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Theorem ordthauslem 22878
Description: Lemma for ordthaus 22879. (Contributed by Mario Carneiro, 13-Sep-2015.)
Hypothesis
Ref Expression
ordthauslem.1 𝑋 = dom 𝑅
Assertion
Ref Expression
ordthauslem ((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (𝐴𝑅𝐡 β†’ (𝐴 β‰  𝐡 β†’ βˆƒπ‘š ∈ (ordTopβ€˜π‘…)βˆƒπ‘› ∈ (ordTopβ€˜π‘…)(𝐴 ∈ π‘š ∧ 𝐡 ∈ 𝑛 ∧ (π‘š ∩ 𝑛) = βˆ…))))
Distinct variable groups:   π‘š,𝑛,𝐴   𝐡,π‘š,𝑛   𝑅,π‘š,𝑛   π‘š,𝑋,𝑛
Proof of Theorem ordthauslem
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpll1 1212 . . . . . 6 ((((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ (𝐴𝑅𝐡 ∧ 𝐴 β‰  𝐡)) ∧ {π‘₯ ∈ 𝑋 ∣ (Β¬ 𝐡𝑅π‘₯ ∧ Β¬ π‘₯𝑅𝐴)} = βˆ…) β†’ 𝑅 ∈ TosetRel )
2 simpll3 1214 . . . . . 6 ((((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ (𝐴𝑅𝐡 ∧ 𝐴 β‰  𝐡)) ∧ {π‘₯ ∈ 𝑋 ∣ (Β¬ 𝐡𝑅π‘₯ ∧ Β¬ π‘₯𝑅𝐴)} = βˆ…) β†’ 𝐡 ∈ 𝑋)
3 ordthauslem.1 . . . . . . 7 𝑋 = dom 𝑅
43ordtopn2 22690 . . . . . 6 ((𝑅 ∈ TosetRel ∧ 𝐡 ∈ 𝑋) β†’ {π‘₯ ∈ 𝑋 ∣ Β¬ 𝐡𝑅π‘₯} ∈ (ordTopβ€˜π‘…))
51, 2, 4syl2anc 584 . . . . 5 ((((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ (𝐴𝑅𝐡 ∧ 𝐴 β‰  𝐡)) ∧ {π‘₯ ∈ 𝑋 ∣ (Β¬ 𝐡𝑅π‘₯ ∧ Β¬ π‘₯𝑅𝐴)} = βˆ…) β†’ {π‘₯ ∈ 𝑋 ∣ Β¬ 𝐡𝑅π‘₯} ∈ (ordTopβ€˜π‘…))
6 simpll2 1213 . . . . . 6 ((((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ (𝐴𝑅𝐡 ∧ 𝐴 β‰  𝐡)) ∧ {π‘₯ ∈ 𝑋 ∣ (Β¬ 𝐡𝑅π‘₯ ∧ Β¬ π‘₯𝑅𝐴)} = βˆ…) β†’ 𝐴 ∈ 𝑋)
73ordtopn1 22689 . . . . . 6 ((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋) β†’ {π‘₯ ∈ 𝑋 ∣ Β¬ π‘₯𝑅𝐴} ∈ (ordTopβ€˜π‘…))
81, 6, 7syl2anc 584 . . . . 5 ((((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ (𝐴𝑅𝐡 ∧ 𝐴 β‰  𝐡)) ∧ {π‘₯ ∈ 𝑋 ∣ (Β¬ 𝐡𝑅π‘₯ ∧ Β¬ π‘₯𝑅𝐴)} = βˆ…) β†’ {π‘₯ ∈ 𝑋 ∣ Β¬ π‘₯𝑅𝐴} ∈ (ordTopβ€˜π‘…))
9 breq2 5151 . . . . . . 7 (π‘₯ = 𝐴 β†’ (𝐡𝑅π‘₯ ↔ 𝐡𝑅𝐴))
109notbid 317 . . . . . 6 (π‘₯ = 𝐴 β†’ (Β¬ 𝐡𝑅π‘₯ ↔ Β¬ 𝐡𝑅𝐴))
11 simprr 771 . . . . . . . 8 (((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ (𝐴𝑅𝐡 ∧ 𝐴 β‰  𝐡)) β†’ 𝐴 β‰  𝐡)
12 simpl1 1191 . . . . . . . . . . 11 (((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ (𝐴𝑅𝐡 ∧ 𝐴 β‰  𝐡)) β†’ 𝑅 ∈ TosetRel )
13 tsrps 18536 . . . . . . . . . . 11 (𝑅 ∈ TosetRel β†’ 𝑅 ∈ PosetRel)
1412, 13syl 17 . . . . . . . . . 10 (((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ (𝐴𝑅𝐡 ∧ 𝐴 β‰  𝐡)) β†’ 𝑅 ∈ PosetRel)
15 simprl 769 . . . . . . . . . 10 (((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ (𝐴𝑅𝐡 ∧ 𝐴 β‰  𝐡)) β†’ 𝐴𝑅𝐡)
16 psasym 18525 . . . . . . . . . . 11 ((𝑅 ∈ PosetRel ∧ 𝐴𝑅𝐡 ∧ 𝐡𝑅𝐴) β†’ 𝐴 = 𝐡)
17163expia 1121 . . . . . . . . . 10 ((𝑅 ∈ PosetRel ∧ 𝐴𝑅𝐡) β†’ (𝐡𝑅𝐴 β†’ 𝐴 = 𝐡))
1814, 15, 17syl2anc 584 . . . . . . . . 9 (((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ (𝐴𝑅𝐡 ∧ 𝐴 β‰  𝐡)) β†’ (𝐡𝑅𝐴 β†’ 𝐴 = 𝐡))
1918necon3ad 2953 . . . . . . . 8 (((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ (𝐴𝑅𝐡 ∧ 𝐴 β‰  𝐡)) β†’ (𝐴 β‰  𝐡 β†’ Β¬ 𝐡𝑅𝐴))
2011, 19mpd 15 . . . . . . 7 (((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ (𝐴𝑅𝐡 ∧ 𝐴 β‰  𝐡)) β†’ Β¬ 𝐡𝑅𝐴)
2120adantr 481 . . . . . 6 ((((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ (𝐴𝑅𝐡 ∧ 𝐴 β‰  𝐡)) ∧ {π‘₯ ∈ 𝑋 ∣ (Β¬ 𝐡𝑅π‘₯ ∧ Β¬ π‘₯𝑅𝐴)} = βˆ…) β†’ Β¬ 𝐡𝑅𝐴)
2210, 6, 21elrabd 3684 . . . . 5 ((((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ (𝐴𝑅𝐡 ∧ 𝐴 β‰  𝐡)) ∧ {π‘₯ ∈ 𝑋 ∣ (Β¬ 𝐡𝑅π‘₯ ∧ Β¬ π‘₯𝑅𝐴)} = βˆ…) β†’ 𝐴 ∈ {π‘₯ ∈ 𝑋 ∣ Β¬ 𝐡𝑅π‘₯})
23 breq1 5150 . . . . . . 7 (π‘₯ = 𝐡 β†’ (π‘₯𝑅𝐴 ↔ 𝐡𝑅𝐴))
2423notbid 317 . . . . . 6 (π‘₯ = 𝐡 β†’ (Β¬ π‘₯𝑅𝐴 ↔ Β¬ 𝐡𝑅𝐴))
2524, 2, 21elrabd 3684 . . . . 5 ((((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ (𝐴𝑅𝐡 ∧ 𝐴 β‰  𝐡)) ∧ {π‘₯ ∈ 𝑋 ∣ (Β¬ 𝐡𝑅π‘₯ ∧ Β¬ π‘₯𝑅𝐴)} = βˆ…) β†’ 𝐡 ∈ {π‘₯ ∈ 𝑋 ∣ Β¬ π‘₯𝑅𝐴})
26 simpr 485 . . . . 5 ((((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ (𝐴𝑅𝐡 ∧ 𝐴 β‰  𝐡)) ∧ {π‘₯ ∈ 𝑋 ∣ (Β¬ 𝐡𝑅π‘₯ ∧ Β¬ π‘₯𝑅𝐴)} = βˆ…) β†’ {π‘₯ ∈ 𝑋 ∣ (Β¬ 𝐡𝑅π‘₯ ∧ Β¬ π‘₯𝑅𝐴)} = βˆ…)
27 eleq2 2822 . . . . . . 7 (π‘š = {π‘₯ ∈ 𝑋 ∣ Β¬ 𝐡𝑅π‘₯} β†’ (𝐴 ∈ π‘š ↔ 𝐴 ∈ {π‘₯ ∈ 𝑋 ∣ Β¬ 𝐡𝑅π‘₯}))
28 ineq1 4204 . . . . . . . 8 (π‘š = {π‘₯ ∈ 𝑋 ∣ Β¬ 𝐡𝑅π‘₯} β†’ (π‘š ∩ 𝑛) = ({π‘₯ ∈ 𝑋 ∣ Β¬ 𝐡𝑅π‘₯} ∩ 𝑛))
2928eqeq1d 2734 . . . . . . 7 (π‘š = {π‘₯ ∈ 𝑋 ∣ Β¬ 𝐡𝑅π‘₯} β†’ ((π‘š ∩ 𝑛) = βˆ… ↔ ({π‘₯ ∈ 𝑋 ∣ Β¬ 𝐡𝑅π‘₯} ∩ 𝑛) = βˆ…))
3027, 293anbi13d 1438 . . . . . 6 (π‘š = {π‘₯ ∈ 𝑋 ∣ Β¬ 𝐡𝑅π‘₯} β†’ ((𝐴 ∈ π‘š ∧ 𝐡 ∈ 𝑛 ∧ (π‘š ∩ 𝑛) = βˆ…) ↔ (𝐴 ∈ {π‘₯ ∈ 𝑋 ∣ Β¬ 𝐡𝑅π‘₯} ∧ 𝐡 ∈ 𝑛 ∧ ({π‘₯ ∈ 𝑋 ∣ Β¬ 𝐡𝑅π‘₯} ∩ 𝑛) = βˆ…)))
31 eleq2 2822 . . . . . . 7 (𝑛 = {π‘₯ ∈ 𝑋 ∣ Β¬ π‘₯𝑅𝐴} β†’ (𝐡 ∈ 𝑛 ↔ 𝐡 ∈ {π‘₯ ∈ 𝑋 ∣ Β¬ π‘₯𝑅𝐴}))
32 ineq2 4205 . . . . . . . . 9 (𝑛 = {π‘₯ ∈ 𝑋 ∣ Β¬ π‘₯𝑅𝐴} β†’ ({π‘₯ ∈ 𝑋 ∣ Β¬ 𝐡𝑅π‘₯} ∩ 𝑛) = ({π‘₯ ∈ 𝑋 ∣ Β¬ 𝐡𝑅π‘₯} ∩ {π‘₯ ∈ 𝑋 ∣ Β¬ π‘₯𝑅𝐴}))
33 inrab 4305 . . . . . . . . 9 ({π‘₯ ∈ 𝑋 ∣ Β¬ 𝐡𝑅π‘₯} ∩ {π‘₯ ∈ 𝑋 ∣ Β¬ π‘₯𝑅𝐴}) = {π‘₯ ∈ 𝑋 ∣ (Β¬ 𝐡𝑅π‘₯ ∧ Β¬ π‘₯𝑅𝐴)}
3432, 33eqtrdi 2788 . . . . . . . 8 (𝑛 = {π‘₯ ∈ 𝑋 ∣ Β¬ π‘₯𝑅𝐴} β†’ ({π‘₯ ∈ 𝑋 ∣ Β¬ 𝐡𝑅π‘₯} ∩ 𝑛) = {π‘₯ ∈ 𝑋 ∣ (Β¬ 𝐡𝑅π‘₯ ∧ Β¬ π‘₯𝑅𝐴)})
3534eqeq1d 2734 . . . . . . 7 (𝑛 = {π‘₯ ∈ 𝑋 ∣ Β¬ π‘₯𝑅𝐴} β†’ (({π‘₯ ∈ 𝑋 ∣ Β¬ 𝐡𝑅π‘₯} ∩ 𝑛) = βˆ… ↔ {π‘₯ ∈ 𝑋 ∣ (Β¬ 𝐡𝑅π‘₯ ∧ Β¬ π‘₯𝑅𝐴)} = βˆ…))
3631, 353anbi23d 1439 . . . . . 6 (𝑛 = {π‘₯ ∈ 𝑋 ∣ Β¬ π‘₯𝑅𝐴} β†’ ((𝐴 ∈ {π‘₯ ∈ 𝑋 ∣ Β¬ 𝐡𝑅π‘₯} ∧ 𝐡 ∈ 𝑛 ∧ ({π‘₯ ∈ 𝑋 ∣ Β¬ 𝐡𝑅π‘₯} ∩ 𝑛) = βˆ…) ↔ (𝐴 ∈ {π‘₯ ∈ 𝑋 ∣ Β¬ 𝐡𝑅π‘₯} ∧ 𝐡 ∈ {π‘₯ ∈ 𝑋 ∣ Β¬ π‘₯𝑅𝐴} ∧ {π‘₯ ∈ 𝑋 ∣ (Β¬ 𝐡𝑅π‘₯ ∧ Β¬ π‘₯𝑅𝐴)} = βˆ…)))
3730, 36rspc2ev 3623 . . . . 5 (({π‘₯ ∈ 𝑋 ∣ Β¬ 𝐡𝑅π‘₯} ∈ (ordTopβ€˜π‘…) ∧ {π‘₯ ∈ 𝑋 ∣ Β¬ π‘₯𝑅𝐴} ∈ (ordTopβ€˜π‘…) ∧ (𝐴 ∈ {π‘₯ ∈ 𝑋 ∣ Β¬ 𝐡𝑅π‘₯} ∧ 𝐡 ∈ {π‘₯ ∈ 𝑋 ∣ Β¬ π‘₯𝑅𝐴} ∧ {π‘₯ ∈ 𝑋 ∣ (Β¬ 𝐡𝑅π‘₯ ∧ Β¬ π‘₯𝑅𝐴)} = βˆ…)) β†’ βˆƒπ‘š ∈ (ordTopβ€˜π‘…)βˆƒπ‘› ∈ (ordTopβ€˜π‘…)(𝐴 ∈ π‘š ∧ 𝐡 ∈ 𝑛 ∧ (π‘š ∩ 𝑛) = βˆ…))
385, 8, 22, 25, 26, 37syl113anc 1382 . . . 4 ((((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ (𝐴𝑅𝐡 ∧ 𝐴 β‰  𝐡)) ∧ {π‘₯ ∈ 𝑋 ∣ (Β¬ 𝐡𝑅π‘₯ ∧ Β¬ π‘₯𝑅𝐴)} = βˆ…) β†’ βˆƒπ‘š ∈ (ordTopβ€˜π‘…)βˆƒπ‘› ∈ (ordTopβ€˜π‘…)(𝐴 ∈ π‘š ∧ 𝐡 ∈ 𝑛 ∧ (π‘š ∩ 𝑛) = βˆ…))
3938ex 413 . . 3 (((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ (𝐴𝑅𝐡 ∧ 𝐴 β‰  𝐡)) β†’ ({π‘₯ ∈ 𝑋 ∣ (Β¬ 𝐡𝑅π‘₯ ∧ Β¬ π‘₯𝑅𝐴)} = βˆ… β†’ βˆƒπ‘š ∈ (ordTopβ€˜π‘…)βˆƒπ‘› ∈ (ordTopβ€˜π‘…)(𝐴 ∈ π‘š ∧ 𝐡 ∈ 𝑛 ∧ (π‘š ∩ 𝑛) = βˆ…)))
40 rabn0 4384 . . . 4 ({π‘₯ ∈ 𝑋 ∣ (Β¬ 𝐡𝑅π‘₯ ∧ Β¬ π‘₯𝑅𝐴)} β‰  βˆ… ↔ βˆƒπ‘₯ ∈ 𝑋 (Β¬ 𝐡𝑅π‘₯ ∧ Β¬ π‘₯𝑅𝐴))
41 simpll1 1212 . . . . . . 7 ((((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ (𝐴𝑅𝐡 ∧ 𝐴 β‰  𝐡)) ∧ (π‘₯ ∈ 𝑋 ∧ (Β¬ 𝐡𝑅π‘₯ ∧ Β¬ π‘₯𝑅𝐴))) β†’ 𝑅 ∈ TosetRel )
42 simprl 769 . . . . . . 7 ((((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ (𝐴𝑅𝐡 ∧ 𝐴 β‰  𝐡)) ∧ (π‘₯ ∈ 𝑋 ∧ (Β¬ 𝐡𝑅π‘₯ ∧ Β¬ π‘₯𝑅𝐴))) β†’ π‘₯ ∈ 𝑋)
433ordtopn2 22690 . . . . . . 7 ((𝑅 ∈ TosetRel ∧ π‘₯ ∈ 𝑋) β†’ {𝑦 ∈ 𝑋 ∣ Β¬ π‘₯𝑅𝑦} ∈ (ordTopβ€˜π‘…))
4441, 42, 43syl2anc 584 . . . . . 6 ((((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ (𝐴𝑅𝐡 ∧ 𝐴 β‰  𝐡)) ∧ (π‘₯ ∈ 𝑋 ∧ (Β¬ 𝐡𝑅π‘₯ ∧ Β¬ π‘₯𝑅𝐴))) β†’ {𝑦 ∈ 𝑋 ∣ Β¬ π‘₯𝑅𝑦} ∈ (ordTopβ€˜π‘…))
453ordtopn1 22689 . . . . . . 7 ((𝑅 ∈ TosetRel ∧ π‘₯ ∈ 𝑋) β†’ {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯} ∈ (ordTopβ€˜π‘…))
4641, 42, 45syl2anc 584 . . . . . 6 ((((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ (𝐴𝑅𝐡 ∧ 𝐴 β‰  𝐡)) ∧ (π‘₯ ∈ 𝑋 ∧ (Β¬ 𝐡𝑅π‘₯ ∧ Β¬ π‘₯𝑅𝐴))) β†’ {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯} ∈ (ordTopβ€˜π‘…))
47 breq2 5151 . . . . . . . 8 (𝑦 = 𝐴 β†’ (π‘₯𝑅𝑦 ↔ π‘₯𝑅𝐴))
4847notbid 317 . . . . . . 7 (𝑦 = 𝐴 β†’ (Β¬ π‘₯𝑅𝑦 ↔ Β¬ π‘₯𝑅𝐴))
49 simpll2 1213 . . . . . . 7 ((((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ (𝐴𝑅𝐡 ∧ 𝐴 β‰  𝐡)) ∧ (π‘₯ ∈ 𝑋 ∧ (Β¬ 𝐡𝑅π‘₯ ∧ Β¬ π‘₯𝑅𝐴))) β†’ 𝐴 ∈ 𝑋)
50 simprrr 780 . . . . . . 7 ((((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ (𝐴𝑅𝐡 ∧ 𝐴 β‰  𝐡)) ∧ (π‘₯ ∈ 𝑋 ∧ (Β¬ 𝐡𝑅π‘₯ ∧ Β¬ π‘₯𝑅𝐴))) β†’ Β¬ π‘₯𝑅𝐴)
5148, 49, 50elrabd 3684 . . . . . 6 ((((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ (𝐴𝑅𝐡 ∧ 𝐴 β‰  𝐡)) ∧ (π‘₯ ∈ 𝑋 ∧ (Β¬ 𝐡𝑅π‘₯ ∧ Β¬ π‘₯𝑅𝐴))) β†’ 𝐴 ∈ {𝑦 ∈ 𝑋 ∣ Β¬ π‘₯𝑅𝑦})
52 breq1 5150 . . . . . . . 8 (𝑦 = 𝐡 β†’ (𝑦𝑅π‘₯ ↔ 𝐡𝑅π‘₯))
5352notbid 317 . . . . . . 7 (𝑦 = 𝐡 β†’ (Β¬ 𝑦𝑅π‘₯ ↔ Β¬ 𝐡𝑅π‘₯))
54 simpll3 1214 . . . . . . 7 ((((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ (𝐴𝑅𝐡 ∧ 𝐴 β‰  𝐡)) ∧ (π‘₯ ∈ 𝑋 ∧ (Β¬ 𝐡𝑅π‘₯ ∧ Β¬ π‘₯𝑅𝐴))) β†’ 𝐡 ∈ 𝑋)
55 simprrl 779 . . . . . . 7 ((((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ (𝐴𝑅𝐡 ∧ 𝐴 β‰  𝐡)) ∧ (π‘₯ ∈ 𝑋 ∧ (Β¬ 𝐡𝑅π‘₯ ∧ Β¬ π‘₯𝑅𝐴))) β†’ Β¬ 𝐡𝑅π‘₯)
5653, 54, 55elrabd 3684 . . . . . 6 ((((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ (𝐴𝑅𝐡 ∧ 𝐴 β‰  𝐡)) ∧ (π‘₯ ∈ 𝑋 ∧ (Β¬ 𝐡𝑅π‘₯ ∧ Β¬ π‘₯𝑅𝐴))) β†’ 𝐡 ∈ {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯})
5741, 42jca 512 . . . . . . . . . 10 ((((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ (𝐴𝑅𝐡 ∧ 𝐴 β‰  𝐡)) ∧ (π‘₯ ∈ 𝑋 ∧ (Β¬ 𝐡𝑅π‘₯ ∧ Β¬ π‘₯𝑅𝐴))) β†’ (𝑅 ∈ TosetRel ∧ π‘₯ ∈ 𝑋))
583tsrlin 18534 . . . . . . . . . . 11 ((𝑅 ∈ TosetRel ∧ π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) β†’ (π‘₯𝑅𝑦 ∨ 𝑦𝑅π‘₯))
59583expa 1118 . . . . . . . . . 10 (((𝑅 ∈ TosetRel ∧ π‘₯ ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) β†’ (π‘₯𝑅𝑦 ∨ 𝑦𝑅π‘₯))
6057, 59sylan 580 . . . . . . . . 9 (((((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ (𝐴𝑅𝐡 ∧ 𝐴 β‰  𝐡)) ∧ (π‘₯ ∈ 𝑋 ∧ (Β¬ 𝐡𝑅π‘₯ ∧ Β¬ π‘₯𝑅𝐴))) ∧ 𝑦 ∈ 𝑋) β†’ (π‘₯𝑅𝑦 ∨ 𝑦𝑅π‘₯))
61 oran 988 . . . . . . . . 9 ((π‘₯𝑅𝑦 ∨ 𝑦𝑅π‘₯) ↔ Β¬ (Β¬ π‘₯𝑅𝑦 ∧ Β¬ 𝑦𝑅π‘₯))
6260, 61sylib 217 . . . . . . . 8 (((((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ (𝐴𝑅𝐡 ∧ 𝐴 β‰  𝐡)) ∧ (π‘₯ ∈ 𝑋 ∧ (Β¬ 𝐡𝑅π‘₯ ∧ Β¬ π‘₯𝑅𝐴))) ∧ 𝑦 ∈ 𝑋) β†’ Β¬ (Β¬ π‘₯𝑅𝑦 ∧ Β¬ 𝑦𝑅π‘₯))
6362ralrimiva 3146 . . . . . . 7 ((((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ (𝐴𝑅𝐡 ∧ 𝐴 β‰  𝐡)) ∧ (π‘₯ ∈ 𝑋 ∧ (Β¬ 𝐡𝑅π‘₯ ∧ Β¬ π‘₯𝑅𝐴))) β†’ βˆ€π‘¦ ∈ 𝑋 Β¬ (Β¬ π‘₯𝑅𝑦 ∧ Β¬ 𝑦𝑅π‘₯))
64 rabeq0 4383 . . . . . . 7 ({𝑦 ∈ 𝑋 ∣ (Β¬ π‘₯𝑅𝑦 ∧ Β¬ 𝑦𝑅π‘₯)} = βˆ… ↔ βˆ€π‘¦ ∈ 𝑋 Β¬ (Β¬ π‘₯𝑅𝑦 ∧ Β¬ 𝑦𝑅π‘₯))
6563, 64sylibr 233 . . . . . 6 ((((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ (𝐴𝑅𝐡 ∧ 𝐴 β‰  𝐡)) ∧ (π‘₯ ∈ 𝑋 ∧ (Β¬ 𝐡𝑅π‘₯ ∧ Β¬ π‘₯𝑅𝐴))) β†’ {𝑦 ∈ 𝑋 ∣ (Β¬ π‘₯𝑅𝑦 ∧ Β¬ 𝑦𝑅π‘₯)} = βˆ…)
66 eleq2 2822 . . . . . . . 8 (π‘š = {𝑦 ∈ 𝑋 ∣ Β¬ π‘₯𝑅𝑦} β†’ (𝐴 ∈ π‘š ↔ 𝐴 ∈ {𝑦 ∈ 𝑋 ∣ Β¬ π‘₯𝑅𝑦}))
67 ineq1 4204 . . . . . . . . 9 (π‘š = {𝑦 ∈ 𝑋 ∣ Β¬ π‘₯𝑅𝑦} β†’ (π‘š ∩ 𝑛) = ({𝑦 ∈ 𝑋 ∣ Β¬ π‘₯𝑅𝑦} ∩ 𝑛))
6867eqeq1d 2734 . . . . . . . 8 (π‘š = {𝑦 ∈ 𝑋 ∣ Β¬ π‘₯𝑅𝑦} β†’ ((π‘š ∩ 𝑛) = βˆ… ↔ ({𝑦 ∈ 𝑋 ∣ Β¬ π‘₯𝑅𝑦} ∩ 𝑛) = βˆ…))
6966, 683anbi13d 1438 . . . . . . 7 (π‘š = {𝑦 ∈ 𝑋 ∣ Β¬ π‘₯𝑅𝑦} β†’ ((𝐴 ∈ π‘š ∧ 𝐡 ∈ 𝑛 ∧ (π‘š ∩ 𝑛) = βˆ…) ↔ (𝐴 ∈ {𝑦 ∈ 𝑋 ∣ Β¬ π‘₯𝑅𝑦} ∧ 𝐡 ∈ 𝑛 ∧ ({𝑦 ∈ 𝑋 ∣ Β¬ π‘₯𝑅𝑦} ∩ 𝑛) = βˆ…)))
70 eleq2 2822 . . . . . . . 8 (𝑛 = {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯} β†’ (𝐡 ∈ 𝑛 ↔ 𝐡 ∈ {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯}))
71 ineq2 4205 . . . . . . . . . 10 (𝑛 = {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯} β†’ ({𝑦 ∈ 𝑋 ∣ Β¬ π‘₯𝑅𝑦} ∩ 𝑛) = ({𝑦 ∈ 𝑋 ∣ Β¬ π‘₯𝑅𝑦} ∩ {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯}))
72 inrab 4305 . . . . . . . . . 10 ({𝑦 ∈ 𝑋 ∣ Β¬ π‘₯𝑅𝑦} ∩ {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯}) = {𝑦 ∈ 𝑋 ∣ (Β¬ π‘₯𝑅𝑦 ∧ Β¬ 𝑦𝑅π‘₯)}
7371, 72eqtrdi 2788 . . . . . . . . 9 (𝑛 = {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯} β†’ ({𝑦 ∈ 𝑋 ∣ Β¬ π‘₯𝑅𝑦} ∩ 𝑛) = {𝑦 ∈ 𝑋 ∣ (Β¬ π‘₯𝑅𝑦 ∧ Β¬ 𝑦𝑅π‘₯)})
7473eqeq1d 2734 . . . . . . . 8 (𝑛 = {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯} β†’ (({𝑦 ∈ 𝑋 ∣ Β¬ π‘₯𝑅𝑦} ∩ 𝑛) = βˆ… ↔ {𝑦 ∈ 𝑋 ∣ (Β¬ π‘₯𝑅𝑦 ∧ Β¬ 𝑦𝑅π‘₯)} = βˆ…))
7570, 743anbi23d 1439 . . . . . . 7 (𝑛 = {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯} β†’ ((𝐴 ∈ {𝑦 ∈ 𝑋 ∣ Β¬ π‘₯𝑅𝑦} ∧ 𝐡 ∈ 𝑛 ∧ ({𝑦 ∈ 𝑋 ∣ Β¬ π‘₯𝑅𝑦} ∩ 𝑛) = βˆ…) ↔ (𝐴 ∈ {𝑦 ∈ 𝑋 ∣ Β¬ π‘₯𝑅𝑦} ∧ 𝐡 ∈ {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯} ∧ {𝑦 ∈ 𝑋 ∣ (Β¬ π‘₯𝑅𝑦 ∧ Β¬ 𝑦𝑅π‘₯)} = βˆ…)))
7669, 75rspc2ev 3623 . . . . . 6 (({𝑦 ∈ 𝑋 ∣ Β¬ π‘₯𝑅𝑦} ∈ (ordTopβ€˜π‘…) ∧ {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯} ∈ (ordTopβ€˜π‘…) ∧ (𝐴 ∈ {𝑦 ∈ 𝑋 ∣ Β¬ π‘₯𝑅𝑦} ∧ 𝐡 ∈ {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯} ∧ {𝑦 ∈ 𝑋 ∣ (Β¬ π‘₯𝑅𝑦 ∧ Β¬ 𝑦𝑅π‘₯)} = βˆ…)) β†’ βˆƒπ‘š ∈ (ordTopβ€˜π‘…)βˆƒπ‘› ∈ (ordTopβ€˜π‘…)(𝐴 ∈ π‘š ∧ 𝐡 ∈ 𝑛 ∧ (π‘š ∩ 𝑛) = βˆ…))
7744, 46, 51, 56, 65, 76syl113anc 1382 . . . . 5 ((((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ (𝐴𝑅𝐡 ∧ 𝐴 β‰  𝐡)) ∧ (π‘₯ ∈ 𝑋 ∧ (Β¬ 𝐡𝑅π‘₯ ∧ Β¬ π‘₯𝑅𝐴))) β†’ βˆƒπ‘š ∈ (ordTopβ€˜π‘…)βˆƒπ‘› ∈ (ordTopβ€˜π‘…)(𝐴 ∈ π‘š ∧ 𝐡 ∈ 𝑛 ∧ (π‘š ∩ 𝑛) = βˆ…))
7877rexlimdvaa 3156 . . . 4 (((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ (𝐴𝑅𝐡 ∧ 𝐴 β‰  𝐡)) β†’ (βˆƒπ‘₯ ∈ 𝑋 (Β¬ 𝐡𝑅π‘₯ ∧ Β¬ π‘₯𝑅𝐴) β†’ βˆƒπ‘š ∈ (ordTopβ€˜π‘…)βˆƒπ‘› ∈ (ordTopβ€˜π‘…)(𝐴 ∈ π‘š ∧ 𝐡 ∈ 𝑛 ∧ (π‘š ∩ 𝑛) = βˆ…)))
7940, 78biimtrid 241 . . 3 (((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ (𝐴𝑅𝐡 ∧ 𝐴 β‰  𝐡)) β†’ ({π‘₯ ∈ 𝑋 ∣ (Β¬ 𝐡𝑅π‘₯ ∧ Β¬ π‘₯𝑅𝐴)} β‰  βˆ… β†’ βˆƒπ‘š ∈ (ordTopβ€˜π‘…)βˆƒπ‘› ∈ (ordTopβ€˜π‘…)(𝐴 ∈ π‘š ∧ 𝐡 ∈ 𝑛 ∧ (π‘š ∩ 𝑛) = βˆ…)))
8039, 79pm2.61dne 3028 . 2 (((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ (𝐴𝑅𝐡 ∧ 𝐴 β‰  𝐡)) β†’ βˆƒπ‘š ∈ (ordTopβ€˜π‘…)βˆƒπ‘› ∈ (ordTopβ€˜π‘…)(𝐴 ∈ π‘š ∧ 𝐡 ∈ 𝑛 ∧ (π‘š ∩ 𝑛) = βˆ…))
8180exp32 421 1 ((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (𝐴𝑅𝐡 β†’ (𝐴 β‰  𝐡 β†’ βˆƒπ‘š ∈ (ordTopβ€˜π‘…)βˆƒπ‘› ∈ (ordTopβ€˜π‘…)(𝐴 ∈ π‘š ∧ 𝐡 ∈ 𝑛 ∧ (π‘š ∩ 𝑛) = βˆ…))))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 396   ∨ wo 845   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   β‰  wne 2940  βˆ€wral 3061  βˆƒwrex 3070  {crab 3432   ∩ cin 3946  βˆ…c0 4321   class class class wbr 5147  dom cdm 5675  β€˜cfv 6540  ordTopcordt 17441  PosetRelcps 18513   TosetRel ctsr 18514
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-om 7852  df-1o 8462  df-en 8936  df-fin 8939  df-fi 9402  df-topgen 17385  df-ordt 17443  df-ps 18515  df-tsr 18516  df-bases 22440
This theorem is referenced by:  ordthaus  22879
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