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Theorem ntruni 35801
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 4935 . . . 4 (π‘œ ∈ 𝑂 β†’ π‘œ βŠ† βˆͺ 𝑂)
2 sspwuni 5097 . . . . 5 (𝑂 βŠ† 𝒫 𝑋 ↔ βˆͺ 𝑂 βŠ† 𝑋)
3 ntruni.1 . . . . . . 7 𝑋 = βˆͺ 𝐽
43ntrss 22952 . . . . . 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 1119 . . . . 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 3140 . 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 5042 . 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 1534   ∈ wcel 2099  βˆ€wral 3056   βŠ† wss 3944  π’« cpw 4598  βˆͺ cuni 4903  βˆͺ ciun 4991  β€˜cfv 6542  Topctop 22788  intcnt 22914
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-iin 4994  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-top 22789  df-cld 22916  df-ntr 22917  df-cls 22918
This theorem is referenced by: (None)
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