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Theorem gneispa 43563
Description: Each point 𝑝 of the neighborhood space has at least one neighborhood; each neighborhood of 𝑝 contains 𝑝. Axiom A of Seifert and Threlfall. (Contributed by RP, 5-Apr-2021.)
Hypothesis
Ref Expression
gneispace.x 𝑋 = βˆͺ 𝐽
Assertion
Ref Expression
gneispa (𝐽 ∈ Top β†’ βˆ€π‘ ∈ 𝑋 (((neiβ€˜π½)β€˜{𝑝}) β‰  βˆ… ∧ βˆ€π‘› ∈ ((neiβ€˜π½)β€˜{𝑝})𝑝 ∈ 𝑛))
Distinct variable groups:   𝑛,𝐽,𝑝   𝑛,𝑋
Allowed substitution hint:   𝑋(𝑝)

Proof of Theorem gneispa
StepHypRef Expression
1 snssi 4814 . . . . . 6 (𝑝 ∈ 𝑋 β†’ {𝑝} βŠ† 𝑋)
2 gneispace.x . . . . . . 7 𝑋 = βˆͺ 𝐽
32tpnei 23043 . . . . . 6 (𝐽 ∈ Top β†’ ({𝑝} βŠ† 𝑋 ↔ 𝑋 ∈ ((neiβ€˜π½)β€˜{𝑝})))
41, 3imbitrid 243 . . . . 5 (𝐽 ∈ Top β†’ (𝑝 ∈ 𝑋 β†’ 𝑋 ∈ ((neiβ€˜π½)β€˜{𝑝})))
54imp 405 . . . 4 ((𝐽 ∈ Top ∧ 𝑝 ∈ 𝑋) β†’ 𝑋 ∈ ((neiβ€˜π½)β€˜{𝑝}))
65ne0d 4337 . . 3 ((𝐽 ∈ Top ∧ 𝑝 ∈ 𝑋) β†’ ((neiβ€˜π½)β€˜{𝑝}) β‰  βˆ…)
7 elnei 23033 . . . . 5 ((𝐽 ∈ Top ∧ 𝑝 ∈ 𝑋 ∧ 𝑛 ∈ ((neiβ€˜π½)β€˜{𝑝})) β†’ 𝑝 ∈ 𝑛)
873expia 1118 . . . 4 ((𝐽 ∈ Top ∧ 𝑝 ∈ 𝑋) β†’ (𝑛 ∈ ((neiβ€˜π½)β€˜{𝑝}) β†’ 𝑝 ∈ 𝑛))
98ralrimiv 3141 . . 3 ((𝐽 ∈ Top ∧ 𝑝 ∈ 𝑋) β†’ βˆ€π‘› ∈ ((neiβ€˜π½)β€˜{𝑝})𝑝 ∈ 𝑛)
106, 9jca 510 . 2 ((𝐽 ∈ Top ∧ 𝑝 ∈ 𝑋) β†’ (((neiβ€˜π½)β€˜{𝑝}) β‰  βˆ… ∧ βˆ€π‘› ∈ ((neiβ€˜π½)β€˜{𝑝})𝑝 ∈ 𝑛))
1110ralrimiva 3142 1 (𝐽 ∈ Top β†’ βˆ€π‘ ∈ 𝑋 (((neiβ€˜π½)β€˜{𝑝}) β‰  βˆ… ∧ βˆ€π‘› ∈ ((neiβ€˜π½)β€˜{𝑝})𝑝 ∈ 𝑛))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098   β‰  wne 2936  βˆ€wral 3057   βŠ† wss 3947  βˆ…c0 4324  {csn 4630  βˆͺ cuni 4910  β€˜cfv 6551  Topctop 22813  neicnei 23019
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 2166  ax-ext 2698  ax-rep 5287  ax-sep 5301  ax-nul 5308  ax-pow 5367  ax-pr 5431
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-ral 3058  df-rex 3067  df-reu 3373  df-rab 3429  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-iun 5000  df-br 5151  df-opab 5213  df-mpt 5234  df-id 5578  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-rn 5691  df-res 5692  df-ima 5693  df-iota 6503  df-fun 6553  df-fn 6554  df-f 6555  df-f1 6556  df-fo 6557  df-f1o 6558  df-fv 6559  df-top 22814  df-nei 23020
This theorem is referenced by: (None)
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