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Theorem gneispa 43438
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 4806 . . . . . 6 (𝑝 ∈ 𝑋 β†’ {𝑝} βŠ† 𝑋)
2 gneispace.x . . . . . . 7 𝑋 = βˆͺ 𝐽
32tpnei 22976 . . . . . 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 406 . . . 4 ((𝐽 ∈ Top ∧ 𝑝 ∈ 𝑋) β†’ 𝑋 ∈ ((neiβ€˜π½)β€˜{𝑝}))
65ne0d 4330 . . 3 ((𝐽 ∈ Top ∧ 𝑝 ∈ 𝑋) β†’ ((neiβ€˜π½)β€˜{𝑝}) β‰  βˆ…)
7 elnei 22966 . . . . 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 3139 . . 3 ((𝐽 ∈ Top ∧ 𝑝 ∈ 𝑋) β†’ βˆ€π‘› ∈ ((neiβ€˜π½)β€˜{𝑝})𝑝 ∈ 𝑛)
106, 9jca 511 . 2 ((𝐽 ∈ Top ∧ 𝑝 ∈ 𝑋) β†’ (((neiβ€˜π½)β€˜{𝑝}) β‰  βˆ… ∧ βˆ€π‘› ∈ ((neiβ€˜π½)β€˜{𝑝})𝑝 ∈ 𝑛))
1110ralrimiva 3140 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 395   = wceq 1533   ∈ wcel 2098   β‰  wne 2934  βˆ€wral 3055   βŠ† wss 3943  βˆ…c0 4317  {csn 4623  βˆͺ cuni 4902  β€˜cfv 6536  Topctop 22746  neicnei 22952
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 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-top 22747  df-nei 22953
This theorem is referenced by: (None)
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