠 ≠ ∅)) ∧ (𝑎 ∈ dom 𝑠 ∧ ∀𝑏 ∈ dom 𝑠 ¬ 𝑏𝑅𝑎)) ∧ (𝑐 ∈ (𝑠 “ {𝑎}) ∧ ∀𝑑 ∈ (𝑠 “ {𝑎}) ¬ 𝑑𝑆𝑐)) ∧ 𝑞 ∈ 𝑠) → ¬ 𝑞𝑇⟨𝑎, 𝑐⟩))
124	123	com12 32	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ (𝑠 ⊆ (𝐴 × 𝐵) ∧ 𝑠 ≠ ∅)) ∧ (𝑎 ∈ dom 𝑠 ∧ ∀𝑏 ∈ dom 𝑠 ¬ 𝑏𝑅𝑎)) ∧ (𝑐 ∈ (𝑠 “ {𝑎}) ∧ ∀𝑑 ∈ (𝑠 “ {𝑎}) ¬ 𝑑𝑆𝑐)) ∧ 𝑞 ∈ 𝑠) → (𝑞 = ⟨𝑒, 𝑓⟩ → ¬ 𝑞𝑇⟨𝑎, 𝑐⟩))
125	124	exlimdvv 1933	. . . . . . . . 9 ⊢ (((((𝜑 ∧ (𝑠 ⊆ (𝐴 × 𝐵) ∧ 𝑠 ≠ ∅)) ∧ (𝑎 ∈ dom 𝑠 ∧ ∀𝑏 ∈ dom 𝑠 ¬ 𝑏𝑅𝑎)) ∧ (𝑐 ∈ (𝑠 “ {𝑎}) ∧ ∀𝑑 ∈ (𝑠 “ {𝑎}) ¬ 𝑑𝑆𝑐)) ∧ 𝑞 ∈ 𝑠) → (∃𝑒∃𝑓 𝑞 = ⟨𝑒, 𝑓⟩ → ¬ 𝑞𝑇⟨𝑎, 𝑐⟩))
126	66, 125	mpd 15	. . . . . . . 8 ⊢ (((((𝜑 ∧ (𝑠 ⊆ (𝐴 × 𝐵) ∧ 𝑠 ≠ ∅)) ∧ (𝑎 ∈ dom 𝑠 ∧ ∀𝑏 ∈ dom 𝑠 ¬ 𝑏𝑅𝑎)) ∧ (𝑐 ∈ (𝑠 “ {𝑎}) ∧ ∀𝑑 ∈ (𝑠 “ {𝑎}) ¬ 𝑑𝑆𝑐)) ∧ 𝑞 ∈ 𝑠) → ¬ 𝑞𝑇⟨𝑎, 𝑐⟩)
127	126	ralrimiva 3152	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑠 ⊆ (𝐴 × 𝐵) ∧ 𝑠 ≠ ∅)) ∧ (𝑎 ∈ dom 𝑠 ∧ ∀𝑏 ∈ dom 𝑠 ¬ 𝑏𝑅𝑎)) ∧ (𝑐 ∈ (𝑠 “ {𝑎}) ∧ ∀𝑑 ∈ (𝑠 “ {𝑎}) ¬ 𝑑𝑆𝑐)) → ∀𝑞 ∈ 𝑠 ¬ 𝑞𝑇⟨𝑎, 𝑐⟩)
128	58, 63, 127	rspcedvdw 3638	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑠 ⊆ (𝐴 × 𝐵) ∧ 𝑠 ≠ ∅)) ∧ (𝑎 ∈ dom 𝑠 ∧ ∀𝑏 ∈ dom 𝑠 ¬ 𝑏𝑅𝑎)) ∧ (𝑐 ∈ (𝑠 “ {𝑎}) ∧ ∀𝑑 ∈ (𝑠 “ {𝑎}) ¬ 𝑑𝑆𝑐)) → ∃𝑝 ∈ 𝑠 ∀𝑞 ∈ 𝑠 ¬ 𝑞𝑇𝑝)
129	55, 128	rexlimddv 3167	. . . . 5 ⊢ (((𝜑 ∧ (𝑠 ⊆ (𝐴 × 𝐵) ∧ 𝑠 ≠ ∅)) ∧ (𝑎 ∈ dom 𝑠 ∧ ∀𝑏 ∈ dom 𝑠 ¬ 𝑏𝑅𝑎)) → ∃𝑝 ∈ 𝑠 ∀𝑞 ∈ 𝑠 ¬ 𝑞𝑇𝑝)
130	28, 129	rexlimddv 3167	. . . 4 ⊢ ((𝜑 ∧ (𝑠 ⊆ (𝐴 × 𝐵) ∧ 𝑠 ≠ ∅)) → ∃𝑝 ∈ 𝑠 ∀𝑞 ∈ 𝑠 ¬ 𝑞𝑇𝑝)
131	130	ex 412	. . 3 ⊢ (𝜑 → ((𝑠 ⊆ (𝐴 × 𝐵) ∧ 𝑠 ≠ ∅) → ∃𝑝 ∈ 𝑠 ∀𝑞 ∈ 𝑠 ¬ 𝑞𝑇𝑝))
132	131	alrimiv 1926	. 2 ⊢ (𝜑 → ∀𝑠((𝑠 ⊆ (𝐴 × 𝐵) ∧ 𝑠 ≠ ∅) → ∃𝑝 ∈ 𝑠 ∀𝑞 ∈ 𝑠 ¬ 𝑞𝑇𝑝))
133		df-fr 5652	. 2 ⊢ (𝑇 Fr (𝐴 × 𝐵) ↔ ∀𝑠((𝑠 ⊆ (𝐴 × 𝐵) ∧ 𝑠 ≠ ∅) → ∃𝑝 ∈ 𝑠 ∀𝑞 ∈ 𝑠 ¬ 𝑞𝑇𝑝))
134	132, 133	sylibr 234	1 ⊢ (𝜑 → 𝑇 Fr (𝐴 × 𝐵))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 846 ∧ w3a 1087 ∀wal 1535 = wceq 1537 ∃wex 1777 ∈ wcel 2108 ≠ wne 2946 ∀wral 3067 ∃wrex 3076 ∩ cin 3975 ⊆ wss 3976 ∅c0 4352 {csn 4648 ⟨cop 4654 class class class wbr 5166 {copab 5228 Fr wfr 5649 × cxp 5698 dom cdm 5700 ran crn 5701 “ cima 5703 Rel wrel 5705 ‘cfv 6573 1^st c1st 8028 2^nd c2nd 8029

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pr 5447 ax-un 7770

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-ral 3068 df-rex 3077 df-rab 3444 df-v 3490 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-br 5167 df-opab 5229 df-mpt 5250 df-id 5593 df-fr 5652 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-iota 6525 df-fun 6575 df-fv 6581 df-1st 8030 df-2nd 8031

This theorem is referenced by: xpord2indlem 8188 on2recsfn 8723 on2recsov 8724 noxpordfr 28002

W3C validator