MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  opnssneib Structured version   Visualization version   GIF version
Theorem opnssneib 22839
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 765 . . . . . 6 (((𝑆 ∈ 𝐽 ∧ 𝑁 βŠ† 𝑋) ∧ 𝑆 βŠ† 𝑁) β†’ 𝑁 βŠ† 𝑋)
2 sseq2 4007 . . . . . . . . . 10 (𝑔 = 𝑆 β†’ (𝑆 βŠ† 𝑔 ↔ 𝑆 βŠ† 𝑆))
3 sseq1 4006 . . . . . . . . . 10 (𝑔 = 𝑆 β†’ (𝑔 βŠ† 𝑁 ↔ 𝑆 βŠ† 𝑁))
42, 3anbi12d 629 . . . . . . . . 9 (𝑔 = 𝑆 β†’ ((𝑆 βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑁) ↔ (𝑆 βŠ† 𝑆 ∧ 𝑆 βŠ† 𝑁)))
5 ssid 4003 . . . . . . . . . 10 𝑆 βŠ† 𝑆
65biantrur 529 . . . . . . . . 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 3611 . . . . . . 7 ((𝑆 ∈ 𝐽 ∧ 𝑆 βŠ† 𝑁) β†’ βˆƒπ‘” ∈ 𝐽 (𝑆 βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑁))
98adantlr 711 . . . . . 6 (((𝑆 ∈ 𝐽 ∧ 𝑁 βŠ† 𝑋) ∧ 𝑆 βŠ† 𝑁) β†’ βˆƒπ‘” ∈ 𝐽 (𝑆 βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑁))
101, 9jca 510 . . . . 5 (((𝑆 ∈ 𝐽 ∧ 𝑁 βŠ† 𝑋) ∧ 𝑆 βŠ† 𝑁) β†’ (𝑁 βŠ† 𝑋 ∧ βˆƒπ‘” ∈ 𝐽 (𝑆 βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑁)))
1110ex 411 . . . 4 ((𝑆 ∈ 𝐽 ∧ 𝑁 βŠ† 𝑋) β†’ (𝑆 βŠ† 𝑁 β†’ (𝑁 βŠ† 𝑋 ∧ βˆƒπ‘” ∈ 𝐽 (𝑆 βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑁))))
12113adant1 1128 . . 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 22629 . . . . 5 ((𝐽 ∈ Top ∧ 𝑆 ∈ 𝐽) β†’ 𝑆 βŠ† 𝑋)
1513isnei 22827 . . . . 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 589 . . . 4 ((𝐽 ∈ Top ∧ 𝑆 ∈ 𝐽) β†’ (𝑁 ∈ ((neiβ€˜π½)β€˜π‘†) ↔ (𝑁 βŠ† 𝑋 ∧ βˆƒπ‘” ∈ 𝐽 (𝑆 βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑁))))
17163adant3 1130 . . 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 22834 . . . 4 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†)) β†’ 𝑆 βŠ† 𝑁)
2019ex 411 . . 3 (𝐽 ∈ Top β†’ (𝑁 ∈ ((neiβ€˜π½)β€˜π‘†) β†’ 𝑆 βŠ† 𝑁))
21203ad2ant1 1131 . 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 394   ∧ w3a 1085   = wceq 1539   ∈ wcel 2104  βˆƒwrex 3068   βŠ† wss 3947  βˆͺ cuni 4907  β€˜cfv 6542  Topctop 22615  neicnei 22821
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
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-nei 22822
This theorem is referenced by:  neissex  22851
  Copyright terms: Public domain W3C validator