Theorem ordthauslem 21187

Description: Lemma for ordthaus 21188. (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 1100	. . . . . 6 ⊢ ((((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝑅𝐵 ∧ 𝐴 ≠ 𝐵)) ∧ {𝑥 ∈ 𝑋 ∣ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴)} = ∅) → 𝑅 ∈ TosetRel )
2		simpll3 1102	. . . . . 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 20999	. . . . . 6 ⊢ ((𝑅 ∈ TosetRel ∧ 𝐵 ∈ 𝑋) → {𝑥 ∈ 𝑋 ∣ ¬ 𝐵𝑅𝑥} ∈ (ordTop‘𝑅))
5	1, 2, 4	syl2anc 693	. . . . 5 ⊢ ((((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝑅𝐵 ∧ 𝐴 ≠ 𝐵)) ∧ {𝑥 ∈ 𝑋 ∣ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴)} = ∅) → {𝑥 ∈ 𝑋 ∣ ¬ 𝐵𝑅𝑥} ∈ (ordTop‘𝑅))
6		simpll2 1101	. . . . . 6 ⊢ ((((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝑅𝐵 ∧ 𝐴 ≠ 𝐵)) ∧ {𝑥 ∈ 𝑋 ∣ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴)} = ∅) → 𝐴 ∈ 𝑋)
7	3	ordtopn1 20998	. . . . . 6 ⊢ ((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋) → {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝐴} ∈ (ordTop‘𝑅))
8	1, 6, 7	syl2anc 693	. . . . 5 ⊢ ((((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝑅𝐵 ∧ 𝐴 ≠ 𝐵)) ∧ {𝑥 ∈ 𝑋 ∣ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴)} = ∅) → {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝐴} ∈ (ordTop‘𝑅))
9		simprr 796	. . . . . . . 8 ⊢ (((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝑅𝐵 ∧ 𝐴 ≠ 𝐵)) → 𝐴 ≠ 𝐵)
10		simpl1 1064	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝑅𝐵 ∧ 𝐴 ≠ 𝐵)) → 𝑅 ∈ TosetRel )
11		tsrps 17221	. . . . . . . . . . 11 ⊢ (𝑅 ∈ TosetRel → 𝑅 ∈ PosetRel)
12	10, 11	syl 17	. . . . . . . . . 10 ⊢ (((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝑅𝐵 ∧ 𝐴 ≠ 𝐵)) → 𝑅 ∈ PosetRel)
13		simprl 794	. . . . . . . . . 10 ⊢ (((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝑅𝐵 ∧ 𝐴 ≠ 𝐵)) → 𝐴𝑅𝐵)
14		psasym 17210	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ PosetRel ∧ 𝐴𝑅𝐵 ∧ 𝐵𝑅𝐴) → 𝐴 = 𝐵)
15	14	3expia 1267	. . . . . . . . . 10 ⊢ ((𝑅 ∈ PosetRel ∧ 𝐴𝑅𝐵) → (𝐵𝑅𝐴 → 𝐴 = 𝐵))
16	12, 13, 15	syl2anc 693	. . . . . . . . 9 ⊢ (((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝑅𝐵 ∧ 𝐴 ≠ 𝐵)) → (𝐵𝑅𝐴 → 𝐴 = 𝐵))
17	16	necon3ad 2807	. . . . . . . 8 ⊢ (((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝑅𝐵 ∧ 𝐴 ≠ 𝐵)) → (𝐴 ≠ 𝐵 → ¬ 𝐵𝑅𝐴))
18	9, 17	mpd 15	. . . . . . 7 ⊢ (((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝑅𝐵 ∧ 𝐴 ≠ 𝐵)) → ¬ 𝐵𝑅𝐴)
19	18	adantr 481	. . . . . 6 ⊢ ((((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝑅𝐵 ∧ 𝐴 ≠ 𝐵)) ∧ {𝑥 ∈ 𝑋 ∣ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴)} = ∅) → ¬ 𝐵𝑅𝐴)
20		breq2 4657	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝐵𝑅𝑥 ↔ 𝐵𝑅𝐴))
21	20	notbid 308	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (¬ 𝐵𝑅𝑥 ↔ ¬ 𝐵𝑅𝐴))
22	21	elrab 3363	. . . . . 6 ⊢ (𝐴 ∈ {𝑥 ∈ 𝑋 ∣ ¬ 𝐵𝑅𝑥} ↔ (𝐴 ∈ 𝑋 ∧ ¬ 𝐵𝑅𝐴))
23	6, 19, 22	sylanbrc 698	. . . . 5 ⊢ ((((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝑅𝐵 ∧ 𝐴 ≠ 𝐵)) ∧ {𝑥 ∈ 𝑋 ∣ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴)} = ∅) → 𝐴 ∈ {𝑥 ∈ 𝑋 ∣ ¬ 𝐵𝑅𝑥})
24		breq1 4656	. . . . . . . 8 ⊢ (𝑥 = 𝐵 → (𝑥𝑅𝐴 ↔ 𝐵𝑅𝐴))
25	24	notbid 308	. . . . . . 7 ⊢ (𝑥 = 𝐵 → (¬ 𝑥𝑅𝐴 ↔ ¬ 𝐵𝑅𝐴))
26	25	elrab 3363	. . . . . 6 ⊢ (𝐵 ∈ {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝐴} ↔ (𝐵 ∈ 𝑋 ∧ ¬ 𝐵𝑅𝐴))
27	2, 19, 26	sylanbrc 698	. . . . 5 ⊢ ((((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝑅𝐵 ∧ 𝐴 ≠ 𝐵)) ∧ {𝑥 ∈ 𝑋 ∣ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴)} = ∅) → 𝐵 ∈ {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝐴})
28		simpr 477	. . . . 5 ⊢ ((((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝑅𝐵 ∧ 𝐴 ≠ 𝐵)) ∧ {𝑥 ∈ 𝑋 ∣ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴)} = ∅) → {𝑥 ∈ 𝑋 ∣ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴)} = ∅)
29		eleq2 2690	. . . . . . 7 ⊢ (𝑚 = {𝑥 ∈ 𝑋 ∣ ¬ 𝐵𝑅𝑥} → (𝐴 ∈ 𝑚 ↔ 𝐴 ∈ {𝑥 ∈ 𝑋 ∣ ¬ 𝐵𝑅𝑥}))
30		ineq1 3807	. . . . . . . 8 ⊢ (𝑚 = {𝑥 ∈ 𝑋 ∣ ¬ 𝐵𝑅𝑥} → (𝑚 ∩ 𝑛) = ({𝑥 ∈ 𝑋 ∣ ¬ 𝐵𝑅𝑥} ∩ 𝑛))
31	30	eqeq1d 2624	. . . . . . 7 ⊢ (𝑚 = {𝑥 ∈ 𝑋 ∣ ¬ 𝐵𝑅𝑥} → ((𝑚 ∩ 𝑛) = ∅ ↔ ({𝑥 ∈ 𝑋 ∣ ¬ 𝐵𝑅𝑥} ∩ 𝑛) = ∅))
32	29, 31	3anbi13d 1401	. . . . . 6 ⊢ (𝑚 = {𝑥 ∈ 𝑋 ∣ ¬ 𝐵𝑅𝑥} → ((𝐴 ∈ 𝑚 ∧ 𝐵 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅) ↔ (𝐴 ∈ {𝑥 ∈ 𝑋 ∣ ¬ 𝐵𝑅𝑥} ∧ 𝐵 ∈ 𝑛 ∧ ({𝑥 ∈ 𝑋 ∣ ¬ 𝐵𝑅𝑥} ∩ 𝑛) = ∅)))
33		eleq2 2690	. . . . . . 7 ⊢ (𝑛 = {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝐴} → (𝐵 ∈ 𝑛 ↔ 𝐵 ∈ {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝐴}))
34		ineq2 3808	. . . . . . . . 9 ⊢ (𝑛 = {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝐴} → ({𝑥 ∈ 𝑋 ∣ ¬ 𝐵𝑅𝑥} ∩ 𝑛) = ({𝑥 ∈ 𝑋 ∣ ¬ 𝐵𝑅𝑥} ∩ {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝐴}))
35		inrab 3899	. . . . . . . . 9 ⊢ ({𝑥 ∈ 𝑋 ∣ ¬ 𝐵𝑅𝑥} ∩ {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝐴}) = {𝑥 ∈ 𝑋 ∣ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴)}
36	34, 35	syl6eq 2672	. . . . . . . 8 ⊢ (𝑛 = {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝐴} → ({𝑥 ∈ 𝑋 ∣ ¬ 𝐵𝑅𝑥} ∩ 𝑛) = {𝑥 ∈ 𝑋 ∣ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴)})
37	36	eqeq1d 2624	. . . . . . 7 ⊢ (𝑛 = {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝐴} → (({𝑥 ∈ 𝑋 ∣ ¬ 𝐵𝑅𝑥} ∩ 𝑛) = ∅ ↔ {𝑥 ∈ 𝑋 ∣ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴)} = ∅))
38	33, 37	3anbi23d 1402	. . . . . 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)} = ∅)))
39	32, 38	rspc2ev 3324	. . . . 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) = ∅))
40	5, 8, 23, 27, 28, 39	syl113anc 1338	. . . 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) = ∅))
41	40	ex 450	. . 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) = ∅)))
42		rabn0 3958	. . . 4 ⊢ ({𝑥 ∈ 𝑋 ∣ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴)} ≠ ∅ ↔ ∃𝑥 ∈ 𝑋 (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴))
43		simpll1 1100	. . . . . . 7 ⊢ ((((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝑅𝐵 ∧ 𝐴 ≠ 𝐵)) ∧ (𝑥 ∈ 𝑋 ∧ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴))) → 𝑅 ∈ TosetRel )
44		simprl 794	. . . . . . 7 ⊢ ((((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝑅𝐵 ∧ 𝐴 ≠ 𝐵)) ∧ (𝑥 ∈ 𝑋 ∧ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴))) → 𝑥 ∈ 𝑋)
45	3	ordtopn2 20999	. . . . . . 7 ⊢ ((𝑅 ∈ TosetRel ∧ 𝑥 ∈ 𝑋) → {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦} ∈ (ordTop‘𝑅))
46	43, 44, 45	syl2anc 693	. . . . . 6 ⊢ ((((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝑅𝐵 ∧ 𝐴 ≠ 𝐵)) ∧ (𝑥 ∈ 𝑋 ∧ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴))) → {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦} ∈ (ordTop‘𝑅))
47	3	ordtopn1 20998	. . . . . . 7 ⊢ ((𝑅 ∈ TosetRel ∧ 𝑥 ∈ 𝑋) → {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥} ∈ (ordTop‘𝑅))
48	43, 44, 47	syl2anc 693	. . . . . 6 ⊢ ((((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝑅𝐵 ∧ 𝐴 ≠ 𝐵)) ∧ (𝑥 ∈ 𝑋 ∧ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴))) → {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥} ∈ (ordTop‘𝑅))
49		simpll2 1101	. . . . . . 7 ⊢ ((((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝑅𝐵 ∧ 𝐴 ≠ 𝐵)) ∧ (𝑥 ∈ 𝑋 ∧ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴))) → 𝐴 ∈ 𝑋)
50		simprrr 805	. . . . . . 7 ⊢ ((((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝑅𝐵 ∧ 𝐴 ≠ 𝐵)) ∧ (𝑥 ∈ 𝑋 ∧ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴))) → ¬ 𝑥𝑅𝐴)
51		breq2 4657	. . . . . . . . 9 ⊢ (𝑦 = 𝐴 → (𝑥𝑅𝑦 ↔ 𝑥𝑅𝐴))
52	51	notbid 308	. . . . . . . 8 ⊢ (𝑦 = 𝐴 → (¬ 𝑥𝑅𝑦 ↔ ¬ 𝑥𝑅𝐴))
53	52	elrab 3363	. . . . . . 7 ⊢ (𝐴 ∈ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦} ↔ (𝐴 ∈ 𝑋 ∧ ¬ 𝑥𝑅𝐴))
54	49, 50, 53	sylanbrc 698	. . . . . 6 ⊢ ((((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝑅𝐵 ∧ 𝐴 ≠ 𝐵)) ∧ (𝑥 ∈ 𝑋 ∧ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴))) → 𝐴 ∈ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦})
55		simpll3 1102	. . . . . . 7 ⊢ ((((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝑅𝐵 ∧ 𝐴 ≠ 𝐵)) ∧ (𝑥 ∈ 𝑋 ∧ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴))) → 𝐵 ∈ 𝑋)
56		simprrl 804	. . . . . . 7 ⊢ ((((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝑅𝐵 ∧ 𝐴 ≠ 𝐵)) ∧ (𝑥 ∈ 𝑋 ∧ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴))) → ¬ 𝐵𝑅𝑥)
57		breq1 4656	. . . . . . . . 9 ⊢ (𝑦 = 𝐵 → (𝑦𝑅𝑥 ↔ 𝐵𝑅𝑥))
58	57	notbid 308	. . . . . . . 8 ⊢ (𝑦 = 𝐵 → (¬ 𝑦𝑅𝑥 ↔ ¬ 𝐵𝑅𝑥))
59	58	elrab 3363	. . . . . . 7 ⊢ (𝐵 ∈ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥} ↔ (𝐵 ∈ 𝑋 ∧ ¬ 𝐵𝑅𝑥))
60	55, 56, 59	sylanbrc 698	. . . . . 6 ⊢ ((((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝑅𝐵 ∧ 𝐴 ≠ 𝐵)) ∧ (𝑥 ∈ 𝑋 ∧ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴))) → 𝐵 ∈ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥})
61	43, 44	jca 554	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝑅𝐵 ∧ 𝐴 ≠ 𝐵)) ∧ (𝑥 ∈ 𝑋 ∧ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴))) → (𝑅 ∈ TosetRel ∧ 𝑥 ∈ 𝑋))
62	3	tsrlin 17219	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ TosetRel ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥))
63	62	3expa 1265	. . . . . . . . . 10 ⊢ (((𝑅 ∈ TosetRel ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → (𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥))
64	61, 63	sylan 488	. . . . . . . . 9 ⊢ (((((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝑅𝐵 ∧ 𝐴 ≠ 𝐵)) ∧ (𝑥 ∈ 𝑋 ∧ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴))) ∧ 𝑦 ∈ 𝑋) → (𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥))
65		oran 517	. . . . . . . . 9 ⊢ ((𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥) ↔ ¬ (¬ 𝑥𝑅𝑦 ∧ ¬ 𝑦𝑅𝑥))
66	64, 65	sylib 208	. . . . . . . 8 ⊢ (((((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝑅𝐵 ∧ 𝐴 ≠ 𝐵)) ∧ (𝑥 ∈ 𝑋 ∧ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴))) ∧ 𝑦 ∈ 𝑋) → ¬ (¬ 𝑥𝑅𝑦 ∧ ¬ 𝑦𝑅𝑥))
67	66	ralrimiva 2966	. . . . . . 7 ⊢ ((((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝑅𝐵 ∧ 𝐴 ≠ 𝐵)) ∧ (𝑥 ∈ 𝑋 ∧ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴))) → ∀𝑦 ∈ 𝑋 ¬ (¬ 𝑥𝑅𝑦 ∧ ¬ 𝑦𝑅𝑥))
68		rabeq0 3957	. . . . . . 7 ⊢ ({𝑦 ∈ 𝑋 ∣ (¬ 𝑥𝑅𝑦 ∧ ¬ 𝑦𝑅𝑥)} = ∅ ↔ ∀𝑦 ∈ 𝑋 ¬ (¬ 𝑥𝑅𝑦 ∧ ¬ 𝑦𝑅𝑥))
69	67, 68	sylibr 224	. . . . . 6 ⊢ ((((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝑅𝐵 ∧ 𝐴 ≠ 𝐵)) ∧ (𝑥 ∈ 𝑋 ∧ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴))) → {𝑦 ∈ 𝑋 ∣ (¬ 𝑥𝑅𝑦 ∧ ¬ 𝑦𝑅𝑥)} = ∅)
70		eleq2 2690	. . . . . . . 8 ⊢ (𝑚 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦} → (𝐴 ∈ 𝑚 ↔ 𝐴 ∈ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}))
71		ineq1 3807	. . . . . . . . 9 ⊢ (𝑚 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦} → (𝑚 ∩ 𝑛) = ({𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦} ∩ 𝑛))
72	71	eqeq1d 2624	. . . . . . . 8 ⊢ (𝑚 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦} → ((𝑚 ∩ 𝑛) = ∅ ↔ ({𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦} ∩ 𝑛) = ∅))
73	70, 72	3anbi13d 1401	. . . . . . 7 ⊢ (𝑚 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦} → ((𝐴 ∈ 𝑚 ∧ 𝐵 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅) ↔ (𝐴 ∈ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦} ∧ 𝐵 ∈ 𝑛 ∧ ({𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦} ∩ 𝑛) = ∅)))
74		eleq2 2690	. . . . . . . 8 ⊢ (𝑛 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥} → (𝐵 ∈ 𝑛 ↔ 𝐵 ∈ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}))
75		ineq2 3808	. . . . . . . . . 10 ⊢ (𝑛 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥} → ({𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦} ∩ 𝑛) = ({𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦} ∩ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}))
76		inrab 3899	. . . . . . . . . 10 ⊢ ({𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦} ∩ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}) = {𝑦 ∈ 𝑋 ∣ (¬ 𝑥𝑅𝑦 ∧ ¬ 𝑦𝑅𝑥)}
77	75, 76	syl6eq 2672	. . . . . . . . 9 ⊢ (𝑛 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥} → ({𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦} ∩ 𝑛) = {𝑦 ∈ 𝑋 ∣ (¬ 𝑥𝑅𝑦 ∧ ¬ 𝑦𝑅𝑥)})
78	77	eqeq1d 2624	. . . . . . . 8 ⊢ (𝑛 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥} → (({𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦} ∩ 𝑛) = ∅ ↔ {𝑦 ∈ 𝑋 ∣ (¬ 𝑥𝑅𝑦 ∧ ¬ 𝑦𝑅𝑥)} = ∅))
79	74, 78	3anbi23d 1402	. . . . . . 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)} = ∅)))
80	73, 79	rspc2ev 3324	. . . . . 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) = ∅))
81	46, 48, 54, 60, 69, 80	syl113anc 1338	. . . . 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) = ∅))
82	81	rexlimdvaa 3032	. . . 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) = ∅)))
83	42, 82	syl5bi 232	. . 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) = ∅)))
84	41, 83	pm2.61dne 2880	. 2 ⊢ (((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝑅𝐵 ∧ 𝐴 ≠ 𝐵)) → ∃𝑚 ∈ (ordTop‘𝑅)∃𝑛 ∈ (ordTop‘𝑅)(𝐴 ∈ 𝑚 ∧ 𝐵 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅))
85	84	exp32 631	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   ∨ wo 383   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  ∀wral 2912  ∃wrex 2913  {crab 2916   ∩ cin 3573  ∅c0 3915   class class class wbr 4653  dom cdm 5114  ‘cfv 5888  ordTopcordt 16159  PosetRelcps 17198   TosetRel ctsr 17199

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-fin 7959  df-fi 8317  df-topgen 16104  df-ordt 16161  df-ps 17200  df-tsr 17201  df-bases 20750

This theorem is referenced by:  ordthaus  21188