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Theorem neindisj2 22627
Description: A point 𝑃 belongs to the closure of a set 𝑆 iff every neighborhood of 𝑃 meets 𝑆. (Contributed by FL, 15-Sep-2013.)
Hypothesis
Ref Expression
tpnei.1 𝑋 = βˆͺ 𝐽
Assertion
Ref Expression
neindisj2 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋 ∧ 𝑃 ∈ 𝑋) β†’ (𝑃 ∈ ((clsβ€˜π½)β€˜π‘†) ↔ βˆ€π‘› ∈ ((neiβ€˜π½)β€˜{𝑃})(𝑛 ∩ 𝑆) β‰  βˆ…))
Distinct variable groups:   𝑛,𝐽   𝑃,𝑛   𝑆,𝑛   𝑛,𝑋
Proof of Theorem neindisj2
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 tpnei.1 . . 3 𝑋 = βˆͺ 𝐽
21elcls 22577 . 2 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋 ∧ 𝑃 ∈ 𝑋) β†’ (𝑃 ∈ ((clsβ€˜π½)β€˜π‘†) ↔ βˆ€π‘₯ ∈ 𝐽 (𝑃 ∈ π‘₯ β†’ (π‘₯ ∩ 𝑆) β‰  βˆ…)))
31isneip 22609 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋) β†’ (𝑛 ∈ ((neiβ€˜π½)β€˜{𝑃}) ↔ (𝑛 βŠ† 𝑋 ∧ βˆƒπ‘₯ ∈ 𝐽 (𝑃 ∈ π‘₯ ∧ π‘₯ βŠ† 𝑛))))
4 r19.29r 3117 . . . . . . . . . . 11 ((βˆƒπ‘₯ ∈ 𝐽 (𝑃 ∈ π‘₯ ∧ π‘₯ βŠ† 𝑛) ∧ βˆ€π‘₯ ∈ 𝐽 (𝑃 ∈ π‘₯ β†’ (π‘₯ ∩ 𝑆) β‰  βˆ…)) β†’ βˆƒπ‘₯ ∈ 𝐽 ((𝑃 ∈ π‘₯ ∧ π‘₯ βŠ† 𝑛) ∧ (𝑃 ∈ π‘₯ β†’ (π‘₯ ∩ 𝑆) β‰  βˆ…)))
5 pm3.35 802 . . . . . . . . . . . . . . . 16 ((𝑃 ∈ π‘₯ ∧ (𝑃 ∈ π‘₯ β†’ (π‘₯ ∩ 𝑆) β‰  βˆ…)) β†’ (π‘₯ ∩ 𝑆) β‰  βˆ…)
6 ssrin 4234 . . . . . . . . . . . . . . . . . 18 (π‘₯ βŠ† 𝑛 β†’ (π‘₯ ∩ 𝑆) βŠ† (𝑛 ∩ 𝑆))
7 sseq2 4009 . . . . . . . . . . . . . . . . . . 19 ((𝑛 ∩ 𝑆) = βˆ… β†’ ((π‘₯ ∩ 𝑆) βŠ† (𝑛 ∩ 𝑆) ↔ (π‘₯ ∩ 𝑆) βŠ† βˆ…))
8 ss0 4399 . . . . . . . . . . . . . . . . . . 19 ((π‘₯ ∩ 𝑆) βŠ† βˆ… β†’ (π‘₯ ∩ 𝑆) = βˆ…)
97, 8syl6bi 253 . . . . . . . . . . . . . . . . . 18 ((𝑛 ∩ 𝑆) = βˆ… β†’ ((π‘₯ ∩ 𝑆) βŠ† (𝑛 ∩ 𝑆) β†’ (π‘₯ ∩ 𝑆) = βˆ…))
106, 9syl5com 31 . . . . . . . . . . . . . . . . 17 (π‘₯ βŠ† 𝑛 β†’ ((𝑛 ∩ 𝑆) = βˆ… β†’ (π‘₯ ∩ 𝑆) = βˆ…))
1110necon3d 2962 . . . . . . . . . . . . . . . 16 (π‘₯ βŠ† 𝑛 β†’ ((π‘₯ ∩ 𝑆) β‰  βˆ… β†’ (𝑛 ∩ 𝑆) β‰  βˆ…))
125, 11syl5com 31 . . . . . . . . . . . . . . 15 ((𝑃 ∈ π‘₯ ∧ (𝑃 ∈ π‘₯ β†’ (π‘₯ ∩ 𝑆) β‰  βˆ…)) β†’ (π‘₯ βŠ† 𝑛 β†’ (𝑛 ∩ 𝑆) β‰  βˆ…))
1312ex 414 . . . . . . . . . . . . . 14 (𝑃 ∈ π‘₯ β†’ ((𝑃 ∈ π‘₯ β†’ (π‘₯ ∩ 𝑆) β‰  βˆ…) β†’ (π‘₯ βŠ† 𝑛 β†’ (𝑛 ∩ 𝑆) β‰  βˆ…)))
1413com23 86 . . . . . . . . . . . . 13 (𝑃 ∈ π‘₯ β†’ (π‘₯ βŠ† 𝑛 β†’ ((𝑃 ∈ π‘₯ β†’ (π‘₯ ∩ 𝑆) β‰  βˆ…) β†’ (𝑛 ∩ 𝑆) β‰  βˆ…)))
1514imp31 419 . . . . . . . . . . . 12 (((𝑃 ∈ π‘₯ ∧ π‘₯ βŠ† 𝑛) ∧ (𝑃 ∈ π‘₯ β†’ (π‘₯ ∩ 𝑆) β‰  βˆ…)) β†’ (𝑛 ∩ 𝑆) β‰  βˆ…)
1615rexlimivw 3152 . . . . . . . . . . 11 (βˆƒπ‘₯ ∈ 𝐽 ((𝑃 ∈ π‘₯ ∧ π‘₯ βŠ† 𝑛) ∧ (𝑃 ∈ π‘₯ β†’ (π‘₯ ∩ 𝑆) β‰  βˆ…)) β†’ (𝑛 ∩ 𝑆) β‰  βˆ…)
174, 16syl 17 . . . . . . . . . 10 ((βˆƒπ‘₯ ∈ 𝐽 (𝑃 ∈ π‘₯ ∧ π‘₯ βŠ† 𝑛) ∧ βˆ€π‘₯ ∈ 𝐽 (𝑃 ∈ π‘₯ β†’ (π‘₯ ∩ 𝑆) β‰  βˆ…)) β†’ (𝑛 ∩ 𝑆) β‰  βˆ…)
1817ex 414 . . . . . . . . 9 (βˆƒπ‘₯ ∈ 𝐽 (𝑃 ∈ π‘₯ ∧ π‘₯ βŠ† 𝑛) β†’ (βˆ€π‘₯ ∈ 𝐽 (𝑃 ∈ π‘₯ β†’ (π‘₯ ∩ 𝑆) β‰  βˆ…) β†’ (𝑛 ∩ 𝑆) β‰  βˆ…))
1918adantl 483 . . . . . . . 8 ((𝑛 βŠ† 𝑋 ∧ βˆƒπ‘₯ ∈ 𝐽 (𝑃 ∈ π‘₯ ∧ π‘₯ βŠ† 𝑛)) β†’ (βˆ€π‘₯ ∈ 𝐽 (𝑃 ∈ π‘₯ β†’ (π‘₯ ∩ 𝑆) β‰  βˆ…) β†’ (𝑛 ∩ 𝑆) β‰  βˆ…))
203, 19syl6bi 253 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋) β†’ (𝑛 ∈ ((neiβ€˜π½)β€˜{𝑃}) β†’ (βˆ€π‘₯ ∈ 𝐽 (𝑃 ∈ π‘₯ β†’ (π‘₯ ∩ 𝑆) β‰  βˆ…) β†’ (𝑛 ∩ 𝑆) β‰  βˆ…)))
21203adant2 1132 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋 ∧ 𝑃 ∈ 𝑋) β†’ (𝑛 ∈ ((neiβ€˜π½)β€˜{𝑃}) β†’ (βˆ€π‘₯ ∈ 𝐽 (𝑃 ∈ π‘₯ β†’ (π‘₯ ∩ 𝑆) β‰  βˆ…) β†’ (𝑛 ∩ 𝑆) β‰  βˆ…)))
2221com23 86 . . . . 5 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋 ∧ 𝑃 ∈ 𝑋) β†’ (βˆ€π‘₯ ∈ 𝐽 (𝑃 ∈ π‘₯ β†’ (π‘₯ ∩ 𝑆) β‰  βˆ…) β†’ (𝑛 ∈ ((neiβ€˜π½)β€˜{𝑃}) β†’ (𝑛 ∩ 𝑆) β‰  βˆ…)))
2322imp 408 . . . 4 (((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋 ∧ 𝑃 ∈ 𝑋) ∧ βˆ€π‘₯ ∈ 𝐽 (𝑃 ∈ π‘₯ β†’ (π‘₯ ∩ 𝑆) β‰  βˆ…)) β†’ (𝑛 ∈ ((neiβ€˜π½)β€˜{𝑃}) β†’ (𝑛 ∩ 𝑆) β‰  βˆ…))
2423ralrimiv 3146 . . 3 (((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋 ∧ 𝑃 ∈ 𝑋) ∧ βˆ€π‘₯ ∈ 𝐽 (𝑃 ∈ π‘₯ β†’ (π‘₯ ∩ 𝑆) β‰  βˆ…)) β†’ βˆ€π‘› ∈ ((neiβ€˜π½)β€˜{𝑃})(𝑛 ∩ 𝑆) β‰  βˆ…)
25 opnneip 22623 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ π‘₯ ∈ 𝐽 ∧ 𝑃 ∈ π‘₯) β†’ π‘₯ ∈ ((neiβ€˜π½)β€˜{𝑃}))
26 ineq1 4206 . . . . . . . . . . . . . . 15 (𝑛 = π‘₯ β†’ (𝑛 ∩ 𝑆) = (π‘₯ ∩ 𝑆))
2726neeq1d 3001 . . . . . . . . . . . . . 14 (𝑛 = π‘₯ β†’ ((𝑛 ∩ 𝑆) β‰  βˆ… ↔ (π‘₯ ∩ 𝑆) β‰  βˆ…))
2827rspccva 3612 . . . . . . . . . . . . 13 ((βˆ€π‘› ∈ ((neiβ€˜π½)β€˜{𝑃})(𝑛 ∩ 𝑆) β‰  βˆ… ∧ π‘₯ ∈ ((neiβ€˜π½)β€˜{𝑃})) β†’ (π‘₯ ∩ 𝑆) β‰  βˆ…)
29 idd 24 . . . . . . . . . . . . . . 15 ((𝑃 ∈ 𝑋 ∧ (𝐽 ∈ Top ∧ π‘₯ ∈ 𝐽 ∧ 𝑃 ∈ π‘₯) ∧ 𝑆 βŠ† 𝑋) β†’ ((π‘₯ ∩ 𝑆) β‰  βˆ… β†’ (π‘₯ ∩ 𝑆) β‰  βˆ…))
30293exp 1120 . . . . . . . . . . . . . 14 (𝑃 ∈ 𝑋 β†’ ((𝐽 ∈ Top ∧ π‘₯ ∈ 𝐽 ∧ 𝑃 ∈ π‘₯) β†’ (𝑆 βŠ† 𝑋 β†’ ((π‘₯ ∩ 𝑆) β‰  βˆ… β†’ (π‘₯ ∩ 𝑆) β‰  βˆ…))))
3130com14 96 . . . . . . . . . . . . 13 ((π‘₯ ∩ 𝑆) β‰  βˆ… β†’ ((𝐽 ∈ Top ∧ π‘₯ ∈ 𝐽 ∧ 𝑃 ∈ π‘₯) β†’ (𝑆 βŠ† 𝑋 β†’ (𝑃 ∈ 𝑋 β†’ (π‘₯ ∩ 𝑆) β‰  βˆ…))))
3228, 31syl 17 . . . . . . . . . . . 12 ((βˆ€π‘› ∈ ((neiβ€˜π½)β€˜{𝑃})(𝑛 ∩ 𝑆) β‰  βˆ… ∧ π‘₯ ∈ ((neiβ€˜π½)β€˜{𝑃})) β†’ ((𝐽 ∈ Top ∧ π‘₯ ∈ 𝐽 ∧ 𝑃 ∈ π‘₯) β†’ (𝑆 βŠ† 𝑋 β†’ (𝑃 ∈ 𝑋 β†’ (π‘₯ ∩ 𝑆) β‰  βˆ…))))
3332ex 414 . . . . . . . . . . 11 (βˆ€π‘› ∈ ((neiβ€˜π½)β€˜{𝑃})(𝑛 ∩ 𝑆) β‰  βˆ… β†’ (π‘₯ ∈ ((neiβ€˜π½)β€˜{𝑃}) β†’ ((𝐽 ∈ Top ∧ π‘₯ ∈ 𝐽 ∧ 𝑃 ∈ π‘₯) β†’ (𝑆 βŠ† 𝑋 β†’ (𝑃 ∈ 𝑋 β†’ (π‘₯ ∩ 𝑆) β‰  βˆ…)))))
3433com3l 89 . . . . . . . . . 10 (π‘₯ ∈ ((neiβ€˜π½)β€˜{𝑃}) β†’ ((𝐽 ∈ Top ∧ π‘₯ ∈ 𝐽 ∧ 𝑃 ∈ π‘₯) β†’ (βˆ€π‘› ∈ ((neiβ€˜π½)β€˜{𝑃})(𝑛 ∩ 𝑆) β‰  βˆ… β†’ (𝑆 βŠ† 𝑋 β†’ (𝑃 ∈ 𝑋 β†’ (π‘₯ ∩ 𝑆) β‰  βˆ…)))))
3525, 34mpcom 38 . . . . . . . . 9 ((𝐽 ∈ Top ∧ π‘₯ ∈ 𝐽 ∧ 𝑃 ∈ π‘₯) β†’ (βˆ€π‘› ∈ ((neiβ€˜π½)β€˜{𝑃})(𝑛 ∩ 𝑆) β‰  βˆ… β†’ (𝑆 βŠ† 𝑋 β†’ (𝑃 ∈ 𝑋 β†’ (π‘₯ ∩ 𝑆) β‰  βˆ…))))
36353expia 1122 . . . . . . . 8 ((𝐽 ∈ Top ∧ π‘₯ ∈ 𝐽) β†’ (𝑃 ∈ π‘₯ β†’ (βˆ€π‘› ∈ ((neiβ€˜π½)β€˜{𝑃})(𝑛 ∩ 𝑆) β‰  βˆ… β†’ (𝑆 βŠ† 𝑋 β†’ (𝑃 ∈ 𝑋 β†’ (π‘₯ ∩ 𝑆) β‰  βˆ…)))))
3736com25 99 . . . . . . 7 ((𝐽 ∈ Top ∧ π‘₯ ∈ 𝐽) β†’ (𝑃 ∈ 𝑋 β†’ (βˆ€π‘› ∈ ((neiβ€˜π½)β€˜{𝑃})(𝑛 ∩ 𝑆) β‰  βˆ… β†’ (𝑆 βŠ† 𝑋 β†’ (𝑃 ∈ π‘₯ β†’ (π‘₯ ∩ 𝑆) β‰  βˆ…)))))
3837ex 414 . . . . . 6 (𝐽 ∈ Top β†’ (π‘₯ ∈ 𝐽 β†’ (𝑃 ∈ 𝑋 β†’ (βˆ€π‘› ∈ ((neiβ€˜π½)β€˜{𝑃})(𝑛 ∩ 𝑆) β‰  βˆ… β†’ (𝑆 βŠ† 𝑋 β†’ (𝑃 ∈ π‘₯ β†’ (π‘₯ ∩ 𝑆) β‰  βˆ…))))))
3938com25 99 . . . . 5 (𝐽 ∈ Top β†’ (𝑆 βŠ† 𝑋 β†’ (𝑃 ∈ 𝑋 β†’ (βˆ€π‘› ∈ ((neiβ€˜π½)β€˜{𝑃})(𝑛 ∩ 𝑆) β‰  βˆ… β†’ (π‘₯ ∈ 𝐽 β†’ (𝑃 ∈ π‘₯ β†’ (π‘₯ ∩ 𝑆) β‰  βˆ…))))))
40393imp1 1348 . . . 4 (((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋 ∧ 𝑃 ∈ 𝑋) ∧ βˆ€π‘› ∈ ((neiβ€˜π½)β€˜{𝑃})(𝑛 ∩ 𝑆) β‰  βˆ…) β†’ (π‘₯ ∈ 𝐽 β†’ (𝑃 ∈ π‘₯ β†’ (π‘₯ ∩ 𝑆) β‰  βˆ…)))
4140ralrimiv 3146 . . 3 (((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋 ∧ 𝑃 ∈ 𝑋) ∧ βˆ€π‘› ∈ ((neiβ€˜π½)β€˜{𝑃})(𝑛 ∩ 𝑆) β‰  βˆ…) β†’ βˆ€π‘₯ ∈ 𝐽 (𝑃 ∈ π‘₯ β†’ (π‘₯ ∩ 𝑆) β‰  βˆ…))
4224, 41impbida 800 . 2 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋 ∧ 𝑃 ∈ 𝑋) β†’ (βˆ€π‘₯ ∈ 𝐽 (𝑃 ∈ π‘₯ β†’ (π‘₯ ∩ 𝑆) β‰  βˆ…) ↔ βˆ€π‘› ∈ ((neiβ€˜π½)β€˜{𝑃})(𝑛 ∩ 𝑆) β‰  βˆ…))
432, 42bitrd 279 1 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋 ∧ 𝑃 ∈ 𝑋) β†’ (𝑃 ∈ ((clsβ€˜π½)β€˜π‘†) ↔ βˆ€π‘› ∈ ((neiβ€˜π½)β€˜{𝑃})(𝑛 ∩ 𝑆) β‰  βˆ…))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   β‰  wne 2941  βˆ€wral 3062  βˆƒwrex 3071   ∩ cin 3948   βŠ† wss 3949  βˆ…c0 4323  {csn 4629  βˆͺ cuni 4909  β€˜cfv 6544  Topctop 22395  clsccl 22522  neicnei 22601
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 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-iin 5001  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-top 22396  df-cld 22523  df-ntr 22524  df-cls 22525  df-nei 22602
This theorem is referenced by:  islp2  22649  trnei  23396  flimclsi  23482
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