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Theorem ordthauslem 21395
Description: Lemma for ordthaus 21396. (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 1262 . . . . . 6 ((((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐴𝐵)) ∧ {𝑥𝑋 ∣ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴)} = ∅) → 𝑅 ∈ TosetRel )
2 simpll3 1266 . . . . . 6 ((((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐴𝐵)) ∧ {𝑥𝑋 ∣ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴)} = ∅) → 𝐵𝑋)
3 ordthauslem.1 . . . . . . 7 𝑋 = dom 𝑅
43ordtopn2 21207 . . . . . 6 ((𝑅 ∈ TosetRel ∧ 𝐵𝑋) → {𝑥𝑋 ∣ ¬ 𝐵𝑅𝑥} ∈ (ordTop‘𝑅))
51, 2, 4syl2anc 575 . . . . 5 ((((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐴𝐵)) ∧ {𝑥𝑋 ∣ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴)} = ∅) → {𝑥𝑋 ∣ ¬ 𝐵𝑅𝑥} ∈ (ordTop‘𝑅))
6 simpll2 1264 . . . . . 6 ((((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐴𝐵)) ∧ {𝑥𝑋 ∣ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴)} = ∅) → 𝐴𝑋)
73ordtopn1 21206 . . . . . 6 ((𝑅 ∈ TosetRel ∧ 𝐴𝑋) → {𝑥𝑋 ∣ ¬ 𝑥𝑅𝐴} ∈ (ordTop‘𝑅))
81, 6, 7syl2anc 575 . . . . 5 ((((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐴𝐵)) ∧ {𝑥𝑋 ∣ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴)} = ∅) → {𝑥𝑋 ∣ ¬ 𝑥𝑅𝐴} ∈ (ordTop‘𝑅))
9 simprr 780 . . . . . . . 8 (((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐴𝐵)) → 𝐴𝐵)
10 simpl1 1235 . . . . . . . . . . 11 (((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐴𝐵)) → 𝑅 ∈ TosetRel )
11 tsrps 17420 . . . . . . . . . . 11 (𝑅 ∈ TosetRel → 𝑅 ∈ PosetRel)
1210, 11syl 17 . . . . . . . . . 10 (((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐴𝐵)) → 𝑅 ∈ PosetRel)
13 simprl 778 . . . . . . . . . 10 (((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐴𝐵)) → 𝐴𝑅𝐵)
14 psasym 17409 . . . . . . . . . . 11 ((𝑅 ∈ PosetRel ∧ 𝐴𝑅𝐵𝐵𝑅𝐴) → 𝐴 = 𝐵)
15143expia 1143 . . . . . . . . . 10 ((𝑅 ∈ PosetRel ∧ 𝐴𝑅𝐵) → (𝐵𝑅𝐴𝐴 = 𝐵))
1612, 13, 15syl2anc 575 . . . . . . . . 9 (((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐴𝐵)) → (𝐵𝑅𝐴𝐴 = 𝐵))
1716necon3ad 2987 . . . . . . . 8 (((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐴𝐵)) → (𝐴𝐵 → ¬ 𝐵𝑅𝐴))
189, 17mpd 15 . . . . . . 7 (((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐴𝐵)) → ¬ 𝐵𝑅𝐴)
1918adantr 468 . . . . . 6 ((((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐴𝐵)) ∧ {𝑥𝑋 ∣ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴)} = ∅) → ¬ 𝐵𝑅𝐴)
20 breq2 4841 . . . . . . . 8 (𝑥 = 𝐴 → (𝐵𝑅𝑥𝐵𝑅𝐴))
2120notbid 309 . . . . . . 7 (𝑥 = 𝐴 → (¬ 𝐵𝑅𝑥 ↔ ¬ 𝐵𝑅𝐴))
2221elrab 3555 . . . . . 6 (𝐴 ∈ {𝑥𝑋 ∣ ¬ 𝐵𝑅𝑥} ↔ (𝐴𝑋 ∧ ¬ 𝐵𝑅𝐴))
236, 19, 22sylanbrc 574 . . . . 5 ((((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐴𝐵)) ∧ {𝑥𝑋 ∣ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴)} = ∅) → 𝐴 ∈ {𝑥𝑋 ∣ ¬ 𝐵𝑅𝑥})
24 breq1 4840 . . . . . . . 8 (𝑥 = 𝐵 → (𝑥𝑅𝐴𝐵𝑅𝐴))
2524notbid 309 . . . . . . 7 (𝑥 = 𝐵 → (¬ 𝑥𝑅𝐴 ↔ ¬ 𝐵𝑅𝐴))
2625elrab 3555 . . . . . 6 (𝐵 ∈ {𝑥𝑋 ∣ ¬ 𝑥𝑅𝐴} ↔ (𝐵𝑋 ∧ ¬ 𝐵𝑅𝐴))
272, 19, 26sylanbrc 574 . . . . 5 ((((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐴𝐵)) ∧ {𝑥𝑋 ∣ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴)} = ∅) → 𝐵 ∈ {𝑥𝑋 ∣ ¬ 𝑥𝑅𝐴})
28 simpr 473 . . . . 5 ((((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐴𝐵)) ∧ {𝑥𝑋 ∣ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴)} = ∅) → {𝑥𝑋 ∣ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴)} = ∅)
29 eleq2 2870 . . . . . . 7 (𝑚 = {𝑥𝑋 ∣ ¬ 𝐵𝑅𝑥} → (𝐴𝑚𝐴 ∈ {𝑥𝑋 ∣ ¬ 𝐵𝑅𝑥}))
30 ineq1 4000 . . . . . . . 8 (𝑚 = {𝑥𝑋 ∣ ¬ 𝐵𝑅𝑥} → (𝑚𝑛) = ({𝑥𝑋 ∣ ¬ 𝐵𝑅𝑥} ∩ 𝑛))
3130eqeq1d 2804 . . . . . . 7 (𝑚 = {𝑥𝑋 ∣ ¬ 𝐵𝑅𝑥} → ((𝑚𝑛) = ∅ ↔ ({𝑥𝑋 ∣ ¬ 𝐵𝑅𝑥} ∩ 𝑛) = ∅))
3229, 313anbi13d 1555 . . . . . 6 (𝑚 = {𝑥𝑋 ∣ ¬ 𝐵𝑅𝑥} → ((𝐴𝑚𝐵𝑛 ∧ (𝑚𝑛) = ∅) ↔ (𝐴 ∈ {𝑥𝑋 ∣ ¬ 𝐵𝑅𝑥} ∧ 𝐵𝑛 ∧ ({𝑥𝑋 ∣ ¬ 𝐵𝑅𝑥} ∩ 𝑛) = ∅)))
33 eleq2 2870 . . . . . . 7 (𝑛 = {𝑥𝑋 ∣ ¬ 𝑥𝑅𝐴} → (𝐵𝑛𝐵 ∈ {𝑥𝑋 ∣ ¬ 𝑥𝑅𝐴}))
34 ineq2 4001 . . . . . . . . 9 (𝑛 = {𝑥𝑋 ∣ ¬ 𝑥𝑅𝐴} → ({𝑥𝑋 ∣ ¬ 𝐵𝑅𝑥} ∩ 𝑛) = ({𝑥𝑋 ∣ ¬ 𝐵𝑅𝑥} ∩ {𝑥𝑋 ∣ ¬ 𝑥𝑅𝐴}))
35 inrab 4094 . . . . . . . . 9 ({𝑥𝑋 ∣ ¬ 𝐵𝑅𝑥} ∩ {𝑥𝑋 ∣ ¬ 𝑥𝑅𝐴}) = {𝑥𝑋 ∣ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴)}
3634, 35syl6eq 2852 . . . . . . . 8 (𝑛 = {𝑥𝑋 ∣ ¬ 𝑥𝑅𝐴} → ({𝑥𝑋 ∣ ¬ 𝐵𝑅𝑥} ∩ 𝑛) = {𝑥𝑋 ∣ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴)})
3736eqeq1d 2804 . . . . . . 7 (𝑛 = {𝑥𝑋 ∣ ¬ 𝑥𝑅𝐴} → (({𝑥𝑋 ∣ ¬ 𝐵𝑅𝑥} ∩ 𝑛) = ∅ ↔ {𝑥𝑋 ∣ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴)} = ∅))
3833, 373anbi23d 1556 . . . . . 6 (𝑛 = {𝑥𝑋 ∣ ¬ 𝑥𝑅𝐴} → ((𝐴 ∈ {𝑥𝑋 ∣ ¬ 𝐵𝑅𝑥} ∧ 𝐵𝑛 ∧ ({𝑥𝑋 ∣ ¬ 𝐵𝑅𝑥} ∩ 𝑛) = ∅) ↔ (𝐴 ∈ {𝑥𝑋 ∣ ¬ 𝐵𝑅𝑥} ∧ 𝐵 ∈ {𝑥𝑋 ∣ ¬ 𝑥𝑅𝐴} ∧ {𝑥𝑋 ∣ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴)} = ∅)))
3932, 38rspc2ev 3513 . . . . 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) = ∅))
405, 8, 23, 27, 28, 39syl113anc 1494 . . . 4 ((((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐴𝐵)) ∧ {𝑥𝑋 ∣ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴)} = ∅) → ∃𝑚 ∈ (ordTop‘𝑅)∃𝑛 ∈ (ordTop‘𝑅)(𝐴𝑚𝐵𝑛 ∧ (𝑚𝑛) = ∅))
4140ex 399 . . 3 (((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐴𝐵)) → ({𝑥𝑋 ∣ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴)} = ∅ → ∃𝑚 ∈ (ordTop‘𝑅)∃𝑛 ∈ (ordTop‘𝑅)(𝐴𝑚𝐵𝑛 ∧ (𝑚𝑛) = ∅)))
42 rabn0 4152 . . . 4 ({𝑥𝑋 ∣ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴)} ≠ ∅ ↔ ∃𝑥𝑋𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴))
43 simpll1 1262 . . . . . . 7 ((((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐴𝐵)) ∧ (𝑥𝑋 ∧ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴))) → 𝑅 ∈ TosetRel )
44 simprl 778 . . . . . . 7 ((((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐴𝐵)) ∧ (𝑥𝑋 ∧ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴))) → 𝑥𝑋)
453ordtopn2 21207 . . . . . . 7 ((𝑅 ∈ TosetRel ∧ 𝑥𝑋) → {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦} ∈ (ordTop‘𝑅))
4643, 44, 45syl2anc 575 . . . . . 6 ((((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐴𝐵)) ∧ (𝑥𝑋 ∧ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴))) → {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦} ∈ (ordTop‘𝑅))
473ordtopn1 21206 . . . . . . 7 ((𝑅 ∈ TosetRel ∧ 𝑥𝑋) → {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥} ∈ (ordTop‘𝑅))
4843, 44, 47syl2anc 575 . . . . . 6 ((((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐴𝐵)) ∧ (𝑥𝑋 ∧ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴))) → {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥} ∈ (ordTop‘𝑅))
49 simpll2 1264 . . . . . . 7 ((((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐴𝐵)) ∧ (𝑥𝑋 ∧ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴))) → 𝐴𝑋)
50 simprrr 791 . . . . . . 7 ((((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐴𝐵)) ∧ (𝑥𝑋 ∧ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴))) → ¬ 𝑥𝑅𝐴)
51 breq2 4841 . . . . . . . . 9 (𝑦 = 𝐴 → (𝑥𝑅𝑦𝑥𝑅𝐴))
5251notbid 309 . . . . . . . 8 (𝑦 = 𝐴 → (¬ 𝑥𝑅𝑦 ↔ ¬ 𝑥𝑅𝐴))
5352elrab 3555 . . . . . . 7 (𝐴 ∈ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦} ↔ (𝐴𝑋 ∧ ¬ 𝑥𝑅𝐴))
5449, 50, 53sylanbrc 574 . . . . . 6 ((((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐴𝐵)) ∧ (𝑥𝑋 ∧ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴))) → 𝐴 ∈ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦})
55 simpll3 1266 . . . . . . 7 ((((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐴𝐵)) ∧ (𝑥𝑋 ∧ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴))) → 𝐵𝑋)
56 simprrl 790 . . . . . . 7 ((((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐴𝐵)) ∧ (𝑥𝑋 ∧ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴))) → ¬ 𝐵𝑅𝑥)
57 breq1 4840 . . . . . . . . 9 (𝑦 = 𝐵 → (𝑦𝑅𝑥𝐵𝑅𝑥))
5857notbid 309 . . . . . . . 8 (𝑦 = 𝐵 → (¬ 𝑦𝑅𝑥 ↔ ¬ 𝐵𝑅𝑥))
5958elrab 3555 . . . . . . 7 (𝐵 ∈ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥} ↔ (𝐵𝑋 ∧ ¬ 𝐵𝑅𝑥))
6055, 56, 59sylanbrc 574 . . . . . 6 ((((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐴𝐵)) ∧ (𝑥𝑋 ∧ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴))) → 𝐵 ∈ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥})
6143, 44jca 503 . . . . . . . . . 10 ((((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐴𝐵)) ∧ (𝑥𝑋 ∧ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴))) → (𝑅 ∈ TosetRel ∧ 𝑥𝑋))
623tsrlin 17418 . . . . . . . . . . 11 ((𝑅 ∈ TosetRel ∧ 𝑥𝑋𝑦𝑋) → (𝑥𝑅𝑦𝑦𝑅𝑥))
63623expa 1140 . . . . . . . . . 10 (((𝑅 ∈ TosetRel ∧ 𝑥𝑋) ∧ 𝑦𝑋) → (𝑥𝑅𝑦𝑦𝑅𝑥))
6461, 63sylan 571 . . . . . . . . 9 (((((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐴𝐵)) ∧ (𝑥𝑋 ∧ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴))) ∧ 𝑦𝑋) → (𝑥𝑅𝑦𝑦𝑅𝑥))
65 oran 1003 . . . . . . . . 9 ((𝑥𝑅𝑦𝑦𝑅𝑥) ↔ ¬ (¬ 𝑥𝑅𝑦 ∧ ¬ 𝑦𝑅𝑥))
6664, 65sylib 209 . . . . . . . 8 (((((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐴𝐵)) ∧ (𝑥𝑋 ∧ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴))) ∧ 𝑦𝑋) → ¬ (¬ 𝑥𝑅𝑦 ∧ ¬ 𝑦𝑅𝑥))
6766ralrimiva 3150 . . . . . . 7 ((((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐴𝐵)) ∧ (𝑥𝑋 ∧ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴))) → ∀𝑦𝑋 ¬ (¬ 𝑥𝑅𝑦 ∧ ¬ 𝑦𝑅𝑥))
68 rabeq0 4151 . . . . . . 7 ({𝑦𝑋 ∣ (¬ 𝑥𝑅𝑦 ∧ ¬ 𝑦𝑅𝑥)} = ∅ ↔ ∀𝑦𝑋 ¬ (¬ 𝑥𝑅𝑦 ∧ ¬ 𝑦𝑅𝑥))
6967, 68sylibr 225 . . . . . 6 ((((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐴𝐵)) ∧ (𝑥𝑋 ∧ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴))) → {𝑦𝑋 ∣ (¬ 𝑥𝑅𝑦 ∧ ¬ 𝑦𝑅𝑥)} = ∅)
70 eleq2 2870 . . . . . . . 8 (𝑚 = {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦} → (𝐴𝑚𝐴 ∈ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦}))
71 ineq1 4000 . . . . . . . . 9 (𝑚 = {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦} → (𝑚𝑛) = ({𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦} ∩ 𝑛))
7271eqeq1d 2804 . . . . . . . 8 (𝑚 = {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦} → ((𝑚𝑛) = ∅ ↔ ({𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦} ∩ 𝑛) = ∅))
7370, 723anbi13d 1555 . . . . . . 7 (𝑚 = {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦} → ((𝐴𝑚𝐵𝑛 ∧ (𝑚𝑛) = ∅) ↔ (𝐴 ∈ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦} ∧ 𝐵𝑛 ∧ ({𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦} ∩ 𝑛) = ∅)))
74 eleq2 2870 . . . . . . . 8 (𝑛 = {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥} → (𝐵𝑛𝐵 ∈ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥}))
75 ineq2 4001 . . . . . . . . . 10 (𝑛 = {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥} → ({𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦} ∩ 𝑛) = ({𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦} ∩ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥}))
76 inrab 4094 . . . . . . . . . 10 ({𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦} ∩ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥}) = {𝑦𝑋 ∣ (¬ 𝑥𝑅𝑦 ∧ ¬ 𝑦𝑅𝑥)}
7775, 76syl6eq 2852 . . . . . . . . 9 (𝑛 = {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥} → ({𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦} ∩ 𝑛) = {𝑦𝑋 ∣ (¬ 𝑥𝑅𝑦 ∧ ¬ 𝑦𝑅𝑥)})
7877eqeq1d 2804 . . . . . . . 8 (𝑛 = {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥} → (({𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦} ∩ 𝑛) = ∅ ↔ {𝑦𝑋 ∣ (¬ 𝑥𝑅𝑦 ∧ ¬ 𝑦𝑅𝑥)} = ∅))
7974, 783anbi23d 1556 . . . . . . 7 (𝑛 = {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥} → ((𝐴 ∈ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦} ∧ 𝐵𝑛 ∧ ({𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦} ∩ 𝑛) = ∅) ↔ (𝐴 ∈ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦} ∧ 𝐵 ∈ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥} ∧ {𝑦𝑋 ∣ (¬ 𝑥𝑅𝑦 ∧ ¬ 𝑦𝑅𝑥)} = ∅)))
8073, 79rspc2ev 3513 . . . . . 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) = ∅))
8146, 48, 54, 60, 69, 80syl113anc 1494 . . . . 5 ((((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐴𝐵)) ∧ (𝑥𝑋 ∧ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴))) → ∃𝑚 ∈ (ordTop‘𝑅)∃𝑛 ∈ (ordTop‘𝑅)(𝐴𝑚𝐵𝑛 ∧ (𝑚𝑛) = ∅))
8281rexlimdvaa 3216 . . . 4 (((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐴𝐵)) → (∃𝑥𝑋𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴) → ∃𝑚 ∈ (ordTop‘𝑅)∃𝑛 ∈ (ordTop‘𝑅)(𝐴𝑚𝐵𝑛 ∧ (𝑚𝑛) = ∅)))
8342, 82syl5bi 233 . . 3 (((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐴𝐵)) → ({𝑥𝑋 ∣ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴)} ≠ ∅ → ∃𝑚 ∈ (ordTop‘𝑅)∃𝑛 ∈ (ordTop‘𝑅)(𝐴𝑚𝐵𝑛 ∧ (𝑚𝑛) = ∅)))
8441, 83pm2.61dne 3060 . 2 (((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐴𝐵)) → ∃𝑚 ∈ (ordTop‘𝑅)∃𝑛 ∈ (ordTop‘𝑅)(𝐴𝑚𝐵𝑛 ∧ (𝑚𝑛) = ∅))
8584exp32 409 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 384  wo 865  w3a 1100   = wceq 1637  wcel 2155  wne 2974  wral 3092  wrex 3093  {crab 3096  cin 3762  c0 4110   class class class wbr 4837  dom cdm 5305  cfv 6095  ordTopcordt 16358  PosetRelcps 17397   TosetRel ctsr 17398
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2067  ax-7 2103  ax-8 2157  ax-9 2164  ax-10 2184  ax-11 2200  ax-12 2213  ax-13 2419  ax-ext 2781  ax-sep 4968  ax-nul 4977  ax-pow 5029  ax-pr 5090  ax-un 7173
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2060  df-eu 2633  df-mo 2634  df-clab 2789  df-cleq 2795  df-clel 2798  df-nfc 2933  df-ne 2975  df-ral 3097  df-rex 3098  df-reu 3099  df-rab 3101  df-v 3389  df-sbc 3628  df-csb 3723  df-dif 3766  df-un 3768  df-in 3770  df-ss 3777  df-pss 3779  df-nul 4111  df-if 4274  df-pw 4347  df-sn 4365  df-pr 4367  df-tp 4369  df-op 4371  df-uni 4624  df-int 4663  df-iun 4707  df-br 4838  df-opab 4900  df-mpt 4917  df-tr 4940  df-id 5213  df-eprel 5218  df-po 5226  df-so 5227  df-fr 5264  df-we 5266  df-xp 5311  df-rel 5312  df-cnv 5313  df-co 5314  df-dm 5315  df-rn 5316  df-res 5317  df-ima 5318  df-pred 5887  df-ord 5933  df-on 5934  df-lim 5935  df-suc 5936  df-iota 6058  df-fun 6097  df-fn 6098  df-f 6099  df-f1 6100  df-fo 6101  df-f1o 6102  df-fv 6103  df-ov 6871  df-oprab 6872  df-mpt2 6873  df-om 7290  df-wrecs 7636  df-recs 7698  df-rdg 7736  df-1o 7790  df-oadd 7794  df-er 7973  df-en 8187  df-fin 8190  df-fi 8550  df-topgen 16303  df-ordt 16360  df-ps 17399  df-tsr 17400  df-bases 20958
This theorem is referenced by:  ordthaus  21396
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