MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  hausnei2 Structured version   Visualization version   GIF version

Theorem hausnei2 23078
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 23076 . 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 22636 . . 3 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝐽 ∈ Top)
3 simp1 1135 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ π‘š ∈ 𝐽 ∧ 𝑛 ∈ 𝐽) β†’ 𝐽 ∈ Top)
4 simp2 1136 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ π‘š ∈ 𝐽 ∧ 𝑛 ∈ 𝐽) β†’ π‘š ∈ 𝐽)
5 simp1 1135 . . . . . . . . . . 11 ((π‘₯ ∈ π‘š ∧ 𝑦 ∈ 𝑛 ∧ (π‘š ∩ 𝑛) = βˆ…) β†’ π‘₯ ∈ π‘š)
6 opnneip 22844 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ π‘š ∈ 𝐽 ∧ π‘₯ ∈ π‘š) β†’ π‘š ∈ ((neiβ€˜π½)β€˜{π‘₯}))
73, 4, 5, 6syl2an3an 1421 . . . . . . . . . 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 1137 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ π‘š ∈ 𝐽 ∧ 𝑛 ∈ 𝐽) β†’ 𝑛 ∈ 𝐽)
9 simp2 1136 . . . . . . . . . . 11 ((π‘₯ ∈ π‘š ∧ 𝑦 ∈ 𝑛 ∧ (π‘š ∩ 𝑛) = βˆ…) β†’ 𝑦 ∈ 𝑛)
10 opnneip 22844 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ 𝑛 ∈ 𝐽 ∧ 𝑦 ∈ 𝑛) β†’ 𝑛 ∈ ((neiβ€˜π½)β€˜{𝑦}))
113, 8, 9, 10syl2an3an 1421 . . . . . . . . . 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 1195 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ π‘š ∈ 𝐽 ∧ 𝑛 ∈ 𝐽) ∧ (π‘₯ ∈ π‘š ∧ 𝑦 ∈ 𝑛 ∧ (π‘š ∩ 𝑛) = βˆ…)) β†’ (π‘š ∩ 𝑛) = βˆ…)
13 ineq1 4205 . . . . . . . . . . . 12 (𝑒 = π‘š β†’ (𝑒 ∩ 𝑣) = (π‘š ∩ 𝑣))
1413eqeq1d 2733 . . . . . . . . . . 11 (𝑒 = π‘š β†’ ((𝑒 ∩ 𝑣) = βˆ… ↔ (π‘š ∩ 𝑣) = βˆ…))
15 ineq2 4206 . . . . . . . . . . . 12 (𝑣 = 𝑛 β†’ (π‘š ∩ 𝑣) = (π‘š ∩ 𝑛))
1615eqeq1d 2733 . . . . . . . . . . 11 (𝑣 = 𝑛 β†’ ((π‘š ∩ 𝑣) = βˆ… ↔ (π‘š ∩ 𝑛) = βˆ…))
1714, 16rspc2ev 3624 . . . . . . . . . 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 1370 . . . . . . . . 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 412 . . . . . . . 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 1121 . . . . . . 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 3209 . . . . . 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 22833 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝑒 ∈ ((neiβ€˜π½)β€˜{π‘₯})) β†’ βˆƒπ‘š ∈ 𝐽 ({π‘₯} βŠ† π‘š ∧ π‘š βŠ† 𝑒))
2322ex 412 . . . . . . . 8 (𝐽 ∈ Top β†’ (𝑒 ∈ ((neiβ€˜π½)β€˜{π‘₯}) β†’ βˆƒπ‘š ∈ 𝐽 ({π‘₯} βŠ† π‘š ∧ π‘š βŠ† 𝑒)))
24 neii2 22833 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝑣 ∈ ((neiβ€˜π½)β€˜{𝑦})) β†’ βˆƒπ‘› ∈ 𝐽 ({𝑦} βŠ† 𝑛 ∧ 𝑛 βŠ† 𝑣))
2524ex 412 . . . . . . . 8 (𝐽 ∈ Top β†’ (𝑣 ∈ ((neiβ€˜π½)β€˜{𝑦}) β†’ βˆƒπ‘› ∈ 𝐽 ({𝑦} βŠ† 𝑛 ∧ 𝑛 βŠ† 𝑣)))
26 vex 3477 . . . . . . . . . . . . . . 15 π‘₯ ∈ V
2726snss 4789 . . . . . . . . . . . . . 14 (π‘₯ ∈ π‘š ↔ {π‘₯} βŠ† π‘š)
2827anbi1i 623 . . . . . . . . . . . . 13 ((π‘₯ ∈ π‘š ∧ π‘š βŠ† 𝑒) ↔ ({π‘₯} βŠ† π‘š ∧ π‘š βŠ† 𝑒))
29 vex 3477 . . . . . . . . . . . . . . . . . . . . . . 23 𝑦 ∈ V
3029snss 4789 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 ∈ 𝑛 ↔ {𝑦} βŠ† 𝑛)
3130anbi1i 623 . . . . . . . . . . . . . . . . . . . . 21 ((𝑦 ∈ 𝑛 ∧ 𝑛 βŠ† 𝑣) ↔ ({𝑦} βŠ† 𝑛 ∧ 𝑛 βŠ† 𝑣))
32 simp1l 1196 . . . . . . . . . . . . . . . . . . . . . . 23 (((π‘₯ ∈ π‘š ∧ π‘š βŠ† 𝑒) ∧ (𝑦 ∈ 𝑛 ∧ 𝑛 βŠ† 𝑣) ∧ (𝑒 ∩ 𝑣) = βˆ…) β†’ π‘₯ ∈ π‘š)
33 simp2l 1198 . . . . . . . . . . . . . . . . . . . . . . 23 (((π‘₯ ∈ π‘š ∧ π‘š βŠ† 𝑒) ∧ (𝑦 ∈ 𝑛 ∧ 𝑛 βŠ† 𝑣) ∧ (𝑒 ∩ 𝑣) = βˆ…) β†’ 𝑦 ∈ 𝑛)
34 ss2in 4236 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((π‘š βŠ† 𝑒 ∧ 𝑛 βŠ† 𝑣) β†’ (π‘š ∩ 𝑛) βŠ† (𝑒 ∩ 𝑣))
35 ssn0 4400 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((π‘š ∩ 𝑛) βŠ† (𝑒 ∩ 𝑣) ∧ (π‘š ∩ 𝑛) β‰  βˆ…) β†’ (𝑒 ∩ 𝑣) β‰  βˆ…)
3635ex 412 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((π‘š ∩ 𝑛) βŠ† (𝑒 ∩ 𝑣) β†’ ((π‘š ∩ 𝑛) β‰  βˆ… β†’ (𝑒 ∩ 𝑣) β‰  βˆ…))
3736necon4d 2963 . . . . . . . . . . . . . . . . . . . . . . . . . 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 743 . . . . . . . . . . . . . . . . . . . . . . . 24 (((π‘₯ ∈ π‘š ∧ π‘š βŠ† 𝑒) ∧ (𝑦 ∈ 𝑛 ∧ 𝑛 βŠ† 𝑣)) β†’ ((𝑒 ∩ 𝑣) = βˆ… β†’ (π‘š ∩ 𝑛) = βˆ…))
40393impia 1116 . . . . . . . . . . . . . . . . . . . . . . 23 (((π‘₯ ∈ π‘š ∧ π‘š βŠ† 𝑒) ∧ (𝑦 ∈ 𝑛 ∧ 𝑛 βŠ† 𝑣) ∧ (𝑒 ∩ 𝑣) = βˆ…) β†’ (π‘š ∩ 𝑛) = βˆ…)
4132, 33, 403jca 1127 . . . . . . . . . . . . . . . . . . . . . 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 1118 . . . . . . . . . . . . . . . . . . . . 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 406 . . . . . . . . . . . . . . . . . 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 1129 . . . . . . . . . . . . . . . . 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 3169 . . . . . . . . . . . . . . . 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 1118 . . . . . . . . . . . . . . 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 1110 . . . . . . . . . . . . 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 3169 . . . . . . . . . . 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 1118 . . . . . . . . . 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 410 . . . . . . . 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 607 . . . . . . 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 3209 . . . . . 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 340 . . . 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 3217 . . 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 279 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 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105   β‰  wne 2939  βˆ€wral 3060  βˆƒwrex 3069   ∩ cin 3947   βŠ† wss 3948  βˆ…c0 4322  {csn 4628  β€˜cfv 6543  Topctop 22616  TopOnctopon 22633  neicnei 22822  Hauscha 23033
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-top 22617  df-topon 22634  df-nei 22823  df-haus 23040
This theorem is referenced by:  hausflim  23706
  Copyright terms: Public domain W3C validator