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Theorem opnneiid 22985
Description: Only an open set is a neighborhood of itself. (Contributed by FL, 2-Oct-2006.)
Assertion
Ref Expression
opnneiid (𝐽 ∈ Top β†’ (𝑁 ∈ ((neiβ€˜π½)β€˜π‘) ↔ 𝑁 ∈ 𝐽))

Proof of Theorem opnneiid
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 neii2 22967 . . . 4 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘)) β†’ βˆƒπ‘₯ ∈ 𝐽 (𝑁 βŠ† π‘₯ ∧ π‘₯ βŠ† 𝑁))
2 eqss 3992 . . . . . 6 (𝑁 = π‘₯ ↔ (𝑁 βŠ† π‘₯ ∧ π‘₯ βŠ† 𝑁))
3 eleq1a 2822 . . . . . 6 (π‘₯ ∈ 𝐽 β†’ (𝑁 = π‘₯ β†’ 𝑁 ∈ 𝐽))
42, 3biimtrrid 242 . . . . 5 (π‘₯ ∈ 𝐽 β†’ ((𝑁 βŠ† π‘₯ ∧ π‘₯ βŠ† 𝑁) β†’ 𝑁 ∈ 𝐽))
54rexlimiv 3142 . . . 4 (βˆƒπ‘₯ ∈ 𝐽 (𝑁 βŠ† π‘₯ ∧ π‘₯ βŠ† 𝑁) β†’ 𝑁 ∈ 𝐽)
61, 5syl 17 . . 3 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘)) β†’ 𝑁 ∈ 𝐽)
76ex 412 . 2 (𝐽 ∈ Top β†’ (𝑁 ∈ ((neiβ€˜π½)β€˜π‘) β†’ 𝑁 ∈ 𝐽))
8 ssid 3999 . . 3 𝑁 βŠ† 𝑁
9 opnneiss 22977 . . . 4 ((𝐽 ∈ Top ∧ 𝑁 ∈ 𝐽 ∧ 𝑁 βŠ† 𝑁) β†’ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘))
1093exp 1116 . . 3 (𝐽 ∈ Top β†’ (𝑁 ∈ 𝐽 β†’ (𝑁 βŠ† 𝑁 β†’ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘))))
118, 10mpii 46 . 2 (𝐽 ∈ Top β†’ (𝑁 ∈ 𝐽 β†’ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘)))
127, 11impbid 211 1 (𝐽 ∈ Top β†’ (𝑁 ∈ ((neiβ€˜π½)β€˜π‘) ↔ 𝑁 ∈ 𝐽))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   = wceq 1533   ∈ wcel 2098  βˆƒwrex 3064   βŠ† wss 3943  β€˜cfv 6537  Topctop 22750  neicnei 22956
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-top 22751  df-nei 22957
This theorem is referenced by:  0nei  22987
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