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Theorem ordthauslem 22002
 Description: Lemma for ordthaus 22003. (Contributed by Mario Carneiro, 13-Sep-2015.)
Hypothesis
Ref Expression
ordthauslem.1 𝑋 = dom 𝑅
Assertion
Ref Expression
ordthauslem ((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝑅𝐵 → (𝐴𝐵 → ∃𝑚 ∈ (ordTop‘𝑅)∃𝑛 ∈ (ordTop‘𝑅)(𝐴𝑚𝐵𝑛 ∧ (𝑚𝑛) = ∅))))
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.
StepHypRef Expression
1 simpll1 1209 . . . . . 6 ((((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐴𝐵)) ∧ {𝑥𝑋 ∣ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴)} = ∅) → 𝑅 ∈ TosetRel )
2 simpll3 1211 . . . . . 6 ((((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐴𝐵)) ∧ {𝑥𝑋 ∣ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴)} = ∅) → 𝐵𝑋)
3 ordthauslem.1 . . . . . . 7 𝑋 = dom 𝑅
43ordtopn2 21814 . . . . . 6 ((𝑅 ∈ TosetRel ∧ 𝐵𝑋) → {𝑥𝑋 ∣ ¬ 𝐵𝑅𝑥} ∈ (ordTop‘𝑅))
51, 2, 4syl2anc 587 . . . . 5 ((((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐴𝐵)) ∧ {𝑥𝑋 ∣ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴)} = ∅) → {𝑥𝑋 ∣ ¬ 𝐵𝑅𝑥} ∈ (ordTop‘𝑅))
6 simpll2 1210 . . . . . 6 ((((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐴𝐵)) ∧ {𝑥𝑋 ∣ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴)} = ∅) → 𝐴𝑋)
73ordtopn1 21813 . . . . . 6 ((𝑅 ∈ TosetRel ∧ 𝐴𝑋) → {𝑥𝑋 ∣ ¬ 𝑥𝑅𝐴} ∈ (ordTop‘𝑅))
81, 6, 7syl2anc 587 . . . . 5 ((((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐴𝐵)) ∧ {𝑥𝑋 ∣ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴)} = ∅) → {𝑥𝑋 ∣ ¬ 𝑥𝑅𝐴} ∈ (ordTop‘𝑅))
9 breq2 5035 . . . . . . 7 (𝑥 = 𝐴 → (𝐵𝑅𝑥𝐵𝑅𝐴))
109notbid 321 . . . . . 6 (𝑥 = 𝐴 → (¬ 𝐵𝑅𝑥 ↔ ¬ 𝐵𝑅𝐴))
11 simprr 772 . . . . . . . 8 (((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐴𝐵)) → 𝐴𝐵)
12 simpl1 1188 . . . . . . . . . . 11 (((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐴𝐵)) → 𝑅 ∈ TosetRel )
13 tsrps 17830 . . . . . . . . . . 11 (𝑅 ∈ TosetRel → 𝑅 ∈ PosetRel)
1412, 13syl 17 . . . . . . . . . 10 (((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐴𝐵)) → 𝑅 ∈ PosetRel)
15 simprl 770 . . . . . . . . . 10 (((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐴𝐵)) → 𝐴𝑅𝐵)
16 psasym 17819 . . . . . . . . . . 11 ((𝑅 ∈ PosetRel ∧ 𝐴𝑅𝐵𝐵𝑅𝐴) → 𝐴 = 𝐵)
17163expia 1118 . . . . . . . . . 10 ((𝑅 ∈ PosetRel ∧ 𝐴𝑅𝐵) → (𝐵𝑅𝐴𝐴 = 𝐵))
1814, 15, 17syl2anc 587 . . . . . . . . 9 (((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐴𝐵)) → (𝐵𝑅𝐴𝐴 = 𝐵))
1918necon3ad 3000 . . . . . . . 8 (((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐴𝐵)) → (𝐴𝐵 → ¬ 𝐵𝑅𝐴))
2011, 19mpd 15 . . . . . . 7 (((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐴𝐵)) → ¬ 𝐵𝑅𝐴)
2120adantr 484 . . . . . 6 ((((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐴𝐵)) ∧ {𝑥𝑋 ∣ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴)} = ∅) → ¬ 𝐵𝑅𝐴)
2210, 6, 21elrabd 3630 . . . . 5 ((((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐴𝐵)) ∧ {𝑥𝑋 ∣ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴)} = ∅) → 𝐴 ∈ {𝑥𝑋 ∣ ¬ 𝐵𝑅𝑥})
23 breq1 5034 . . . . . . 7 (𝑥 = 𝐵 → (𝑥𝑅𝐴𝐵𝑅𝐴))
2423notbid 321 . . . . . 6 (𝑥 = 𝐵 → (¬ 𝑥𝑅𝐴 ↔ ¬ 𝐵𝑅𝐴))
2524, 2, 21elrabd 3630 . . . . 5 ((((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐴𝐵)) ∧ {𝑥𝑋 ∣ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴)} = ∅) → 𝐵 ∈ {𝑥𝑋 ∣ ¬ 𝑥𝑅𝐴})
26 simpr 488 . . . . 5 ((((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐴𝐵)) ∧ {𝑥𝑋 ∣ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴)} = ∅) → {𝑥𝑋 ∣ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴)} = ∅)
27 eleq2 2878 . . . . . . 7 (𝑚 = {𝑥𝑋 ∣ ¬ 𝐵𝑅𝑥} → (𝐴𝑚𝐴 ∈ {𝑥𝑋 ∣ ¬ 𝐵𝑅𝑥}))
28 ineq1 4131 . . . . . . . 8 (𝑚 = {𝑥𝑋 ∣ ¬ 𝐵𝑅𝑥} → (𝑚𝑛) = ({𝑥𝑋 ∣ ¬ 𝐵𝑅𝑥} ∩ 𝑛))
2928eqeq1d 2800 . . . . . . 7 (𝑚 = {𝑥𝑋 ∣ ¬ 𝐵𝑅𝑥} → ((𝑚𝑛) = ∅ ↔ ({𝑥𝑋 ∣ ¬ 𝐵𝑅𝑥} ∩ 𝑛) = ∅))
3027, 293anbi13d 1435 . . . . . 6 (𝑚 = {𝑥𝑋 ∣ ¬ 𝐵𝑅𝑥} → ((𝐴𝑚𝐵𝑛 ∧ (𝑚𝑛) = ∅) ↔ (𝐴 ∈ {𝑥𝑋 ∣ ¬ 𝐵𝑅𝑥} ∧ 𝐵𝑛 ∧ ({𝑥𝑋 ∣ ¬ 𝐵𝑅𝑥} ∩ 𝑛) = ∅)))
31 eleq2 2878 . . . . . . 7 (𝑛 = {𝑥𝑋 ∣ ¬ 𝑥𝑅𝐴} → (𝐵𝑛𝐵 ∈ {𝑥𝑋 ∣ ¬ 𝑥𝑅𝐴}))
32 ineq2 4133 . . . . . . . . 9 (𝑛 = {𝑥𝑋 ∣ ¬ 𝑥𝑅𝐴} → ({𝑥𝑋 ∣ ¬ 𝐵𝑅𝑥} ∩ 𝑛) = ({𝑥𝑋 ∣ ¬ 𝐵𝑅𝑥} ∩ {𝑥𝑋 ∣ ¬ 𝑥𝑅𝐴}))
33 inrab 4227 . . . . . . . . 9 ({𝑥𝑋 ∣ ¬ 𝐵𝑅𝑥} ∩ {𝑥𝑋 ∣ ¬ 𝑥𝑅𝐴}) = {𝑥𝑋 ∣ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴)}
3432, 33eqtrdi 2849 . . . . . . . 8 (𝑛 = {𝑥𝑋 ∣ ¬ 𝑥𝑅𝐴} → ({𝑥𝑋 ∣ ¬ 𝐵𝑅𝑥} ∩ 𝑛) = {𝑥𝑋 ∣ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴)})
3534eqeq1d 2800 . . . . . . 7 (𝑛 = {𝑥𝑋 ∣ ¬ 𝑥𝑅𝐴} → (({𝑥𝑋 ∣ ¬ 𝐵𝑅𝑥} ∩ 𝑛) = ∅ ↔ {𝑥𝑋 ∣ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴)} = ∅))
3631, 353anbi23d 1436 . . . . . 6 (𝑛 = {𝑥𝑋 ∣ ¬ 𝑥𝑅𝐴} → ((𝐴 ∈ {𝑥𝑋 ∣ ¬ 𝐵𝑅𝑥} ∧ 𝐵𝑛 ∧ ({𝑥𝑋 ∣ ¬ 𝐵𝑅𝑥} ∩ 𝑛) = ∅) ↔ (𝐴 ∈ {𝑥𝑋 ∣ ¬ 𝐵𝑅𝑥} ∧ 𝐵 ∈ {𝑥𝑋 ∣ ¬ 𝑥𝑅𝐴} ∧ {𝑥𝑋 ∣ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴)} = ∅)))
3730, 36rspc2ev 3583 . . . . 5 (({𝑥𝑋 ∣ ¬ 𝐵𝑅𝑥} ∈ (ordTop‘𝑅) ∧ {𝑥𝑋 ∣ ¬ 𝑥𝑅𝐴} ∈ (ordTop‘𝑅) ∧ (𝐴 ∈ {𝑥𝑋 ∣ ¬ 𝐵𝑅𝑥} ∧ 𝐵 ∈ {𝑥𝑋 ∣ ¬ 𝑥𝑅𝐴} ∧ {𝑥𝑋 ∣ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴)} = ∅)) → ∃𝑚 ∈ (ordTop‘𝑅)∃𝑛 ∈ (ordTop‘𝑅)(𝐴𝑚𝐵𝑛 ∧ (𝑚𝑛) = ∅))
385, 8, 22, 25, 26, 37syl113anc 1379 . . . 4 ((((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐴𝐵)) ∧ {𝑥𝑋 ∣ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴)} = ∅) → ∃𝑚 ∈ (ordTop‘𝑅)∃𝑛 ∈ (ordTop‘𝑅)(𝐴𝑚𝐵𝑛 ∧ (𝑚𝑛) = ∅))
3938ex 416 . . 3 (((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐴𝐵)) → ({𝑥𝑋 ∣ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴)} = ∅ → ∃𝑚 ∈ (ordTop‘𝑅)∃𝑛 ∈ (ordTop‘𝑅)(𝐴𝑚𝐵𝑛 ∧ (𝑚𝑛) = ∅)))
40 rabn0 4293 . . . 4 ({𝑥𝑋 ∣ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴)} ≠ ∅ ↔ ∃𝑥𝑋𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴))
41 simpll1 1209 . . . . . . 7 ((((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐴𝐵)) ∧ (𝑥𝑋 ∧ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴))) → 𝑅 ∈ TosetRel )
42 simprl 770 . . . . . . 7 ((((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐴𝐵)) ∧ (𝑥𝑋 ∧ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴))) → 𝑥𝑋)
433ordtopn2 21814 . . . . . . 7 ((𝑅 ∈ TosetRel ∧ 𝑥𝑋) → {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦} ∈ (ordTop‘𝑅))
4441, 42, 43syl2anc 587 . . . . . 6 ((((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐴𝐵)) ∧ (𝑥𝑋 ∧ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴))) → {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦} ∈ (ordTop‘𝑅))
453ordtopn1 21813 . . . . . . 7 ((𝑅 ∈ TosetRel ∧ 𝑥𝑋) → {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥} ∈ (ordTop‘𝑅))
4641, 42, 45syl2anc 587 . . . . . 6 ((((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐴𝐵)) ∧ (𝑥𝑋 ∧ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴))) → {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥} ∈ (ordTop‘𝑅))
47 breq2 5035 . . . . . . . 8 (𝑦 = 𝐴 → (𝑥𝑅𝑦𝑥𝑅𝐴))
4847notbid 321 . . . . . . 7 (𝑦 = 𝐴 → (¬ 𝑥𝑅𝑦 ↔ ¬ 𝑥𝑅𝐴))
49 simpll2 1210 . . . . . . 7 ((((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐴𝐵)) ∧ (𝑥𝑋 ∧ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴))) → 𝐴𝑋)
50 simprrr 781 . . . . . . 7 ((((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐴𝐵)) ∧ (𝑥𝑋 ∧ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴))) → ¬ 𝑥𝑅𝐴)
5148, 49, 50elrabd 3630 . . . . . 6 ((((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐴𝐵)) ∧ (𝑥𝑋 ∧ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴))) → 𝐴 ∈ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦})
52 breq1 5034 . . . . . . . 8 (𝑦 = 𝐵 → (𝑦𝑅𝑥𝐵𝑅𝑥))
5352notbid 321 . . . . . . 7 (𝑦 = 𝐵 → (¬ 𝑦𝑅𝑥 ↔ ¬ 𝐵𝑅𝑥))
54 simpll3 1211 . . . . . . 7 ((((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐴𝐵)) ∧ (𝑥𝑋 ∧ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴))) → 𝐵𝑋)
55 simprrl 780 . . . . . . 7 ((((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐴𝐵)) ∧ (𝑥𝑋 ∧ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴))) → ¬ 𝐵𝑅𝑥)
5653, 54, 55elrabd 3630 . . . . . 6 ((((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐴𝐵)) ∧ (𝑥𝑋 ∧ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴))) → 𝐵 ∈ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥})
5741, 42jca 515 . . . . . . . . . 10 ((((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐴𝐵)) ∧ (𝑥𝑋 ∧ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴))) → (𝑅 ∈ TosetRel ∧ 𝑥𝑋))
583tsrlin 17828 . . . . . . . . . . 11 ((𝑅 ∈ TosetRel ∧ 𝑥𝑋𝑦𝑋) → (𝑥𝑅𝑦𝑦𝑅𝑥))
59583expa 1115 . . . . . . . . . 10 (((𝑅 ∈ TosetRel ∧ 𝑥𝑋) ∧ 𝑦𝑋) → (𝑥𝑅𝑦𝑦𝑅𝑥))
6057, 59sylan 583 . . . . . . . . 9 (((((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐴𝐵)) ∧ (𝑥𝑋 ∧ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴))) ∧ 𝑦𝑋) → (𝑥𝑅𝑦𝑦𝑅𝑥))
61 oran 987 . . . . . . . . 9 ((𝑥𝑅𝑦𝑦𝑅𝑥) ↔ ¬ (¬ 𝑥𝑅𝑦 ∧ ¬ 𝑦𝑅𝑥))
6260, 61sylib 221 . . . . . . . 8 (((((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐴𝐵)) ∧ (𝑥𝑋 ∧ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴))) ∧ 𝑦𝑋) → ¬ (¬ 𝑥𝑅𝑦 ∧ ¬ 𝑦𝑅𝑥))
6362ralrimiva 3149 . . . . . . 7 ((((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐴𝐵)) ∧ (𝑥𝑋 ∧ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴))) → ∀𝑦𝑋 ¬ (¬ 𝑥𝑅𝑦 ∧ ¬ 𝑦𝑅𝑥))
64 rabeq0 4292 . . . . . . 7 ({𝑦𝑋 ∣ (¬ 𝑥𝑅𝑦 ∧ ¬ 𝑦𝑅𝑥)} = ∅ ↔ ∀𝑦𝑋 ¬ (¬ 𝑥𝑅𝑦 ∧ ¬ 𝑦𝑅𝑥))
6563, 64sylibr 237 . . . . . 6 ((((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐴𝐵)) ∧ (𝑥𝑋 ∧ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴))) → {𝑦𝑋 ∣ (¬ 𝑥𝑅𝑦 ∧ ¬ 𝑦𝑅𝑥)} = ∅)
66 eleq2 2878 . . . . . . . 8 (𝑚 = {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦} → (𝐴𝑚𝐴 ∈ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦}))
67 ineq1 4131 . . . . . . . . 9 (𝑚 = {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦} → (𝑚𝑛) = ({𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦} ∩ 𝑛))
6867eqeq1d 2800 . . . . . . . 8 (𝑚 = {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦} → ((𝑚𝑛) = ∅ ↔ ({𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦} ∩ 𝑛) = ∅))
6966, 683anbi13d 1435 . . . . . . 7 (𝑚 = {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦} → ((𝐴𝑚𝐵𝑛 ∧ (𝑚𝑛) = ∅) ↔ (𝐴 ∈ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦} ∧ 𝐵𝑛 ∧ ({𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦} ∩ 𝑛) = ∅)))
70 eleq2 2878 . . . . . . . 8 (𝑛 = {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥} → (𝐵𝑛𝐵 ∈ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥}))
71 ineq2 4133 . . . . . . . . . 10 (𝑛 = {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥} → ({𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦} ∩ 𝑛) = ({𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦} ∩ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥}))
72 inrab 4227 . . . . . . . . . 10 ({𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦} ∩ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥}) = {𝑦𝑋 ∣ (¬ 𝑥𝑅𝑦 ∧ ¬ 𝑦𝑅𝑥)}
7371, 72eqtrdi 2849 . . . . . . . . 9 (𝑛 = {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥} → ({𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦} ∩ 𝑛) = {𝑦𝑋 ∣ (¬ 𝑥𝑅𝑦 ∧ ¬ 𝑦𝑅𝑥)})
7473eqeq1d 2800 . . . . . . . 8 (𝑛 = {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥} → (({𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦} ∩ 𝑛) = ∅ ↔ {𝑦𝑋 ∣ (¬ 𝑥𝑅𝑦 ∧ ¬ 𝑦𝑅𝑥)} = ∅))
7570, 743anbi23d 1436 . . . . . . 7 (𝑛 = {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥} → ((𝐴 ∈ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦} ∧ 𝐵𝑛 ∧ ({𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦} ∩ 𝑛) = ∅) ↔ (𝐴 ∈ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦} ∧ 𝐵 ∈ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥} ∧ {𝑦𝑋 ∣ (¬ 𝑥𝑅𝑦 ∧ ¬ 𝑦𝑅𝑥)} = ∅)))
7669, 75rspc2ev 3583 . . . . . 6 (({𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦} ∈ (ordTop‘𝑅) ∧ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥} ∈ (ordTop‘𝑅) ∧ (𝐴 ∈ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦} ∧ 𝐵 ∈ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥} ∧ {𝑦𝑋 ∣ (¬ 𝑥𝑅𝑦 ∧ ¬ 𝑦𝑅𝑥)} = ∅)) → ∃𝑚 ∈ (ordTop‘𝑅)∃𝑛 ∈ (ordTop‘𝑅)(𝐴𝑚𝐵𝑛 ∧ (𝑚𝑛) = ∅))
7744, 46, 51, 56, 65, 76syl113anc 1379 . . . . 5 ((((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐴𝐵)) ∧ (𝑥𝑋 ∧ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴))) → ∃𝑚 ∈ (ordTop‘𝑅)∃𝑛 ∈ (ordTop‘𝑅)(𝐴𝑚𝐵𝑛 ∧ (𝑚𝑛) = ∅))
7877rexlimdvaa 3244 . . . 4 (((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐴𝐵)) → (∃𝑥𝑋𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴) → ∃𝑚 ∈ (ordTop‘𝑅)∃𝑛 ∈ (ordTop‘𝑅)(𝐴𝑚𝐵𝑛 ∧ (𝑚𝑛) = ∅)))
7940, 78syl5bi 245 . . 3 (((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐴𝐵)) → ({𝑥𝑋 ∣ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴)} ≠ ∅ → ∃𝑚 ∈ (ordTop‘𝑅)∃𝑛 ∈ (ordTop‘𝑅)(𝐴𝑚𝐵𝑛 ∧ (𝑚𝑛) = ∅)))
8039, 79pm2.61dne 3073 . 2 (((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐴𝐵)) → ∃𝑚 ∈ (ordTop‘𝑅)∃𝑛 ∈ (ordTop‘𝑅)(𝐴𝑚𝐵𝑛 ∧ (𝑚𝑛) = ∅))
8180exp32 424 1 ((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝑅𝐵 → (𝐴𝐵 → ∃𝑚 ∈ (ordTop‘𝑅)∃𝑛 ∈ (ordTop‘𝑅)(𝐴𝑚𝐵𝑛 ∧ (𝑚𝑛) = ∅))))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   ∨ wo 844   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   ≠ wne 2987  ∀wral 3106  ∃wrex 3107  {crab 3110   ∩ cin 3880  ∅c0 4243   class class class wbr 5031  dom cdm 5520  ‘cfv 6327  ordTopcordt 16771  PosetRelcps 17807   TosetRel ctsr 17808 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7448 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-int 4840  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-tr 5138  df-id 5426  df-eprel 5431  df-po 5439  df-so 5440  df-fr 5479  df-we 5481  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-pred 6119  df-ord 6165  df-on 6166  df-lim 6167  df-suc 6168  df-iota 6286  df-fun 6329  df-fn 6330  df-f 6331  df-f1 6332  df-fo 6333  df-f1o 6334  df-fv 6335  df-ov 7143  df-oprab 7144  df-mpo 7145  df-om 7568  df-wrecs 7937  df-recs 7998  df-rdg 8036  df-1o 8092  df-oadd 8096  df-er 8279  df-en 8500  df-fin 8503  df-fi 8866  df-topgen 16716  df-ordt 16773  df-ps 17809  df-tsr 17810  df-bases 21565 This theorem is referenced by:  ordthaus  22003
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