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Theorem opnssneib 22518
Description: Any superset of an open set is a neighborhood of it. (Contributed by NM, 14-Feb-2007.)
Hypothesis
Ref Expression
neips.1 𝑋 = βˆͺ 𝐽
Assertion
Ref Expression
opnssneib ((𝐽 ∈ Top ∧ 𝑆 ∈ 𝐽 ∧ 𝑁 βŠ† 𝑋) β†’ (𝑆 βŠ† 𝑁 ↔ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†)))
Proof of Theorem opnssneib
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 simplr 767 . . . . . 6 (((𝑆 ∈ 𝐽 ∧ 𝑁 βŠ† 𝑋) ∧ 𝑆 βŠ† 𝑁) β†’ 𝑁 βŠ† 𝑋)
2 sseq2 3988 . . . . . . . . . 10 (𝑔 = 𝑆 β†’ (𝑆 βŠ† 𝑔 ↔ 𝑆 βŠ† 𝑆))
3 sseq1 3987 . . . . . . . . . 10 (𝑔 = 𝑆 β†’ (𝑔 βŠ† 𝑁 ↔ 𝑆 βŠ† 𝑁))
42, 3anbi12d 631 . . . . . . . . 9 (𝑔 = 𝑆 β†’ ((𝑆 βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑁) ↔ (𝑆 βŠ† 𝑆 ∧ 𝑆 βŠ† 𝑁)))
5 ssid 3984 . . . . . . . . . 10 𝑆 βŠ† 𝑆
65biantrur 531 . . . . . . . . 9 (𝑆 βŠ† 𝑁 ↔ (𝑆 βŠ† 𝑆 ∧ 𝑆 βŠ† 𝑁))
74, 6bitr4di 288 . . . . . . . 8 (𝑔 = 𝑆 β†’ ((𝑆 βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑁) ↔ 𝑆 βŠ† 𝑁))
87rspcev 3595 . . . . . . 7 ((𝑆 ∈ 𝐽 ∧ 𝑆 βŠ† 𝑁) β†’ βˆƒπ‘” ∈ 𝐽 (𝑆 βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑁))
98adantlr 713 . . . . . 6 (((𝑆 ∈ 𝐽 ∧ 𝑁 βŠ† 𝑋) ∧ 𝑆 βŠ† 𝑁) β†’ βˆƒπ‘” ∈ 𝐽 (𝑆 βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑁))
101, 9jca 512 . . . . 5 (((𝑆 ∈ 𝐽 ∧ 𝑁 βŠ† 𝑋) ∧ 𝑆 βŠ† 𝑁) β†’ (𝑁 βŠ† 𝑋 ∧ βˆƒπ‘” ∈ 𝐽 (𝑆 βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑁)))
1110ex 413 . . . 4 ((𝑆 ∈ 𝐽 ∧ 𝑁 βŠ† 𝑋) β†’ (𝑆 βŠ† 𝑁 β†’ (𝑁 βŠ† 𝑋 ∧ βˆƒπ‘” ∈ 𝐽 (𝑆 βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑁))))
12113adant1 1130 . . 3 ((𝐽 ∈ Top ∧ 𝑆 ∈ 𝐽 ∧ 𝑁 βŠ† 𝑋) β†’ (𝑆 βŠ† 𝑁 β†’ (𝑁 βŠ† 𝑋 ∧ βˆƒπ‘” ∈ 𝐽 (𝑆 βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑁))))
13 neips.1 . . . . . 6 𝑋 = βˆͺ 𝐽
1413eltopss 22308 . . . . 5 ((𝐽 ∈ Top ∧ 𝑆 ∈ 𝐽) β†’ 𝑆 βŠ† 𝑋)
1513isnei 22506 . . . . 5 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ (𝑁 ∈ ((neiβ€˜π½)β€˜π‘†) ↔ (𝑁 βŠ† 𝑋 ∧ βˆƒπ‘” ∈ 𝐽 (𝑆 βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑁))))
1614, 15syldan 591 . . . 4 ((𝐽 ∈ Top ∧ 𝑆 ∈ 𝐽) β†’ (𝑁 ∈ ((neiβ€˜π½)β€˜π‘†) ↔ (𝑁 βŠ† 𝑋 ∧ βˆƒπ‘” ∈ 𝐽 (𝑆 βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑁))))
17163adant3 1132 . . 3 ((𝐽 ∈ Top ∧ 𝑆 ∈ 𝐽 ∧ 𝑁 βŠ† 𝑋) β†’ (𝑁 ∈ ((neiβ€˜π½)β€˜π‘†) ↔ (𝑁 βŠ† 𝑋 ∧ βˆƒπ‘” ∈ 𝐽 (𝑆 βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑁))))
1812, 17sylibrd 258 . 2 ((𝐽 ∈ Top ∧ 𝑆 ∈ 𝐽 ∧ 𝑁 βŠ† 𝑋) β†’ (𝑆 βŠ† 𝑁 β†’ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†)))
19 ssnei 22513 . . . 4 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†)) β†’ 𝑆 βŠ† 𝑁)
2019ex 413 . . 3 (𝐽 ∈ Top β†’ (𝑁 ∈ ((neiβ€˜π½)β€˜π‘†) β†’ 𝑆 βŠ† 𝑁))
21203ad2ant1 1133 . 2 ((𝐽 ∈ Top ∧ 𝑆 ∈ 𝐽 ∧ 𝑁 βŠ† 𝑋) β†’ (𝑁 ∈ ((neiβ€˜π½)β€˜π‘†) β†’ 𝑆 βŠ† 𝑁))
2218, 21impbid 211 1 ((𝐽 ∈ Top ∧ 𝑆 ∈ 𝐽 ∧ 𝑁 βŠ† 𝑋) β†’ (𝑆 βŠ† 𝑁 ↔ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  βˆƒwrex 3069   βŠ† wss 3928  βˆͺ cuni 4885  β€˜cfv 6516  Topctop 22294  neicnei 22500
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-rep 5262  ax-sep 5276  ax-nul 5283  ax-pow 5340  ax-pr 5404
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  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 3365  df-rab 3419  df-v 3461  df-sbc 3758  df-csb 3874  df-dif 3931  df-un 3933  df-in 3935  df-ss 3945  df-nul 4303  df-if 4507  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4886  df-iun 4976  df-br 5126  df-opab 5188  df-mpt 5209  df-id 5551  df-xp 5659  df-rel 5660  df-cnv 5661  df-co 5662  df-dm 5663  df-rn 5664  df-res 5665  df-ima 5666  df-iota 6468  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523  df-fv 6524  df-top 22295  df-nei 22501
This theorem is referenced by:  neissex  22530
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