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Theorem hausnei2 23077
Description: The Hausdorff condition still holds if one considers general neighborhoods instead of open sets. (Contributed by Jeff Hankins, 5-Sep-2009.)
Assertion
Ref Expression
hausnei2 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ (𝐽 ∈ Haus ↔ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (π‘₯ β‰  𝑦 β†’ βˆƒπ‘’ ∈ ((neiβ€˜π½)β€˜{π‘₯})βˆƒπ‘£ ∈ ((neiβ€˜π½)β€˜{𝑦})(𝑒 ∩ 𝑣) = βˆ…)))
Distinct variable groups:   π‘₯,𝑦   𝑣,𝑒,π‘₯,𝑦,𝐽   𝑒,𝑋,𝑣,π‘₯,𝑦

Proof of Theorem hausnei2
Dummy variables π‘š 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ishaus2 23075 . 2 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ (𝐽 ∈ Haus ↔ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (π‘₯ β‰  𝑦 β†’ βˆƒπ‘š ∈ 𝐽 βˆƒπ‘› ∈ 𝐽 (π‘₯ ∈ π‘š ∧ 𝑦 ∈ 𝑛 ∧ (π‘š ∩ 𝑛) = βˆ…))))
2 topontop 22635 . . 3 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝐽 ∈ Top)
3 simp1 1134 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ π‘š ∈ 𝐽 ∧ 𝑛 ∈ 𝐽) β†’ 𝐽 ∈ Top)
4 simp2 1135 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ π‘š ∈ 𝐽 ∧ 𝑛 ∈ 𝐽) β†’ π‘š ∈ 𝐽)
5 simp1 1134 . . . . . . . . . . 11 ((π‘₯ ∈ π‘š ∧ 𝑦 ∈ 𝑛 ∧ (π‘š ∩ 𝑛) = βˆ…) β†’ π‘₯ ∈ π‘š)
6 opnneip 22843 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ π‘š ∈ 𝐽 ∧ π‘₯ ∈ π‘š) β†’ π‘š ∈ ((neiβ€˜π½)β€˜{π‘₯}))
73, 4, 5, 6syl2an3an 1420 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ π‘š ∈ 𝐽 ∧ 𝑛 ∈ 𝐽) ∧ (π‘₯ ∈ π‘š ∧ 𝑦 ∈ 𝑛 ∧ (π‘š ∩ 𝑛) = βˆ…)) β†’ π‘š ∈ ((neiβ€˜π½)β€˜{π‘₯}))
8 simp3 1136 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ π‘š ∈ 𝐽 ∧ 𝑛 ∈ 𝐽) β†’ 𝑛 ∈ 𝐽)
9 simp2 1135 . . . . . . . . . . 11 ((π‘₯ ∈ π‘š ∧ 𝑦 ∈ 𝑛 ∧ (π‘š ∩ 𝑛) = βˆ…) β†’ 𝑦 ∈ 𝑛)
10 opnneip 22843 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ 𝑛 ∈ 𝐽 ∧ 𝑦 ∈ 𝑛) β†’ 𝑛 ∈ ((neiβ€˜π½)β€˜{𝑦}))
113, 8, 9, 10syl2an3an 1420 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ π‘š ∈ 𝐽 ∧ 𝑛 ∈ 𝐽) ∧ (π‘₯ ∈ π‘š ∧ 𝑦 ∈ 𝑛 ∧ (π‘š ∩ 𝑛) = βˆ…)) β†’ 𝑛 ∈ ((neiβ€˜π½)β€˜{𝑦}))
12 simpr3 1194 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ π‘š ∈ 𝐽 ∧ 𝑛 ∈ 𝐽) ∧ (π‘₯ ∈ π‘š ∧ 𝑦 ∈ 𝑛 ∧ (π‘š ∩ 𝑛) = βˆ…)) β†’ (π‘š ∩ 𝑛) = βˆ…)
13 ineq1 4204 . . . . . . . . . . . 12 (𝑒 = π‘š β†’ (𝑒 ∩ 𝑣) = (π‘š ∩ 𝑣))
1413eqeq1d 2732 . . . . . . . . . . 11 (𝑒 = π‘š β†’ ((𝑒 ∩ 𝑣) = βˆ… ↔ (π‘š ∩ 𝑣) = βˆ…))
15 ineq2 4205 . . . . . . . . . . . 12 (𝑣 = 𝑛 β†’ (π‘š ∩ 𝑣) = (π‘š ∩ 𝑛))
1615eqeq1d 2732 . . . . . . . . . . 11 (𝑣 = 𝑛 β†’ ((π‘š ∩ 𝑣) = βˆ… ↔ (π‘š ∩ 𝑛) = βˆ…))
1714, 16rspc2ev 3623 . . . . . . . . . 10 ((π‘š ∈ ((neiβ€˜π½)β€˜{π‘₯}) ∧ 𝑛 ∈ ((neiβ€˜π½)β€˜{𝑦}) ∧ (π‘š ∩ 𝑛) = βˆ…) β†’ βˆƒπ‘’ ∈ ((neiβ€˜π½)β€˜{π‘₯})βˆƒπ‘£ ∈ ((neiβ€˜π½)β€˜{𝑦})(𝑒 ∩ 𝑣) = βˆ…)
187, 11, 12, 17syl3anc 1369 . . . . . . . . 9 (((𝐽 ∈ Top ∧ π‘š ∈ 𝐽 ∧ 𝑛 ∈ 𝐽) ∧ (π‘₯ ∈ π‘š ∧ 𝑦 ∈ 𝑛 ∧ (π‘š ∩ 𝑛) = βˆ…)) β†’ βˆƒπ‘’ ∈ ((neiβ€˜π½)β€˜{π‘₯})βˆƒπ‘£ ∈ ((neiβ€˜π½)β€˜{𝑦})(𝑒 ∩ 𝑣) = βˆ…)
1918ex 411 . . . . . . . 8 ((𝐽 ∈ Top ∧ π‘š ∈ 𝐽 ∧ 𝑛 ∈ 𝐽) β†’ ((π‘₯ ∈ π‘š ∧ 𝑦 ∈ 𝑛 ∧ (π‘š ∩ 𝑛) = βˆ…) β†’ βˆƒπ‘’ ∈ ((neiβ€˜π½)β€˜{π‘₯})βˆƒπ‘£ ∈ ((neiβ€˜π½)β€˜{𝑦})(𝑒 ∩ 𝑣) = βˆ…))
20193expib 1120 . . . . . . 7 (𝐽 ∈ Top β†’ ((π‘š ∈ 𝐽 ∧ 𝑛 ∈ 𝐽) β†’ ((π‘₯ ∈ π‘š ∧ 𝑦 ∈ 𝑛 ∧ (π‘š ∩ 𝑛) = βˆ…) β†’ βˆƒπ‘’ ∈ ((neiβ€˜π½)β€˜{π‘₯})βˆƒπ‘£ ∈ ((neiβ€˜π½)β€˜{𝑦})(𝑒 ∩ 𝑣) = βˆ…)))
2120rexlimdvv 3208 . . . . . 6 (𝐽 ∈ Top β†’ (βˆƒπ‘š ∈ 𝐽 βˆƒπ‘› ∈ 𝐽 (π‘₯ ∈ π‘š ∧ 𝑦 ∈ 𝑛 ∧ (π‘š ∩ 𝑛) = βˆ…) β†’ βˆƒπ‘’ ∈ ((neiβ€˜π½)β€˜{π‘₯})βˆƒπ‘£ ∈ ((neiβ€˜π½)β€˜{𝑦})(𝑒 ∩ 𝑣) = βˆ…))
22 neii2 22832 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝑒 ∈ ((neiβ€˜π½)β€˜{π‘₯})) β†’ βˆƒπ‘š ∈ 𝐽 ({π‘₯} βŠ† π‘š ∧ π‘š βŠ† 𝑒))
2322ex 411 . . . . . . . 8 (𝐽 ∈ Top β†’ (𝑒 ∈ ((neiβ€˜π½)β€˜{π‘₯}) β†’ βˆƒπ‘š ∈ 𝐽 ({π‘₯} βŠ† π‘š ∧ π‘š βŠ† 𝑒)))
24 neii2 22832 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝑣 ∈ ((neiβ€˜π½)β€˜{𝑦})) β†’ βˆƒπ‘› ∈ 𝐽 ({𝑦} βŠ† 𝑛 ∧ 𝑛 βŠ† 𝑣))
2524ex 411 . . . . . . . 8 (𝐽 ∈ Top β†’ (𝑣 ∈ ((neiβ€˜π½)β€˜{𝑦}) β†’ βˆƒπ‘› ∈ 𝐽 ({𝑦} βŠ† 𝑛 ∧ 𝑛 βŠ† 𝑣)))
26 vex 3476 . . . . . . . . . . . . . . 15 π‘₯ ∈ V
2726snss 4788 . . . . . . . . . . . . . 14 (π‘₯ ∈ π‘š ↔ {π‘₯} βŠ† π‘š)
2827anbi1i 622 . . . . . . . . . . . . 13 ((π‘₯ ∈ π‘š ∧ π‘š βŠ† 𝑒) ↔ ({π‘₯} βŠ† π‘š ∧ π‘š βŠ† 𝑒))
29 vex 3476 . . . . . . . . . . . . . . . . . . . . . . 23 𝑦 ∈ V
3029snss 4788 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 ∈ 𝑛 ↔ {𝑦} βŠ† 𝑛)
3130anbi1i 622 . . . . . . . . . . . . . . . . . . . . 21 ((𝑦 ∈ 𝑛 ∧ 𝑛 βŠ† 𝑣) ↔ ({𝑦} βŠ† 𝑛 ∧ 𝑛 βŠ† 𝑣))
32 simp1l 1195 . . . . . . . . . . . . . . . . . . . . . . 23 (((π‘₯ ∈ π‘š ∧ π‘š βŠ† 𝑒) ∧ (𝑦 ∈ 𝑛 ∧ 𝑛 βŠ† 𝑣) ∧ (𝑒 ∩ 𝑣) = βˆ…) β†’ π‘₯ ∈ π‘š)
33 simp2l 1197 . . . . . . . . . . . . . . . . . . . . . . 23 (((π‘₯ ∈ π‘š ∧ π‘š βŠ† 𝑒) ∧ (𝑦 ∈ 𝑛 ∧ 𝑛 βŠ† 𝑣) ∧ (𝑒 ∩ 𝑣) = βˆ…) β†’ 𝑦 ∈ 𝑛)
34 ss2in 4235 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((π‘š βŠ† 𝑒 ∧ 𝑛 βŠ† 𝑣) β†’ (π‘š ∩ 𝑛) βŠ† (𝑒 ∩ 𝑣))
35 ssn0 4399 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((π‘š ∩ 𝑛) βŠ† (𝑒 ∩ 𝑣) ∧ (π‘š ∩ 𝑛) β‰  βˆ…) β†’ (𝑒 ∩ 𝑣) β‰  βˆ…)
3635ex 411 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((π‘š ∩ 𝑛) βŠ† (𝑒 ∩ 𝑣) β†’ ((π‘š ∩ 𝑛) β‰  βˆ… β†’ (𝑒 ∩ 𝑣) β‰  βˆ…))
3736necon4d 2962 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((π‘š ∩ 𝑛) βŠ† (𝑒 ∩ 𝑣) β†’ ((𝑒 ∩ 𝑣) = βˆ… β†’ (π‘š ∩ 𝑛) = βˆ…))
3834, 37syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((π‘š βŠ† 𝑒 ∧ 𝑛 βŠ† 𝑣) β†’ ((𝑒 ∩ 𝑣) = βˆ… β†’ (π‘š ∩ 𝑛) = βˆ…))
3938ad2ant2l 742 . . . . . . . . . . . . . . . . . . . . . . . 24 (((π‘₯ ∈ π‘š ∧ π‘š βŠ† 𝑒) ∧ (𝑦 ∈ 𝑛 ∧ 𝑛 βŠ† 𝑣)) β†’ ((𝑒 ∩ 𝑣) = βˆ… β†’ (π‘š ∩ 𝑛) = βˆ…))
40393impia 1115 . . . . . . . . . . . . . . . . . . . . . . 23 (((π‘₯ ∈ π‘š ∧ π‘š βŠ† 𝑒) ∧ (𝑦 ∈ 𝑛 ∧ 𝑛 βŠ† 𝑣) ∧ (𝑒 ∩ 𝑣) = βˆ…) β†’ (π‘š ∩ 𝑛) = βˆ…)
4132, 33, 403jca 1126 . . . . . . . . . . . . . . . . . . . . . 22 (((π‘₯ ∈ π‘š ∧ π‘š βŠ† 𝑒) ∧ (𝑦 ∈ 𝑛 ∧ 𝑛 βŠ† 𝑣) ∧ (𝑒 ∩ 𝑣) = βˆ…) β†’ (π‘₯ ∈ π‘š ∧ 𝑦 ∈ 𝑛 ∧ (π‘š ∩ 𝑛) = βˆ…))
42413exp 1117 . . . . . . . . . . . . . . . . . . . . 21 ((π‘₯ ∈ π‘š ∧ π‘š βŠ† 𝑒) β†’ ((𝑦 ∈ 𝑛 ∧ 𝑛 βŠ† 𝑣) β†’ ((𝑒 ∩ 𝑣) = βˆ… β†’ (π‘₯ ∈ π‘š ∧ 𝑦 ∈ 𝑛 ∧ (π‘š ∩ 𝑛) = βˆ…))))
4331, 42biimtrrid 242 . . . . . . . . . . . . . . . . . . . 20 ((π‘₯ ∈ π‘š ∧ π‘š βŠ† 𝑒) β†’ (({𝑦} βŠ† 𝑛 ∧ 𝑛 βŠ† 𝑣) β†’ ((𝑒 ∩ 𝑣) = βˆ… β†’ (π‘₯ ∈ π‘š ∧ 𝑦 ∈ 𝑛 ∧ (π‘š ∩ 𝑛) = βˆ…))))
4443com3r 87 . . . . . . . . . . . . . . . . . . 19 ((𝑒 ∩ 𝑣) = βˆ… β†’ ((π‘₯ ∈ π‘š ∧ π‘š βŠ† 𝑒) β†’ (({𝑦} βŠ† 𝑛 ∧ 𝑛 βŠ† 𝑣) β†’ (π‘₯ ∈ π‘š ∧ 𝑦 ∈ 𝑛 ∧ (π‘š ∩ 𝑛) = βˆ…))))
4544imp 405 . . . . . . . . . . . . . . . . . 18 (((𝑒 ∩ 𝑣) = βˆ… ∧ (π‘₯ ∈ π‘š ∧ π‘š βŠ† 𝑒)) β†’ (({𝑦} βŠ† 𝑛 ∧ 𝑛 βŠ† 𝑣) β†’ (π‘₯ ∈ π‘š ∧ 𝑦 ∈ 𝑛 ∧ (π‘š ∩ 𝑛) = βˆ…)))
46453adant1 1128 . . . . . . . . . . . . . . . . 17 ((𝐽 ∈ Top ∧ (𝑒 ∩ 𝑣) = βˆ… ∧ (π‘₯ ∈ π‘š ∧ π‘š βŠ† 𝑒)) β†’ (({𝑦} βŠ† 𝑛 ∧ 𝑛 βŠ† 𝑣) β†’ (π‘₯ ∈ π‘š ∧ 𝑦 ∈ 𝑛 ∧ (π‘š ∩ 𝑛) = βˆ…)))
4746reximdv 3168 . . . . . . . . . . . . . . . 16 ((𝐽 ∈ Top ∧ (𝑒 ∩ 𝑣) = βˆ… ∧ (π‘₯ ∈ π‘š ∧ π‘š βŠ† 𝑒)) β†’ (βˆƒπ‘› ∈ 𝐽 ({𝑦} βŠ† 𝑛 ∧ 𝑛 βŠ† 𝑣) β†’ βˆƒπ‘› ∈ 𝐽 (π‘₯ ∈ π‘š ∧ 𝑦 ∈ 𝑛 ∧ (π‘š ∩ 𝑛) = βˆ…)))
48473exp 1117 . . . . . . . . . . . . . . 15 (𝐽 ∈ Top β†’ ((𝑒 ∩ 𝑣) = βˆ… β†’ ((π‘₯ ∈ π‘š ∧ π‘š βŠ† 𝑒) β†’ (βˆƒπ‘› ∈ 𝐽 ({𝑦} βŠ† 𝑛 ∧ 𝑛 βŠ† 𝑣) β†’ βˆƒπ‘› ∈ 𝐽 (π‘₯ ∈ π‘š ∧ 𝑦 ∈ 𝑛 ∧ (π‘š ∩ 𝑛) = βˆ…)))))
4948com34 91 . . . . . . . . . . . . . 14 (𝐽 ∈ Top β†’ ((𝑒 ∩ 𝑣) = βˆ… β†’ (βˆƒπ‘› ∈ 𝐽 ({𝑦} βŠ† 𝑛 ∧ 𝑛 βŠ† 𝑣) β†’ ((π‘₯ ∈ π‘š ∧ π‘š βŠ† 𝑒) β†’ βˆƒπ‘› ∈ 𝐽 (π‘₯ ∈ π‘š ∧ 𝑦 ∈ 𝑛 ∧ (π‘š ∩ 𝑛) = βˆ…)))))
50493imp 1109 . . . . . . . . . . . . 13 ((𝐽 ∈ Top ∧ (𝑒 ∩ 𝑣) = βˆ… ∧ βˆƒπ‘› ∈ 𝐽 ({𝑦} βŠ† 𝑛 ∧ 𝑛 βŠ† 𝑣)) β†’ ((π‘₯ ∈ π‘š ∧ π‘š βŠ† 𝑒) β†’ βˆƒπ‘› ∈ 𝐽 (π‘₯ ∈ π‘š ∧ 𝑦 ∈ 𝑛 ∧ (π‘š ∩ 𝑛) = βˆ…)))
5128, 50biimtrrid 242 . . . . . . . . . . . 12 ((𝐽 ∈ Top ∧ (𝑒 ∩ 𝑣) = βˆ… ∧ βˆƒπ‘› ∈ 𝐽 ({𝑦} βŠ† 𝑛 ∧ 𝑛 βŠ† 𝑣)) β†’ (({π‘₯} βŠ† π‘š ∧ π‘š βŠ† 𝑒) β†’ βˆƒπ‘› ∈ 𝐽 (π‘₯ ∈ π‘š ∧ 𝑦 ∈ 𝑛 ∧ (π‘š ∩ 𝑛) = βˆ…)))
5251reximdv 3168 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ (𝑒 ∩ 𝑣) = βˆ… ∧ βˆƒπ‘› ∈ 𝐽 ({𝑦} βŠ† 𝑛 ∧ 𝑛 βŠ† 𝑣)) β†’ (βˆƒπ‘š ∈ 𝐽 ({π‘₯} βŠ† π‘š ∧ π‘š βŠ† 𝑒) β†’ βˆƒπ‘š ∈ 𝐽 βˆƒπ‘› ∈ 𝐽 (π‘₯ ∈ π‘š ∧ 𝑦 ∈ 𝑛 ∧ (π‘š ∩ 𝑛) = βˆ…)))
53523exp 1117 . . . . . . . . . 10 (𝐽 ∈ Top β†’ ((𝑒 ∩ 𝑣) = βˆ… β†’ (βˆƒπ‘› ∈ 𝐽 ({𝑦} βŠ† 𝑛 ∧ 𝑛 βŠ† 𝑣) β†’ (βˆƒπ‘š ∈ 𝐽 ({π‘₯} βŠ† π‘š ∧ π‘š βŠ† 𝑒) β†’ βˆƒπ‘š ∈ 𝐽 βˆƒπ‘› ∈ 𝐽 (π‘₯ ∈ π‘š ∧ 𝑦 ∈ 𝑛 ∧ (π‘š ∩ 𝑛) = βˆ…)))))
5453com24 95 . . . . . . . . 9 (𝐽 ∈ Top β†’ (βˆƒπ‘š ∈ 𝐽 ({π‘₯} βŠ† π‘š ∧ π‘š βŠ† 𝑒) β†’ (βˆƒπ‘› ∈ 𝐽 ({𝑦} βŠ† 𝑛 ∧ 𝑛 βŠ† 𝑣) β†’ ((𝑒 ∩ 𝑣) = βˆ… β†’ βˆƒπ‘š ∈ 𝐽 βˆƒπ‘› ∈ 𝐽 (π‘₯ ∈ π‘š ∧ 𝑦 ∈ 𝑛 ∧ (π‘š ∩ 𝑛) = βˆ…)))))
5554impd 409 . . . . . . . 8 (𝐽 ∈ Top β†’ ((βˆƒπ‘š ∈ 𝐽 ({π‘₯} βŠ† π‘š ∧ π‘š βŠ† 𝑒) ∧ βˆƒπ‘› ∈ 𝐽 ({𝑦} βŠ† 𝑛 ∧ 𝑛 βŠ† 𝑣)) β†’ ((𝑒 ∩ 𝑣) = βˆ… β†’ βˆƒπ‘š ∈ 𝐽 βˆƒπ‘› ∈ 𝐽 (π‘₯ ∈ π‘š ∧ 𝑦 ∈ 𝑛 ∧ (π‘š ∩ 𝑛) = βˆ…))))
5623, 25, 55syl2and 606 . . . . . . 7 (𝐽 ∈ Top β†’ ((𝑒 ∈ ((neiβ€˜π½)β€˜{π‘₯}) ∧ 𝑣 ∈ ((neiβ€˜π½)β€˜{𝑦})) β†’ ((𝑒 ∩ 𝑣) = βˆ… β†’ βˆƒπ‘š ∈ 𝐽 βˆƒπ‘› ∈ 𝐽 (π‘₯ ∈ π‘š ∧ 𝑦 ∈ 𝑛 ∧ (π‘š ∩ 𝑛) = βˆ…))))
5756rexlimdvv 3208 . . . . . 6 (𝐽 ∈ Top β†’ (βˆƒπ‘’ ∈ ((neiβ€˜π½)β€˜{π‘₯})βˆƒπ‘£ ∈ ((neiβ€˜π½)β€˜{𝑦})(𝑒 ∩ 𝑣) = βˆ… β†’ βˆƒπ‘š ∈ 𝐽 βˆƒπ‘› ∈ 𝐽 (π‘₯ ∈ π‘š ∧ 𝑦 ∈ 𝑛 ∧ (π‘š ∩ 𝑛) = βˆ…)))
5821, 57impbid 211 . . . . 5 (𝐽 ∈ Top β†’ (βˆƒπ‘š ∈ 𝐽 βˆƒπ‘› ∈ 𝐽 (π‘₯ ∈ π‘š ∧ 𝑦 ∈ 𝑛 ∧ (π‘š ∩ 𝑛) = βˆ…) ↔ βˆƒπ‘’ ∈ ((neiβ€˜π½)β€˜{π‘₯})βˆƒπ‘£ ∈ ((neiβ€˜π½)β€˜{𝑦})(𝑒 ∩ 𝑣) = βˆ…))
5958imbi2d 339 . . . 4 (𝐽 ∈ Top β†’ ((π‘₯ β‰  𝑦 β†’ βˆƒπ‘š ∈ 𝐽 βˆƒπ‘› ∈ 𝐽 (π‘₯ ∈ π‘š ∧ 𝑦 ∈ 𝑛 ∧ (π‘š ∩ 𝑛) = βˆ…)) ↔ (π‘₯ β‰  𝑦 β†’ βˆƒπ‘’ ∈ ((neiβ€˜π½)β€˜{π‘₯})βˆƒπ‘£ ∈ ((neiβ€˜π½)β€˜{𝑦})(𝑒 ∩ 𝑣) = βˆ…)))
60592ralbidv 3216 . . 3 (𝐽 ∈ Top β†’ (βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (π‘₯ β‰  𝑦 β†’ βˆƒπ‘š ∈ 𝐽 βˆƒπ‘› ∈ 𝐽 (π‘₯ ∈ π‘š ∧ 𝑦 ∈ 𝑛 ∧ (π‘š ∩ 𝑛) = βˆ…)) ↔ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (π‘₯ β‰  𝑦 β†’ βˆƒπ‘’ ∈ ((neiβ€˜π½)β€˜{π‘₯})βˆƒπ‘£ ∈ ((neiβ€˜π½)β€˜{𝑦})(𝑒 ∩ 𝑣) = βˆ…)))
612, 60syl 17 . 2 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ (βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (π‘₯ β‰  𝑦 β†’ βˆƒπ‘š ∈ 𝐽 βˆƒπ‘› ∈ 𝐽 (π‘₯ ∈ π‘š ∧ 𝑦 ∈ 𝑛 ∧ (π‘š ∩ 𝑛) = βˆ…)) ↔ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (π‘₯ β‰  𝑦 β†’ βˆƒπ‘’ ∈ ((neiβ€˜π½)β€˜{π‘₯})βˆƒπ‘£ ∈ ((neiβ€˜π½)β€˜{𝑦})(𝑒 ∩ 𝑣) = βˆ…)))
621, 61bitrd 278 1 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ (𝐽 ∈ Haus ↔ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (π‘₯ β‰  𝑦 β†’ βˆƒπ‘’ ∈ ((neiβ€˜π½)β€˜{π‘₯})βˆƒπ‘£ ∈ ((neiβ€˜π½)β€˜{𝑦})(𝑒 ∩ 𝑣) = βˆ…)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1085   = wceq 1539   ∈ wcel 2104   β‰  wne 2938  βˆ€wral 3059  βˆƒwrex 3068   ∩ cin 3946   βŠ† wss 3947  βˆ…c0 4321  {csn 4627  β€˜cfv 6542  Topctop 22615  TopOnctopon 22632  neicnei 22821  Hauscha 23032
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  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-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-top 22616  df-topon 22633  df-nei 22822  df-haus 23039
This theorem is referenced by:  hausflim  23705
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