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Theorem ntruni 35207
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 4941 . . . 4 (π‘œ ∈ 𝑂 β†’ π‘œ βŠ† βˆͺ 𝑂)
2 sspwuni 5103 . . . . 5 (𝑂 βŠ† 𝒫 𝑋 ↔ βˆͺ 𝑂 βŠ† 𝑋)
3 ntruni.1 . . . . . . 7 𝑋 = βˆͺ 𝐽
43ntrss 22558 . . . . . 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 1121 . . . . 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 594 . . . 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 3145 . 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 5048 . 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 396   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061   βŠ† wss 3948  π’« cpw 4602  βˆͺ cuni 4908  βˆͺ ciun 4997  β€˜cfv 6543  Topctop 22394  intcnt 22520
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-iin 5000  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-top 22395  df-cld 22522  df-ntr 22523  df-cls 22524
This theorem is referenced by: (None)
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