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

Theorem ssnei 21718
 Description: A set is included in any of its neighborhoods. Generalization to subsets of elnei 21719. (Contributed by FL, 16-Nov-2006.)
Assertion
Ref Expression
ssnei ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → 𝑆𝑁)

Proof of Theorem ssnei
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 neii2 21716 . 2 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → ∃𝑔𝐽 (𝑆𝑔𝑔𝑁))
2 sstr 3961 . . 3 ((𝑆𝑔𝑔𝑁) → 𝑆𝑁)
32rexlimivw 3274 . 2 (∃𝑔𝐽 (𝑆𝑔𝑔𝑁) → 𝑆𝑁)
41, 3syl 17 1 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → 𝑆𝑁)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∈ wcel 2115  ∃wrex 3134   ⊆ wss 3919  ‘cfv 6343  Topctop 21501  neicnei 21705 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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-top 21502  df-nei 21706 This theorem is referenced by:  elnei  21719  0nnei  21720  opnneissb  21722  opnssneib  21723  tpnei  21729  cvmlift2lem1  32606
 Copyright terms: Public domain W3C validator