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Theorem gneispb 42495
Description: Given a neighborhood 𝑁 of 𝑃, each subset of the neighborhood space containing this neighborhood is also a neighborhood of 𝑃. Axiom B of Seifert and Threlfall. (Contributed by RP, 5-Apr-2021.)
Hypothesis
Ref Expression
gneispace.x 𝑋 = βˆͺ 𝐽
Assertion
Ref Expression
gneispb ((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋 ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜{𝑃})) β†’ βˆ€π‘  ∈ 𝒫 𝑋(𝑁 βŠ† 𝑠 β†’ 𝑠 ∈ ((neiβ€˜π½)β€˜{𝑃})))
Distinct variable groups:   𝐽,𝑠   𝑁,𝑠   𝑃,𝑠   𝑋,𝑠

Proof of Theorem gneispb
StepHypRef Expression
1 3simpb 1150 . . . . 5 ((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋 ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜{𝑃})) β†’ (𝐽 ∈ Top ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜{𝑃})))
21ad2antrr 725 . . . 4 ((((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋 ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜{𝑃})) ∧ 𝑠 ∈ 𝒫 𝑋) ∧ 𝑁 βŠ† 𝑠) β†’ (𝐽 ∈ Top ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜{𝑃})))
3 simpr 486 . . . 4 ((((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋 ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜{𝑃})) ∧ 𝑠 ∈ 𝒫 𝑋) ∧ 𝑁 βŠ† 𝑠) β†’ 𝑁 βŠ† 𝑠)
4 simplr 768 . . . . 5 ((((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋 ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜{𝑃})) ∧ 𝑠 ∈ 𝒫 𝑋) ∧ 𝑁 βŠ† 𝑠) β†’ 𝑠 ∈ 𝒫 𝑋)
54elpwid 4573 . . . 4 ((((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋 ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜{𝑃})) ∧ 𝑠 ∈ 𝒫 𝑋) ∧ 𝑁 βŠ† 𝑠) β†’ 𝑠 βŠ† 𝑋)
6 gneispace.x . . . . 5 𝑋 = βˆͺ 𝐽
76ssnei2 22490 . . . 4 (((𝐽 ∈ Top ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜{𝑃})) ∧ (𝑁 βŠ† 𝑠 ∧ 𝑠 βŠ† 𝑋)) β†’ 𝑠 ∈ ((neiβ€˜π½)β€˜{𝑃}))
82, 3, 5, 7syl12anc 836 . . 3 ((((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋 ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜{𝑃})) ∧ 𝑠 ∈ 𝒫 𝑋) ∧ 𝑁 βŠ† 𝑠) β†’ 𝑠 ∈ ((neiβ€˜π½)β€˜{𝑃}))
98exp31 421 . 2 ((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋 ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜{𝑃})) β†’ (𝑠 ∈ 𝒫 𝑋 β†’ (𝑁 βŠ† 𝑠 β†’ 𝑠 ∈ ((neiβ€˜π½)β€˜{𝑃}))))
109ralrimiv 3139 1 ((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋 ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜{𝑃})) β†’ βˆ€π‘  ∈ 𝒫 𝑋(𝑁 βŠ† 𝑠 β†’ 𝑠 ∈ ((neiβ€˜π½)β€˜{𝑃})))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3061   βŠ† wss 3914  π’« cpw 4564  {csn 4590  βˆͺ cuni 4869  β€˜cfv 6500  Topctop 22265  neicnei 22471
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-top 22266  df-nei 22472
This theorem is referenced by: (None)
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