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Theorem ordthauslem 21097
Description: Lemma for ordthaus 21098. (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 1098 . . . . . 6 ((((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐴𝐵)) ∧ {𝑥𝑋 ∣ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴)} = ∅) → 𝑅 ∈ TosetRel )
2 simpll3 1100 . . . . . 6 ((((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐴𝐵)) ∧ {𝑥𝑋 ∣ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴)} = ∅) → 𝐵𝑋)
3 ordthauslem.1 . . . . . . 7 𝑋 = dom 𝑅
43ordtopn2 20909 . . . . . 6 ((𝑅 ∈ TosetRel ∧ 𝐵𝑋) → {𝑥𝑋 ∣ ¬ 𝐵𝑅𝑥} ∈ (ordTop‘𝑅))
51, 2, 4syl2anc 692 . . . . 5 ((((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐴𝐵)) ∧ {𝑥𝑋 ∣ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴)} = ∅) → {𝑥𝑋 ∣ ¬ 𝐵𝑅𝑥} ∈ (ordTop‘𝑅))
6 simpll2 1099 . . . . . 6 ((((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐴𝐵)) ∧ {𝑥𝑋 ∣ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴)} = ∅) → 𝐴𝑋)
73ordtopn1 20908 . . . . . 6 ((𝑅 ∈ TosetRel ∧ 𝐴𝑋) → {𝑥𝑋 ∣ ¬ 𝑥𝑅𝐴} ∈ (ordTop‘𝑅))
81, 6, 7syl2anc 692 . . . . 5 ((((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐴𝐵)) ∧ {𝑥𝑋 ∣ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴)} = ∅) → {𝑥𝑋 ∣ ¬ 𝑥𝑅𝐴} ∈ (ordTop‘𝑅))
9 simprr 795 . . . . . . . 8 (((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐴𝐵)) → 𝐴𝐵)
10 simpl1 1062 . . . . . . . . . . 11 (((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐴𝐵)) → 𝑅 ∈ TosetRel )
11 tsrps 17142 . . . . . . . . . . 11 (𝑅 ∈ TosetRel → 𝑅 ∈ PosetRel)
1210, 11syl 17 . . . . . . . . . 10 (((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐴𝐵)) → 𝑅 ∈ PosetRel)
13 simprl 793 . . . . . . . . . 10 (((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐴𝐵)) → 𝐴𝑅𝐵)
14 psasym 17131 . . . . . . . . . . 11 ((𝑅 ∈ PosetRel ∧ 𝐴𝑅𝐵𝐵𝑅𝐴) → 𝐴 = 𝐵)
15143expia 1264 . . . . . . . . . 10 ((𝑅 ∈ PosetRel ∧ 𝐴𝑅𝐵) → (𝐵𝑅𝐴𝐴 = 𝐵))
1612, 13, 15syl2anc 692 . . . . . . . . 9 (((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐴𝐵)) → (𝐵𝑅𝐴𝐴 = 𝐵))
1716necon3ad 2803 . . . . . . . 8 (((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐴𝐵)) → (𝐴𝐵 → ¬ 𝐵𝑅𝐴))
189, 17mpd 15 . . . . . . 7 (((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐴𝐵)) → ¬ 𝐵𝑅𝐴)
1918adantr 481 . . . . . 6 ((((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐴𝐵)) ∧ {𝑥𝑋 ∣ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴)} = ∅) → ¬ 𝐵𝑅𝐴)
20 breq2 4617 . . . . . . . 8 (𝑥 = 𝐴 → (𝐵𝑅𝑥𝐵𝑅𝐴))
2120notbid 308 . . . . . . 7 (𝑥 = 𝐴 → (¬ 𝐵𝑅𝑥 ↔ ¬ 𝐵𝑅𝐴))
2221elrab 3346 . . . . . 6 (𝐴 ∈ {𝑥𝑋 ∣ ¬ 𝐵𝑅𝑥} ↔ (𝐴𝑋 ∧ ¬ 𝐵𝑅𝐴))
236, 19, 22sylanbrc 697 . . . . 5 ((((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐴𝐵)) ∧ {𝑥𝑋 ∣ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴)} = ∅) → 𝐴 ∈ {𝑥𝑋 ∣ ¬ 𝐵𝑅𝑥})
24 breq1 4616 . . . . . . . 8 (𝑥 = 𝐵 → (𝑥𝑅𝐴𝐵𝑅𝐴))
2524notbid 308 . . . . . . 7 (𝑥 = 𝐵 → (¬ 𝑥𝑅𝐴 ↔ ¬ 𝐵𝑅𝐴))
2625elrab 3346 . . . . . 6 (𝐵 ∈ {𝑥𝑋 ∣ ¬ 𝑥𝑅𝐴} ↔ (𝐵𝑋 ∧ ¬ 𝐵𝑅𝐴))
272, 19, 26sylanbrc 697 . . . . 5 ((((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐴𝐵)) ∧ {𝑥𝑋 ∣ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴)} = ∅) → 𝐵 ∈ {𝑥𝑋 ∣ ¬ 𝑥𝑅𝐴})
28 simpr 477 . . . . 5 ((((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐴𝐵)) ∧ {𝑥𝑋 ∣ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴)} = ∅) → {𝑥𝑋 ∣ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴)} = ∅)
29 eleq2 2687 . . . . . . 7 (𝑚 = {𝑥𝑋 ∣ ¬ 𝐵𝑅𝑥} → (𝐴𝑚𝐴 ∈ {𝑥𝑋 ∣ ¬ 𝐵𝑅𝑥}))
30 ineq1 3785 . . . . . . . 8 (𝑚 = {𝑥𝑋 ∣ ¬ 𝐵𝑅𝑥} → (𝑚𝑛) = ({𝑥𝑋 ∣ ¬ 𝐵𝑅𝑥} ∩ 𝑛))
3130eqeq1d 2623 . . . . . . 7 (𝑚 = {𝑥𝑋 ∣ ¬ 𝐵𝑅𝑥} → ((𝑚𝑛) = ∅ ↔ ({𝑥𝑋 ∣ ¬ 𝐵𝑅𝑥} ∩ 𝑛) = ∅))
3229, 313anbi13d 1398 . . . . . 6 (𝑚 = {𝑥𝑋 ∣ ¬ 𝐵𝑅𝑥} → ((𝐴𝑚𝐵𝑛 ∧ (𝑚𝑛) = ∅) ↔ (𝐴 ∈ {𝑥𝑋 ∣ ¬ 𝐵𝑅𝑥} ∧ 𝐵𝑛 ∧ ({𝑥𝑋 ∣ ¬ 𝐵𝑅𝑥} ∩ 𝑛) = ∅)))
33 eleq2 2687 . . . . . . 7 (𝑛 = {𝑥𝑋 ∣ ¬ 𝑥𝑅𝐴} → (𝐵𝑛𝐵 ∈ {𝑥𝑋 ∣ ¬ 𝑥𝑅𝐴}))
34 ineq2 3786 . . . . . . . . 9 (𝑛 = {𝑥𝑋 ∣ ¬ 𝑥𝑅𝐴} → ({𝑥𝑋 ∣ ¬ 𝐵𝑅𝑥} ∩ 𝑛) = ({𝑥𝑋 ∣ ¬ 𝐵𝑅𝑥} ∩ {𝑥𝑋 ∣ ¬ 𝑥𝑅𝐴}))
35 inrab 3875 . . . . . . . . 9 ({𝑥𝑋 ∣ ¬ 𝐵𝑅𝑥} ∩ {𝑥𝑋 ∣ ¬ 𝑥𝑅𝐴}) = {𝑥𝑋 ∣ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴)}
3634, 35syl6eq 2671 . . . . . . . 8 (𝑛 = {𝑥𝑋 ∣ ¬ 𝑥𝑅𝐴} → ({𝑥𝑋 ∣ ¬ 𝐵𝑅𝑥} ∩ 𝑛) = {𝑥𝑋 ∣ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴)})
3736eqeq1d 2623 . . . . . . 7 (𝑛 = {𝑥𝑋 ∣ ¬ 𝑥𝑅𝐴} → (({𝑥𝑋 ∣ ¬ 𝐵𝑅𝑥} ∩ 𝑛) = ∅ ↔ {𝑥𝑋 ∣ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴)} = ∅))
3833, 373anbi23d 1399 . . . . . 6 (𝑛 = {𝑥𝑋 ∣ ¬ 𝑥𝑅𝐴} → ((𝐴 ∈ {𝑥𝑋 ∣ ¬ 𝐵𝑅𝑥} ∧ 𝐵𝑛 ∧ ({𝑥𝑋 ∣ ¬ 𝐵𝑅𝑥} ∩ 𝑛) = ∅) ↔ (𝐴 ∈ {𝑥𝑋 ∣ ¬ 𝐵𝑅𝑥} ∧ 𝐵 ∈ {𝑥𝑋 ∣ ¬ 𝑥𝑅𝐴} ∧ {𝑥𝑋 ∣ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴)} = ∅)))
3932, 38rspc2ev 3308 . . . . 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 1335 . . . 4 ((((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐴𝐵)) ∧ {𝑥𝑋 ∣ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴)} = ∅) → ∃𝑚 ∈ (ordTop‘𝑅)∃𝑛 ∈ (ordTop‘𝑅)(𝐴𝑚𝐵𝑛 ∧ (𝑚𝑛) = ∅))
4140ex 450 . . 3 (((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐴𝐵)) → ({𝑥𝑋 ∣ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴)} = ∅ → ∃𝑚 ∈ (ordTop‘𝑅)∃𝑛 ∈ (ordTop‘𝑅)(𝐴𝑚𝐵𝑛 ∧ (𝑚𝑛) = ∅)))
42 rabn0 3932 . . . 4 ({𝑥𝑋 ∣ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴)} ≠ ∅ ↔ ∃𝑥𝑋𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴))
43 simpll1 1098 . . . . . . 7 ((((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐴𝐵)) ∧ (𝑥𝑋 ∧ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴))) → 𝑅 ∈ TosetRel )
44 simprl 793 . . . . . . 7 ((((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐴𝐵)) ∧ (𝑥𝑋 ∧ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴))) → 𝑥𝑋)
453ordtopn2 20909 . . . . . . 7 ((𝑅 ∈ TosetRel ∧ 𝑥𝑋) → {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦} ∈ (ordTop‘𝑅))
4643, 44, 45syl2anc 692 . . . . . 6 ((((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐴𝐵)) ∧ (𝑥𝑋 ∧ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴))) → {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦} ∈ (ordTop‘𝑅))
473ordtopn1 20908 . . . . . . 7 ((𝑅 ∈ TosetRel ∧ 𝑥𝑋) → {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥} ∈ (ordTop‘𝑅))
4843, 44, 47syl2anc 692 . . . . . 6 ((((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐴𝐵)) ∧ (𝑥𝑋 ∧ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴))) → {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥} ∈ (ordTop‘𝑅))
49 simpll2 1099 . . . . . . 7 ((((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐴𝐵)) ∧ (𝑥𝑋 ∧ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴))) → 𝐴𝑋)
50 simprrr 804 . . . . . . 7 ((((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐴𝐵)) ∧ (𝑥𝑋 ∧ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴))) → ¬ 𝑥𝑅𝐴)
51 breq2 4617 . . . . . . . . 9 (𝑦 = 𝐴 → (𝑥𝑅𝑦𝑥𝑅𝐴))
5251notbid 308 . . . . . . . 8 (𝑦 = 𝐴 → (¬ 𝑥𝑅𝑦 ↔ ¬ 𝑥𝑅𝐴))
5352elrab 3346 . . . . . . 7 (𝐴 ∈ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦} ↔ (𝐴𝑋 ∧ ¬ 𝑥𝑅𝐴))
5449, 50, 53sylanbrc 697 . . . . . 6 ((((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐴𝐵)) ∧ (𝑥𝑋 ∧ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴))) → 𝐴 ∈ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦})
55 simpll3 1100 . . . . . . 7 ((((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐴𝐵)) ∧ (𝑥𝑋 ∧ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴))) → 𝐵𝑋)
56 simprrl 803 . . . . . . 7 ((((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐴𝐵)) ∧ (𝑥𝑋 ∧ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴))) → ¬ 𝐵𝑅𝑥)
57 breq1 4616 . . . . . . . . 9 (𝑦 = 𝐵 → (𝑦𝑅𝑥𝐵𝑅𝑥))
5857notbid 308 . . . . . . . 8 (𝑦 = 𝐵 → (¬ 𝑦𝑅𝑥 ↔ ¬ 𝐵𝑅𝑥))
5958elrab 3346 . . . . . . 7 (𝐵 ∈ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥} ↔ (𝐵𝑋 ∧ ¬ 𝐵𝑅𝑥))
6055, 56, 59sylanbrc 697 . . . . . 6 ((((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐴𝐵)) ∧ (𝑥𝑋 ∧ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴))) → 𝐵 ∈ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥})
6143, 44jca 554 . . . . . . . . . 10 ((((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐴𝐵)) ∧ (𝑥𝑋 ∧ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴))) → (𝑅 ∈ TosetRel ∧ 𝑥𝑋))
623tsrlin 17140 . . . . . . . . . . 11 ((𝑅 ∈ TosetRel ∧ 𝑥𝑋𝑦𝑋) → (𝑥𝑅𝑦𝑦𝑅𝑥))
63623expa 1262 . . . . . . . . . 10 (((𝑅 ∈ TosetRel ∧ 𝑥𝑋) ∧ 𝑦𝑋) → (𝑥𝑅𝑦𝑦𝑅𝑥))
6461, 63sylan 488 . . . . . . . . 9 (((((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐴𝐵)) ∧ (𝑥𝑋 ∧ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴))) ∧ 𝑦𝑋) → (𝑥𝑅𝑦𝑦𝑅𝑥))
65 oran 517 . . . . . . . . 9 ((𝑥𝑅𝑦𝑦𝑅𝑥) ↔ ¬ (¬ 𝑥𝑅𝑦 ∧ ¬ 𝑦𝑅𝑥))
6664, 65sylib 208 . . . . . . . 8 (((((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐴𝐵)) ∧ (𝑥𝑋 ∧ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴))) ∧ 𝑦𝑋) → ¬ (¬ 𝑥𝑅𝑦 ∧ ¬ 𝑦𝑅𝑥))
6766ralrimiva 2960 . . . . . . 7 ((((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐴𝐵)) ∧ (𝑥𝑋 ∧ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴))) → ∀𝑦𝑋 ¬ (¬ 𝑥𝑅𝑦 ∧ ¬ 𝑦𝑅𝑥))
68 rabeq0 3931 . . . . . . 7 ({𝑦𝑋 ∣ (¬ 𝑥𝑅𝑦 ∧ ¬ 𝑦𝑅𝑥)} = ∅ ↔ ∀𝑦𝑋 ¬ (¬ 𝑥𝑅𝑦 ∧ ¬ 𝑦𝑅𝑥))
6967, 68sylibr 224 . . . . . 6 ((((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐴𝐵)) ∧ (𝑥𝑋 ∧ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴))) → {𝑦𝑋 ∣ (¬ 𝑥𝑅𝑦 ∧ ¬ 𝑦𝑅𝑥)} = ∅)
70 eleq2 2687 . . . . . . . 8 (𝑚 = {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦} → (𝐴𝑚𝐴 ∈ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦}))
71 ineq1 3785 . . . . . . . . 9 (𝑚 = {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦} → (𝑚𝑛) = ({𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦} ∩ 𝑛))
7271eqeq1d 2623 . . . . . . . 8 (𝑚 = {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦} → ((𝑚𝑛) = ∅ ↔ ({𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦} ∩ 𝑛) = ∅))
7370, 723anbi13d 1398 . . . . . . 7 (𝑚 = {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦} → ((𝐴𝑚𝐵𝑛 ∧ (𝑚𝑛) = ∅) ↔ (𝐴 ∈ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦} ∧ 𝐵𝑛 ∧ ({𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦} ∩ 𝑛) = ∅)))
74 eleq2 2687 . . . . . . . 8 (𝑛 = {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥} → (𝐵𝑛𝐵 ∈ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥}))
75 ineq2 3786 . . . . . . . . . 10 (𝑛 = {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥} → ({𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦} ∩ 𝑛) = ({𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦} ∩ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥}))
76 inrab 3875 . . . . . . . . . 10 ({𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦} ∩ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥}) = {𝑦𝑋 ∣ (¬ 𝑥𝑅𝑦 ∧ ¬ 𝑦𝑅𝑥)}
7775, 76syl6eq 2671 . . . . . . . . 9 (𝑛 = {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥} → ({𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦} ∩ 𝑛) = {𝑦𝑋 ∣ (¬ 𝑥𝑅𝑦 ∧ ¬ 𝑦𝑅𝑥)})
7877eqeq1d 2623 . . . . . . . 8 (𝑛 = {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥} → (({𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦} ∩ 𝑛) = ∅ ↔ {𝑦𝑋 ∣ (¬ 𝑥𝑅𝑦 ∧ ¬ 𝑦𝑅𝑥)} = ∅))
7974, 783anbi23d 1399 . . . . . . 7 (𝑛 = {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥} → ((𝐴 ∈ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦} ∧ 𝐵𝑛 ∧ ({𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦} ∩ 𝑛) = ∅) ↔ (𝐴 ∈ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦} ∧ 𝐵 ∈ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥} ∧ {𝑦𝑋 ∣ (¬ 𝑥𝑅𝑦 ∧ ¬ 𝑦𝑅𝑥)} = ∅)))
8073, 79rspc2ev 3308 . . . . . 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 1335 . . . . 5 ((((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐴𝐵)) ∧ (𝑥𝑋 ∧ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴))) → ∃𝑚 ∈ (ordTop‘𝑅)∃𝑛 ∈ (ordTop‘𝑅)(𝐴𝑚𝐵𝑛 ∧ (𝑚𝑛) = ∅))
8281rexlimdvaa 3025 . . . 4 (((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐴𝐵)) → (∃𝑥𝑋𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴) → ∃𝑚 ∈ (ordTop‘𝑅)∃𝑛 ∈ (ordTop‘𝑅)(𝐴𝑚𝐵𝑛 ∧ (𝑚𝑛) = ∅)))
8342, 82syl5bi 232 . . 3 (((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐴𝐵)) → ({𝑥𝑋 ∣ (¬ 𝐵𝑅𝑥 ∧ ¬ 𝑥𝑅𝐴)} ≠ ∅ → ∃𝑚 ∈ (ordTop‘𝑅)∃𝑛 ∈ (ordTop‘𝑅)(𝐴𝑚𝐵𝑛 ∧ (𝑚𝑛) = ∅)))
8441, 83pm2.61dne 2876 . 2 (((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐴𝐵)) → ∃𝑚 ∈ (ordTop‘𝑅)∃𝑛 ∈ (ordTop‘𝑅)(𝐴𝑚𝐵𝑛 ∧ (𝑚𝑛) = ∅))
8584exp32 630 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  wo 383  wa 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790  wral 2907  wrex 2908  {crab 2911  cin 3554  c0 3891   class class class wbr 4613  dom cdm 5074  cfv 5847  ordTopcordt 16080  PosetRelcps 17119   TosetRel ctsr 17120
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-oadd 7509  df-er 7687  df-en 7900  df-fin 7903  df-fi 8261  df-topgen 16025  df-ordt 16082  df-ps 17121  df-tsr 17122  df-bases 20622
This theorem is referenced by:  ordthaus  21098
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