Theorem ordthauslem 22280

Description: Lemma for ordthaus 22281. (Contributed by Mario Carneiro, 13-Sep-2015.)

Hypothesis

Ref	Expression
ordthauslem.1	⊢ 𝑋 = dom 𝑅

Assertion

Ref	Expression
ordthauslem	⊢ ((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝑅𝐵 → (𝐴 ≠ 𝐵 → ∃𝑚 ∈ (ordTop‘𝑅)∃𝑛 ∈ (ordTop‘𝑅)(𝐴 ∈ 𝑚 ∧ 𝐵 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅))))

Distinct variable groups:   𝑚,𝑛,𝐴   𝐵,𝑚,𝑛   𝑅,𝑚,𝑛   𝑚,𝑋,𝑛

Proof of Theorem ordthauslem

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

Step	Hyp	Ref	Expression
1		simpll1 1214	. . . . . 6 ⊢ ((((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝑅𝐵 ∧ 𝐴 ≠ 𝐵)) ∧ {𝑥 ∈ 𝑋 ∣ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴)} = ∅) → 𝑅 ∈ TosetRel )
2		simpll3 1216	. . . . . 6 ⊢ ((((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝑅𝐵 ∧ 𝐴 ≠ 𝐵)) ∧ {𝑥 ∈ 𝑋 ∣ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴)} = ∅) → 𝐵 ∈ 𝑋)
3		ordthauslem.1	. . . . . . 7 ⊢ 𝑋 = dom 𝑅
4	3	ordtopn2 22092	. . . . . 6 ⊢ ((𝑅 ∈ TosetRel ∧ 𝐵 ∈ 𝑋) → {𝑥 ∈ 𝑋 ∣ ¬ 𝐵𝑅𝑥} ∈ (ordTop‘𝑅))
5	1, 2, 4	syl2anc 587	. . . . 5 ⊢ ((((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝑅𝐵 ∧ 𝐴 ≠ 𝐵)) ∧ {𝑥 ∈ 𝑋 ∣ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴)} = ∅) → {𝑥 ∈ 𝑋 ∣ ¬ 𝐵𝑅𝑥} ∈ (ordTop‘𝑅))
6		simpll2 1215	. . . . . 6 ⊢ ((((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝑅𝐵 ∧ 𝐴 ≠ 𝐵)) ∧ {𝑥 ∈ 𝑋 ∣ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴)} = ∅) → 𝐴 ∈ 𝑋)
7	3	ordtopn1 22091	. . . . . 6 ⊢ ((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋) → {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝐴} ∈ (ordTop‘𝑅))
8	1, 6, 7	syl2anc 587	. . . . 5 ⊢ ((((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝑅𝐵 ∧ 𝐴 ≠ 𝐵)) ∧ {𝑥 ∈ 𝑋 ∣ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴)} = ∅) → {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝐴} ∈ (ordTop‘𝑅))
9		breq2 5057	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝐵𝑅𝑥 ↔ 𝐵𝑅𝐴))
10	9	notbid 321	. . . . . 6 ⊢ (𝑥 = 𝐴 → (¬ 𝐵𝑅𝑥 ↔ ¬ 𝐵𝑅𝐴))
11		simprr 773	. . . . . . . 8 ⊢ (((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝑅𝐵 ∧ 𝐴 ≠ 𝐵)) → 𝐴 ≠ 𝐵)
12		simpl1 1193	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝑅𝐵 ∧ 𝐴 ≠ 𝐵)) → 𝑅 ∈ TosetRel )
13		tsrps 18093	. . . . . . . . . . 11 ⊢ (𝑅 ∈ TosetRel → 𝑅 ∈ PosetRel)
14	12, 13	syl 17	. . . . . . . . . 10 ⊢ (((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝑅𝐵 ∧ 𝐴 ≠ 𝐵)) → 𝑅 ∈ PosetRel)
15		simprl 771	. . . . . . . . . 10 ⊢ (((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝑅𝐵 ∧ 𝐴 ≠ 𝐵)) → 𝐴𝑅𝐵)
16		psasym 18082	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ PosetRel ∧ 𝐴𝑅𝐵 ∧ 𝐵𝑅𝐴) → 𝐴 = 𝐵)
17	16	3expia 1123	. . . . . . . . . 10 ⊢ ((𝑅 ∈ PosetRel ∧ 𝐴𝑅𝐵) → (𝐵𝑅𝐴 → 𝐴 = 𝐵))
18	14, 15, 17	syl2anc 587	. . . . . . . . 9 ⊢ (((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝑅𝐵 ∧ 𝐴 ≠ 𝐵)) → (𝐵𝑅𝐴 → 𝐴 = 𝐵))
19	18	necon3ad 2953	. . . . . . . 8 ⊢ (((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝑅𝐵 ∧ 𝐴 ≠ 𝐵)) → (𝐴 ≠ 𝐵 → ¬ 𝐵𝑅𝐴))
20	11, 19	mpd 15	. . . . . . 7 ⊢ (((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝑅𝐵 ∧ 𝐴 ≠ 𝐵)) → ¬ 𝐵𝑅𝐴)
21	20	adantr 484	. . . . . 6 ⊢ ((((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝑅𝐵 ∧ 𝐴 ≠ 𝐵)) ∧ {𝑥 ∈ 𝑋 ∣ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴)} = ∅) → ¬ 𝐵𝑅𝐴)
22	10, 6, 21	elrabd 3604	. . . . 5 ⊢ ((((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝑅𝐵 ∧ 𝐴 ≠ 𝐵)) ∧ {𝑥 ∈ 𝑋 ∣ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴)} = ∅) → 𝐴 ∈ {𝑥 ∈ 𝑋 ∣ ¬ 𝐵𝑅𝑥})
23		breq1 5056	. . . . . . 7 ⊢ (𝑥 = 𝐵 → (𝑥𝑅𝐴 ↔ 𝐵𝑅𝐴))
24	23	notbid 321	. . . . . 6 ⊢ (𝑥 = 𝐵 → (¬ 𝑥𝑅𝐴 ↔ ¬ 𝐵𝑅𝐴))
25	24, 2, 21	elrabd 3604	. . . . 5 ⊢ ((((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝑅𝐵 ∧ 𝐴 ≠ 𝐵)) ∧ {𝑥 ∈ 𝑋 ∣ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴)} = ∅) → 𝐵 ∈ {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝐴})
26		simpr 488	. . . . 5 ⊢ ((((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝑅𝐵 ∧ 𝐴 ≠ 𝐵)) ∧ {𝑥 ∈ 𝑋 ∣ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴)} = ∅) → {𝑥 ∈ 𝑋 ∣ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴)} = ∅)
27		eleq2 2826	. . . . . . 7 ⊢ (𝑚 = {𝑥 ∈ 𝑋 ∣ ¬ 𝐵𝑅𝑥} → (𝐴 ∈ 𝑚 ↔ 𝐴 ∈ {𝑥 ∈ 𝑋 ∣ ¬ 𝐵𝑅𝑥}))
28		ineq1 4120	. . . . . . . 8 ⊢ (𝑚 = {𝑥 ∈ 𝑋 ∣ ¬ 𝐵𝑅𝑥} → (𝑚 ∩ 𝑛) = ({𝑥 ∈ 𝑋 ∣ ¬ 𝐵𝑅𝑥} ∩ 𝑛))
29	28	eqeq1d 2739	. . . . . . 7 ⊢ (𝑚 = {𝑥 ∈ 𝑋 ∣ ¬ 𝐵𝑅𝑥} → ((𝑚 ∩ 𝑛) = ∅ ↔ ({𝑥 ∈ 𝑋 ∣ ¬ 𝐵𝑅𝑥} ∩ 𝑛) = ∅))
30	27, 29	3anbi13d 1440	. . . . . 6 ⊢ (𝑚 = {𝑥 ∈ 𝑋 ∣ ¬ 𝐵𝑅𝑥} → ((𝐴 ∈ 𝑚 ∧ 𝐵 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅) ↔ (𝐴 ∈ {𝑥 ∈ 𝑋 ∣ ¬ 𝐵𝑅𝑥} ∧ 𝐵 ∈ 𝑛 ∧ ({𝑥 ∈ 𝑋 ∣ ¬ 𝐵𝑅𝑥} ∩ 𝑛) = ∅)))
31		eleq2 2826	. . . . . . 7 ⊢ (𝑛 = {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝐴} → (𝐵 ∈ 𝑛 ↔ 𝐵 ∈ {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝐴}))
32		ineq2 4121	. . . . . . . . 9 ⊢ (𝑛 = {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝐴} → ({𝑥 ∈ 𝑋 ∣ ¬ 𝐵𝑅𝑥} ∩ 𝑛) = ({𝑥 ∈ 𝑋 ∣ ¬ 𝐵𝑅𝑥} ∩ {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝐴}))
33		inrab 4221	. . . . . . . . 9 ⊢ ({𝑥 ∈ 𝑋 ∣ ¬ 𝐵𝑅𝑥} ∩ {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝐴}) = {𝑥 ∈ 𝑋 ∣ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴)}
34	32, 33	eqtrdi 2794	. . . . . . . 8 ⊢ (𝑛 = {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝐴} → ({𝑥 ∈ 𝑋 ∣ ¬ 𝐵𝑅𝑥} ∩ 𝑛) = {𝑥 ∈ 𝑋 ∣ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴)})
35	34	eqeq1d 2739	. . . . . . 7 ⊢ (𝑛 = {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝐴} → (({𝑥 ∈ 𝑋 ∣ ¬ 𝐵𝑅𝑥} ∩ 𝑛) = ∅ ↔ {𝑥 ∈ 𝑋 ∣ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴)} = ∅))
36	31, 35	3anbi23d 1441	. . . . . 6 ⊢ (𝑛 = {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝐴} → ((𝐴 ∈ {𝑥 ∈ 𝑋 ∣ ¬ 𝐵𝑅𝑥} ∧ 𝐵 ∈ 𝑛 ∧ ({𝑥 ∈ 𝑋 ∣ ¬ 𝐵𝑅𝑥} ∩ 𝑛) = ∅) ↔ (𝐴 ∈ {𝑥 ∈ 𝑋 ∣ ¬ 𝐵𝑅𝑥} ∧ 𝐵 ∈ {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝐴} ∧ {𝑥 ∈ 𝑋 ∣ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴)} = ∅)))
37	30, 36	rspc2ev 3549	. . . . 5 ⊢ (({𝑥 ∈ 𝑋 ∣ ¬ 𝐵𝑅𝑥} ∈ (ordTop‘𝑅) ∧ {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝐴} ∈ (ordTop‘𝑅) ∧ (𝐴 ∈ {𝑥 ∈ 𝑋 ∣ ¬ 𝐵𝑅𝑥} ∧ 𝐵 ∈ {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝐴} ∧ {𝑥 ∈ 𝑋 ∣ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴)} = ∅)) → ∃𝑚 ∈ (ordTop‘𝑅)∃𝑛 ∈ (ordTop‘𝑅)(𝐴 ∈ 𝑚 ∧ 𝐵 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅))
38	5, 8, 22, 25, 26, 37	syl113anc 1384	. . . 4 ⊢ ((((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝑅𝐵 ∧ 𝐴 ≠ 𝐵)) ∧ {𝑥 ∈ 𝑋 ∣ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴)} = ∅) → ∃𝑚 ∈ (ordTop‘𝑅)∃𝑛 ∈ (ordTop‘𝑅)(𝐴 ∈ 𝑚 ∧ 𝐵 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅))
39	38	ex 416	. . 3 ⊢ (((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝑅𝐵 ∧ 𝐴 ≠ 𝐵)) → ({𝑥 ∈ 𝑋 ∣ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴)} = ∅ → ∃𝑚 ∈ (ordTop‘𝑅)∃𝑛 ∈ (ordTop‘𝑅)(𝐴 ∈ 𝑚 ∧ 𝐵 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅)))
40		rabn0 4300	. . . 4 ⊢ ({𝑥 ∈ 𝑋 ∣ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴)} ≠ ∅ ↔ ∃𝑥 ∈ 𝑋 (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴))
41		simpll1 1214	. . . . . . 7 ⊢ ((((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝑅𝐵 ∧ 𝐴 ≠ 𝐵)) ∧ (𝑥 ∈ 𝑋 ∧ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴))) → 𝑅 ∈ TosetRel )
42		simprl 771	. . . . . . 7 ⊢ ((((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝑅𝐵 ∧ 𝐴 ≠ 𝐵)) ∧ (𝑥 ∈ 𝑋 ∧ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴))) → 𝑥 ∈ 𝑋)
43	3	ordtopn2 22092	. . . . . . 7 ⊢ ((𝑅 ∈ TosetRel ∧ 𝑥 ∈ 𝑋) → {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦} ∈ (ordTop‘𝑅))
44	41, 42, 43	syl2anc 587	. . . . . 6 ⊢ ((((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝑅𝐵 ∧ 𝐴 ≠ 𝐵)) ∧ (𝑥 ∈ 𝑋 ∧ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴))) → {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦} ∈ (ordTop‘𝑅))
45	3	ordtopn1 22091	. . . . . . 7 ⊢ ((𝑅 ∈ TosetRel ∧ 𝑥 ∈ 𝑋) → {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥} ∈ (ordTop‘𝑅))
46	41, 42, 45	syl2anc 587	. . . . . 6 ⊢ ((((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝑅𝐵 ∧ 𝐴 ≠ 𝐵)) ∧ (𝑥 ∈ 𝑋 ∧ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴))) → {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥} ∈ (ordTop‘𝑅))
47		breq2 5057	. . . . . . . 8 ⊢ (𝑦 = 𝐴 → (𝑥𝑅𝑦 ↔ 𝑥𝑅𝐴))
48	47	notbid 321	. . . . . . 7 ⊢ (𝑦 = 𝐴 → (¬ 𝑥𝑅𝑦 ↔ ¬ 𝑥𝑅𝐴))
49		simpll2 1215	. . . . . . 7 ⊢ ((((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝑅𝐵 ∧ 𝐴 ≠ 𝐵)) ∧ (𝑥 ∈ 𝑋 ∧ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴))) → 𝐴 ∈ 𝑋)
50		simprrr 782	. . . . . . 7 ⊢ ((((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝑅𝐵 ∧ 𝐴 ≠ 𝐵)) ∧ (𝑥 ∈ 𝑋 ∧ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴))) → ¬ 𝑥𝑅𝐴)
51	48, 49, 50	elrabd 3604	. . . . . 6 ⊢ ((((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝑅𝐵 ∧ 𝐴 ≠ 𝐵)) ∧ (𝑥 ∈ 𝑋 ∧ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴))) → 𝐴 ∈ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦})
52		breq1 5056	. . . . . . . 8 ⊢ (𝑦 = 𝐵 → (𝑦𝑅𝑥 ↔ 𝐵𝑅𝑥))
53	52	notbid 321	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (¬ 𝑦𝑅𝑥 ↔ ¬ 𝐵𝑅𝑥))
54		simpll3 1216	. . . . . . 7 ⊢ ((((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝑅𝐵 ∧ 𝐴 ≠ 𝐵)) ∧ (𝑥 ∈ 𝑋 ∧ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴))) → 𝐵 ∈ 𝑋)
55		simprrl 781	. . . . . . 7 ⊢ ((((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝑅𝐵 ∧ 𝐴 ≠ 𝐵)) ∧ (𝑥 ∈ 𝑋 ∧ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴))) → ¬ 𝐵𝑅𝑥)
56	53, 54, 55	elrabd 3604	. . . . . 6 ⊢ ((((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝑅𝐵 ∧ 𝐴 ≠ 𝐵)) ∧ (𝑥 ∈ 𝑋 ∧ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴))) → 𝐵 ∈ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥})
57	41, 42	jca 515	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝑅𝐵 ∧ 𝐴 ≠ 𝐵)) ∧ (𝑥 ∈ 𝑋 ∧ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴))) → (𝑅 ∈ TosetRel ∧ 𝑥 ∈ 𝑋))
58	3	tsrlin 18091	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ TosetRel ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥))
59	58	3expa 1120	. . . . . . . . . 10 ⊢ (((𝑅 ∈ TosetRel ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → (𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥))
60	57, 59	sylan 583	. . . . . . . . 9 ⊢ (((((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝑅𝐵 ∧ 𝐴 ≠ 𝐵)) ∧ (𝑥 ∈ 𝑋 ∧ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴))) ∧ 𝑦 ∈ 𝑋) → (𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥))
61		oran 990	. . . . . . . . 9 ⊢ ((𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥) ↔ ¬ (¬ 𝑥𝑅𝑦 ∧ ¬ 𝑦𝑅𝑥))
62	60, 61	sylib 221	. . . . . . . 8 ⊢ (((((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝑅𝐵 ∧ 𝐴 ≠ 𝐵)) ∧ (𝑥 ∈ 𝑋 ∧ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴))) ∧ 𝑦 ∈ 𝑋) → ¬ (¬ 𝑥𝑅𝑦 ∧ ¬ 𝑦𝑅𝑥))
63	62	ralrimiva 3105	. . . . . . 7 ⊢ ((((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝑅𝐵 ∧ 𝐴 ≠ 𝐵)) ∧ (𝑥 ∈ 𝑋 ∧ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴))) → ∀𝑦 ∈ 𝑋 ¬ (¬ 𝑥𝑅𝑦 ∧ ¬ 𝑦𝑅𝑥))
64		rabeq0 4299	. . . . . . 7 ⊢ ({𝑦 ∈ 𝑋 ∣ (¬ 𝑥𝑅𝑦 ∧ ¬ 𝑦𝑅𝑥)} = ∅ ↔ ∀𝑦 ∈ 𝑋 ¬ (¬ 𝑥𝑅𝑦 ∧ ¬ 𝑦𝑅𝑥))
65	63, 64	sylibr 237	. . . . . 6 ⊢ ((((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝑅𝐵 ∧ 𝐴 ≠ 𝐵)) ∧ (𝑥 ∈ 𝑋 ∧ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴))) → {𝑦 ∈ 𝑋 ∣ (¬ 𝑥𝑅𝑦 ∧ ¬ 𝑦𝑅𝑥)} = ∅)
66		eleq2 2826	. . . . . . . 8 ⊢ (𝑚 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦} → (𝐴 ∈ 𝑚 ↔ 𝐴 ∈ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}))
67		ineq1 4120	. . . . . . . . 9 ⊢ (𝑚 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦} → (𝑚 ∩ 𝑛) = ({𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦} ∩ 𝑛))
68	67	eqeq1d 2739	. . . . . . . 8 ⊢ (𝑚 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦} → ((𝑚 ∩ 𝑛) = ∅ ↔ ({𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦} ∩ 𝑛) = ∅))
69	66, 68	3anbi13d 1440	. . . . . . 7 ⊢ (𝑚 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦} → ((𝐴 ∈ 𝑚 ∧ 𝐵 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅) ↔ (𝐴 ∈ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦} ∧ 𝐵 ∈ 𝑛 ∧ ({𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦} ∩ 𝑛) = ∅)))
70		eleq2 2826	. . . . . . . 8 ⊢ (𝑛 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥} → (𝐵 ∈ 𝑛 ↔ 𝐵 ∈ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}))
71		ineq2 4121	. . . . . . . . . 10 ⊢ (𝑛 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥} → ({𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦} ∩ 𝑛) = ({𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦} ∩ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}))
72		inrab 4221	. . . . . . . . . 10 ⊢ ({𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦} ∩ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}) = {𝑦 ∈ 𝑋 ∣ (¬ 𝑥𝑅𝑦 ∧ ¬ 𝑦𝑅𝑥)}
73	71, 72	eqtrdi 2794	. . . . . . . . 9 ⊢ (𝑛 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥} → ({𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦} ∩ 𝑛) = {𝑦 ∈ 𝑋 ∣ (¬ 𝑥𝑅𝑦 ∧ ¬ 𝑦𝑅𝑥)})
74	73	eqeq1d 2739	. . . . . . . 8 ⊢ (𝑛 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥} → (({𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦} ∩ 𝑛) = ∅ ↔ {𝑦 ∈ 𝑋 ∣ (¬ 𝑥𝑅𝑦 ∧ ¬ 𝑦𝑅𝑥)} = ∅))
75	70, 74	3anbi23d 1441	. . . . . . 7 ⊢ (𝑛 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥} → ((𝐴 ∈ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦} ∧ 𝐵 ∈ 𝑛 ∧ ({𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦} ∩ 𝑛) = ∅) ↔ (𝐴 ∈ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦} ∧ 𝐵 ∈ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥} ∧ {𝑦 ∈ 𝑋 ∣ (¬ 𝑥𝑅𝑦 ∧ ¬ 𝑦𝑅𝑥)} = ∅)))
76	69, 75	rspc2ev 3549	. . . . . 6 ⊢ (({𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦} ∈ (ordTop‘𝑅) ∧ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥} ∈ (ordTop‘𝑅) ∧ (𝐴 ∈ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦} ∧ 𝐵 ∈ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥} ∧ {𝑦 ∈ 𝑋 ∣ (¬ 𝑥𝑅𝑦 ∧ ¬ 𝑦𝑅𝑥)} = ∅)) → ∃𝑚 ∈ (ordTop‘𝑅)∃𝑛 ∈ (ordTop‘𝑅)(𝐴 ∈ 𝑚 ∧ 𝐵 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅))
77	44, 46, 51, 56, 65, 76	syl113anc 1384	. . . . 5 ⊢ ((((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝑅𝐵 ∧ 𝐴 ≠ 𝐵)) ∧ (𝑥 ∈ 𝑋 ∧ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴))) → ∃𝑚 ∈ (ordTop‘𝑅)∃𝑛 ∈ (ordTop‘𝑅)(𝐴 ∈ 𝑚 ∧ 𝐵 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅))
78	77	rexlimdvaa 3204	. . . 4 ⊢ (((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝑅𝐵 ∧ 𝐴 ≠ 𝐵)) → (∃𝑥 ∈ 𝑋 (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴) → ∃𝑚 ∈ (ordTop‘𝑅)∃𝑛 ∈ (ordTop‘𝑅)(𝐴 ∈ 𝑚 ∧ 𝐵 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅)))
79	40, 78	syl5bi 245	. . 3 ⊢ (((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝑅𝐵 ∧ 𝐴 ≠ 𝐵)) → ({𝑥 ∈ 𝑋 ∣ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴)} ≠ ∅ → ∃𝑚 ∈ (ordTop‘𝑅)∃𝑛 ∈ (ordTop‘𝑅)(𝐴 ∈ 𝑚 ∧ 𝐵 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅)))
80	39, 79	pm2.61dne 3028	. 2 ⊢ (((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝑅𝐵 ∧ 𝐴 ≠ 𝐵)) → ∃𝑚 ∈ (ordTop‘𝑅)∃𝑛 ∈ (ordTop‘𝑅)(𝐴 ∈ 𝑚 ∧ 𝐵 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅))
81	80	exp32 424	1 ⊢ ((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝑅𝐵 → (𝐴 ≠ 𝐵 → ∃𝑚 ∈ (ordTop‘𝑅)∃𝑛 ∈ (ordTop‘𝑅)(𝐴 ∈ 𝑚 ∧ 𝐵 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   ∨ wo 847   ∧ w3a 1089   = wceq 1543   ∈ wcel 2110   ≠ wne 2940  ∀wral 3061  ∃wrex 3062  {crab 3065   ∩ cin 3865  ∅c0 4237   class class class wbr 5053  dom cdm 5551  ‘cfv 6380  ordTopcordt 17004  PosetRelcps 18070   TosetRel ctsr 18071

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-int 4860  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-om 7645  df-1o 8202  df-en 8627  df-fin 8630  df-fi 9027  df-topgen 16948  df-ordt 17006  df-ps 18072  df-tsr 18073  df-bases 21843

This theorem is referenced by:  ordthaus  22281