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Theorem ordthauslem 22750
Description: Lemma for ordthaus 22751. (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 1213 . . . . . 6 ((((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ (𝐴𝑅𝐡 ∧ 𝐴 β‰  𝐡)) ∧ {π‘₯ ∈ 𝑋 ∣ (Β¬ 𝐡𝑅π‘₯ ∧ Β¬ π‘₯𝑅𝐴)} = βˆ…) β†’ 𝑅 ∈ TosetRel )
2 simpll3 1215 . . . . . 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 22562 . . . . . 6 ((𝑅 ∈ TosetRel ∧ 𝐡 ∈ 𝑋) β†’ {π‘₯ ∈ 𝑋 ∣ Β¬ 𝐡𝑅π‘₯} ∈ (ordTopβ€˜π‘…))
51, 2, 4syl2anc 585 . . . . 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 1214 . . . . . 6 ((((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ (𝐴𝑅𝐡 ∧ 𝐴 β‰  𝐡)) ∧ {π‘₯ ∈ 𝑋 ∣ (Β¬ 𝐡𝑅π‘₯ ∧ Β¬ π‘₯𝑅𝐴)} = βˆ…) β†’ 𝐴 ∈ 𝑋)
73ordtopn1 22561 . . . . . 6 ((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋) β†’ {π‘₯ ∈ 𝑋 ∣ Β¬ π‘₯𝑅𝐴} ∈ (ordTopβ€˜π‘…))
81, 6, 7syl2anc 585 . . . . 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 5114 . . . . . . 7 (π‘₯ = 𝐴 β†’ (𝐡𝑅π‘₯ ↔ 𝐡𝑅𝐴))
109notbid 318 . . . . . 6 (π‘₯ = 𝐴 β†’ (Β¬ 𝐡𝑅π‘₯ ↔ Β¬ 𝐡𝑅𝐴))
11 simprr 772 . . . . . . . 8 (((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ (𝐴𝑅𝐡 ∧ 𝐴 β‰  𝐡)) β†’ 𝐴 β‰  𝐡)
12 simpl1 1192 . . . . . . . . . . 11 (((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ (𝐴𝑅𝐡 ∧ 𝐴 β‰  𝐡)) β†’ 𝑅 ∈ TosetRel )
13 tsrps 18483 . . . . . . . . . . 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 770 . . . . . . . . . 10 (((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ (𝐴𝑅𝐡 ∧ 𝐴 β‰  𝐡)) β†’ 𝐴𝑅𝐡)
16 psasym 18472 . . . . . . . . . . 11 ((𝑅 ∈ PosetRel ∧ 𝐴𝑅𝐡 ∧ 𝐡𝑅𝐴) β†’ 𝐴 = 𝐡)
17163expia 1122 . . . . . . . . . 10 ((𝑅 ∈ PosetRel ∧ 𝐴𝑅𝐡) β†’ (𝐡𝑅𝐴 β†’ 𝐴 = 𝐡))
1814, 15, 17syl2anc 585 . . . . . . . . 9 (((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ (𝐴𝑅𝐡 ∧ 𝐴 β‰  𝐡)) β†’ (𝐡𝑅𝐴 β†’ 𝐴 = 𝐡))
1918necon3ad 2957 . . . . . . . 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 482 . . . . . 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 3652 . . . . 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 5113 . . . . . . 7 (π‘₯ = 𝐡 β†’ (π‘₯𝑅𝐴 ↔ 𝐡𝑅𝐴))
2423notbid 318 . . . . . 6 (π‘₯ = 𝐡 β†’ (Β¬ π‘₯𝑅𝐴 ↔ Β¬ 𝐡𝑅𝐴))
2524, 2, 21elrabd 3652 . . . . 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 486 . . . . 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 2827 . . . . . . 7 (π‘š = {π‘₯ ∈ 𝑋 ∣ Β¬ 𝐡𝑅π‘₯} β†’ (𝐴 ∈ π‘š ↔ 𝐴 ∈ {π‘₯ ∈ 𝑋 ∣ Β¬ 𝐡𝑅π‘₯}))
28 ineq1 4170 . . . . . . . 8 (π‘š = {π‘₯ ∈ 𝑋 ∣ Β¬ 𝐡𝑅π‘₯} β†’ (π‘š ∩ 𝑛) = ({π‘₯ ∈ 𝑋 ∣ Β¬ 𝐡𝑅π‘₯} ∩ 𝑛))
2928eqeq1d 2739 . . . . . . 7 (π‘š = {π‘₯ ∈ 𝑋 ∣ Β¬ 𝐡𝑅π‘₯} β†’ ((π‘š ∩ 𝑛) = βˆ… ↔ ({π‘₯ ∈ 𝑋 ∣ Β¬ 𝐡𝑅π‘₯} ∩ 𝑛) = βˆ…))
3027, 293anbi13d 1439 . . . . . 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 2827 . . . . . . 7 (𝑛 = {π‘₯ ∈ 𝑋 ∣ Β¬ π‘₯𝑅𝐴} β†’ (𝐡 ∈ 𝑛 ↔ 𝐡 ∈ {π‘₯ ∈ 𝑋 ∣ Β¬ π‘₯𝑅𝐴}))
32 ineq2 4171 . . . . . . . . 9 (𝑛 = {π‘₯ ∈ 𝑋 ∣ Β¬ π‘₯𝑅𝐴} β†’ ({π‘₯ ∈ 𝑋 ∣ Β¬ 𝐡𝑅π‘₯} ∩ 𝑛) = ({π‘₯ ∈ 𝑋 ∣ Β¬ 𝐡𝑅π‘₯} ∩ {π‘₯ ∈ 𝑋 ∣ Β¬ π‘₯𝑅𝐴}))
33 inrab 4271 . . . . . . . . 9 ({π‘₯ ∈ 𝑋 ∣ Β¬ 𝐡𝑅π‘₯} ∩ {π‘₯ ∈ 𝑋 ∣ Β¬ π‘₯𝑅𝐴}) = {π‘₯ ∈ 𝑋 ∣ (Β¬ 𝐡𝑅π‘₯ ∧ Β¬ π‘₯𝑅𝐴)}
3432, 33eqtrdi 2793 . . . . . . . 8 (𝑛 = {π‘₯ ∈ 𝑋 ∣ Β¬ π‘₯𝑅𝐴} β†’ ({π‘₯ ∈ 𝑋 ∣ Β¬ 𝐡𝑅π‘₯} ∩ 𝑛) = {π‘₯ ∈ 𝑋 ∣ (Β¬ 𝐡𝑅π‘₯ ∧ Β¬ π‘₯𝑅𝐴)})
3534eqeq1d 2739 . . . . . . 7 (𝑛 = {π‘₯ ∈ 𝑋 ∣ Β¬ π‘₯𝑅𝐴} β†’ (({π‘₯ ∈ 𝑋 ∣ Β¬ 𝐡𝑅π‘₯} ∩ 𝑛) = βˆ… ↔ {π‘₯ ∈ 𝑋 ∣ (Β¬ 𝐡𝑅π‘₯ ∧ Β¬ π‘₯𝑅𝐴)} = βˆ…))
3631, 353anbi23d 1440 . . . . . 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 3595 . . . . 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 1383 . . . 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 414 . . 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 4350 . . . 4 ({π‘₯ ∈ 𝑋 ∣ (Β¬ 𝐡𝑅π‘₯ ∧ Β¬ π‘₯𝑅𝐴)} β‰  βˆ… ↔ βˆƒπ‘₯ ∈ 𝑋 (Β¬ 𝐡𝑅π‘₯ ∧ Β¬ π‘₯𝑅𝐴))
41 simpll1 1213 . . . . . . 7 ((((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ (𝐴𝑅𝐡 ∧ 𝐴 β‰  𝐡)) ∧ (π‘₯ ∈ 𝑋 ∧ (Β¬ 𝐡𝑅π‘₯ ∧ Β¬ π‘₯𝑅𝐴))) β†’ 𝑅 ∈ TosetRel )
42 simprl 770 . . . . . . 7 ((((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ (𝐴𝑅𝐡 ∧ 𝐴 β‰  𝐡)) ∧ (π‘₯ ∈ 𝑋 ∧ (Β¬ 𝐡𝑅π‘₯ ∧ Β¬ π‘₯𝑅𝐴))) β†’ π‘₯ ∈ 𝑋)
433ordtopn2 22562 . . . . . . 7 ((𝑅 ∈ TosetRel ∧ π‘₯ ∈ 𝑋) β†’ {𝑦 ∈ 𝑋 ∣ Β¬ π‘₯𝑅𝑦} ∈ (ordTopβ€˜π‘…))
4441, 42, 43syl2anc 585 . . . . . 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 22561 . . . . . . 7 ((𝑅 ∈ TosetRel ∧ π‘₯ ∈ 𝑋) β†’ {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯} ∈ (ordTopβ€˜π‘…))
4641, 42, 45syl2anc 585 . . . . . 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 5114 . . . . . . . 8 (𝑦 = 𝐴 β†’ (π‘₯𝑅𝑦 ↔ π‘₯𝑅𝐴))
4847notbid 318 . . . . . . 7 (𝑦 = 𝐴 β†’ (Β¬ π‘₯𝑅𝑦 ↔ Β¬ π‘₯𝑅𝐴))
49 simpll2 1214 . . . . . . 7 ((((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ (𝐴𝑅𝐡 ∧ 𝐴 β‰  𝐡)) ∧ (π‘₯ ∈ 𝑋 ∧ (Β¬ 𝐡𝑅π‘₯ ∧ Β¬ π‘₯𝑅𝐴))) β†’ 𝐴 ∈ 𝑋)
50 simprrr 781 . . . . . . 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 3652 . . . . . 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 5113 . . . . . . . 8 (𝑦 = 𝐡 β†’ (𝑦𝑅π‘₯ ↔ 𝐡𝑅π‘₯))
5352notbid 318 . . . . . . 7 (𝑦 = 𝐡 β†’ (Β¬ 𝑦𝑅π‘₯ ↔ Β¬ 𝐡𝑅π‘₯))
54 simpll3 1215 . . . . . . 7 ((((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ (𝐴𝑅𝐡 ∧ 𝐴 β‰  𝐡)) ∧ (π‘₯ ∈ 𝑋 ∧ (Β¬ 𝐡𝑅π‘₯ ∧ Β¬ π‘₯𝑅𝐴))) β†’ 𝐡 ∈ 𝑋)
55 simprrl 780 . . . . . . 7 ((((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ (𝐴𝑅𝐡 ∧ 𝐴 β‰  𝐡)) ∧ (π‘₯ ∈ 𝑋 ∧ (Β¬ 𝐡𝑅π‘₯ ∧ Β¬ π‘₯𝑅𝐴))) β†’ Β¬ 𝐡𝑅π‘₯)
5653, 54, 55elrabd 3652 . . . . . 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 513 . . . . . . . . . 10 ((((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ (𝐴𝑅𝐡 ∧ 𝐴 β‰  𝐡)) ∧ (π‘₯ ∈ 𝑋 ∧ (Β¬ 𝐡𝑅π‘₯ ∧ Β¬ π‘₯𝑅𝐴))) β†’ (𝑅 ∈ TosetRel ∧ π‘₯ ∈ 𝑋))
583tsrlin 18481 . . . . . . . . . . 11 ((𝑅 ∈ TosetRel ∧ π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) β†’ (π‘₯𝑅𝑦 ∨ 𝑦𝑅π‘₯))
59583expa 1119 . . . . . . . . . 10 (((𝑅 ∈ TosetRel ∧ π‘₯ ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) β†’ (π‘₯𝑅𝑦 ∨ 𝑦𝑅π‘₯))
6057, 59sylan 581 . . . . . . . . 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 989 . . . . . . . . 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 3144 . . . . . . 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 4349 . . . . . . 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 2827 . . . . . . . 8 (π‘š = {𝑦 ∈ 𝑋 ∣ Β¬ π‘₯𝑅𝑦} β†’ (𝐴 ∈ π‘š ↔ 𝐴 ∈ {𝑦 ∈ 𝑋 ∣ Β¬ π‘₯𝑅𝑦}))
67 ineq1 4170 . . . . . . . . 9 (π‘š = {𝑦 ∈ 𝑋 ∣ Β¬ π‘₯𝑅𝑦} β†’ (π‘š ∩ 𝑛) = ({𝑦 ∈ 𝑋 ∣ Β¬ π‘₯𝑅𝑦} ∩ 𝑛))
6867eqeq1d 2739 . . . . . . . 8 (π‘š = {𝑦 ∈ 𝑋 ∣ Β¬ π‘₯𝑅𝑦} β†’ ((π‘š ∩ 𝑛) = βˆ… ↔ ({𝑦 ∈ 𝑋 ∣ Β¬ π‘₯𝑅𝑦} ∩ 𝑛) = βˆ…))
6966, 683anbi13d 1439 . . . . . . 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 2827 . . . . . . . 8 (𝑛 = {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯} β†’ (𝐡 ∈ 𝑛 ↔ 𝐡 ∈ {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯}))
71 ineq2 4171 . . . . . . . . . 10 (𝑛 = {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯} β†’ ({𝑦 ∈ 𝑋 ∣ Β¬ π‘₯𝑅𝑦} ∩ 𝑛) = ({𝑦 ∈ 𝑋 ∣ Β¬ π‘₯𝑅𝑦} ∩ {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯}))
72 inrab 4271 . . . . . . . . . 10 ({𝑦 ∈ 𝑋 ∣ Β¬ π‘₯𝑅𝑦} ∩ {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯}) = {𝑦 ∈ 𝑋 ∣ (Β¬ π‘₯𝑅𝑦 ∧ Β¬ 𝑦𝑅π‘₯)}
7371, 72eqtrdi 2793 . . . . . . . . 9 (𝑛 = {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯} β†’ ({𝑦 ∈ 𝑋 ∣ Β¬ π‘₯𝑅𝑦} ∩ 𝑛) = {𝑦 ∈ 𝑋 ∣ (Β¬ π‘₯𝑅𝑦 ∧ Β¬ 𝑦𝑅π‘₯)})
7473eqeq1d 2739 . . . . . . . 8 (𝑛 = {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯} β†’ (({𝑦 ∈ 𝑋 ∣ Β¬ π‘₯𝑅𝑦} ∩ 𝑛) = βˆ… ↔ {𝑦 ∈ 𝑋 ∣ (Β¬ π‘₯𝑅𝑦 ∧ Β¬ 𝑦𝑅π‘₯)} = βˆ…))
7570, 743anbi23d 1440 . . . . . . 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 3595 . . . . . 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 1383 . . . . 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 3154 . . . 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 3032 . 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 422 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 397   ∨ wo 846   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   β‰  wne 2944  βˆ€wral 3065  βˆƒwrex 3074  {crab 3410   ∩ cin 3914  βˆ…c0 4287   class class class wbr 5110  dom cdm 5638  β€˜cfv 6501  ordTopcordt 17388  PosetRelcps 18460   TosetRel ctsr 18461
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-int 4913  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-om 7808  df-1o 8417  df-en 8891  df-fin 8894  df-fi 9354  df-topgen 17332  df-ordt 17390  df-ps 18462  df-tsr 18463  df-bases 22312
This theorem is referenced by:  ordthaus  22751
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