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Theorem ntruni 35702
Description: A union of interiors is a subset of the interior of the union. The reverse inclusion may not hold. (Contributed by Jeff Hankins, 31-Aug-2009.)
Hypothesis
Ref Expression
ntruni.1 𝑋 = βˆͺ 𝐽
Assertion
Ref Expression
ntruni ((𝐽 ∈ Top ∧ 𝑂 βŠ† 𝒫 𝑋) β†’ βˆͺ π‘œ ∈ 𝑂 ((intβ€˜π½)β€˜π‘œ) βŠ† ((intβ€˜π½)β€˜βˆͺ 𝑂))
Distinct variable groups:   π‘œ,𝐽   π‘œ,𝑂   π‘œ,𝑋
Proof of Theorem ntruni
StepHypRef Expression
1 elssuni 4931 . . . 4 (π‘œ ∈ 𝑂 β†’ π‘œ βŠ† βˆͺ 𝑂)
2 sspwuni 5093 . . . . 5 (𝑂 βŠ† 𝒫 𝑋 ↔ βˆͺ 𝑂 βŠ† 𝑋)
3 ntruni.1 . . . . . . 7 𝑋 = βˆͺ 𝐽
43ntrss 22881 . . . . . 6 ((𝐽 ∈ Top ∧ βˆͺ 𝑂 βŠ† 𝑋 ∧ π‘œ βŠ† βˆͺ 𝑂) β†’ ((intβ€˜π½)β€˜π‘œ) βŠ† ((intβ€˜π½)β€˜βˆͺ 𝑂))
543expia 1118 . . . . 5 ((𝐽 ∈ Top ∧ βˆͺ 𝑂 βŠ† 𝑋) β†’ (π‘œ βŠ† βˆͺ 𝑂 β†’ ((intβ€˜π½)β€˜π‘œ) βŠ† ((intβ€˜π½)β€˜βˆͺ 𝑂)))
62, 5sylan2b 593 . . . 4 ((𝐽 ∈ Top ∧ 𝑂 βŠ† 𝒫 𝑋) β†’ (π‘œ βŠ† βˆͺ 𝑂 β†’ ((intβ€˜π½)β€˜π‘œ) βŠ† ((intβ€˜π½)β€˜βˆͺ 𝑂)))
71, 6syl5 34 . . 3 ((𝐽 ∈ Top ∧ 𝑂 βŠ† 𝒫 𝑋) β†’ (π‘œ ∈ 𝑂 β†’ ((intβ€˜π½)β€˜π‘œ) βŠ† ((intβ€˜π½)β€˜βˆͺ 𝑂)))
87ralrimiv 3137 . 2 ((𝐽 ∈ Top ∧ 𝑂 βŠ† 𝒫 𝑋) β†’ βˆ€π‘œ ∈ 𝑂 ((intβ€˜π½)β€˜π‘œ) βŠ† ((intβ€˜π½)β€˜βˆͺ 𝑂))
9 iunss 5038 . 2 (βˆͺ π‘œ ∈ 𝑂 ((intβ€˜π½)β€˜π‘œ) βŠ† ((intβ€˜π½)β€˜βˆͺ 𝑂) ↔ βˆ€π‘œ ∈ 𝑂 ((intβ€˜π½)β€˜π‘œ) βŠ† ((intβ€˜π½)β€˜βˆͺ 𝑂))
108, 9sylibr 233 1 ((𝐽 ∈ Top ∧ 𝑂 βŠ† 𝒫 𝑋) β†’ βˆͺ π‘œ ∈ 𝑂 ((intβ€˜π½)β€˜π‘œ) βŠ† ((intβ€˜π½)β€˜βˆͺ 𝑂))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098  βˆ€wral 3053   βŠ† wss 3940  π’« cpw 4594  βˆͺ cuni 4899  βˆͺ ciun 4987  β€˜cfv 6533  Topctop 22717  intcnt 22843
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 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718
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 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-int 4941  df-iun 4989  df-iin 4990  df-br 5139  df-opab 5201  df-mpt 5222  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-top 22718  df-cld 22845  df-ntr 22846  df-cls 22847
This theorem is referenced by: (None)
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