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Theorem ntruni 30350
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  = 
U. J
Assertion
Ref Expression
ntruni  |-  ( ( J  e.  Top  /\  O  C_  ~P X )  ->  U_ o  e.  O  ( ( int `  J
) `  o )  C_  ( ( int `  J
) `  U. O ) )
Distinct variable groups:    o, J    o, O    o, X

Proof of Theorem ntruni
StepHypRef Expression
1 elssuni 4281 . . . 4  |-  ( o  e.  O  ->  o  C_ 
U. O )
2 sspwuni 4421 . . . . 5  |-  ( O 
C_  ~P X  <->  U. O  C_  X )
3 ntruni.1 . . . . . . 7  |-  X  = 
U. J
43ntrss 19683 . . . . . 6  |-  ( ( J  e.  Top  /\  U. O  C_  X  /\  o  C_  U. O )  ->  ( ( int `  J ) `  o
)  C_  ( ( int `  J ) `  U. O ) )
543expia 1198 . . . . 5  |-  ( ( J  e.  Top  /\  U. O  C_  X )  ->  ( o  C_  U. O  ->  ( ( int `  J
) `  o )  C_  ( ( int `  J
) `  U. O ) ) )
62, 5sylan2b 475 . . . 4  |-  ( ( J  e.  Top  /\  O  C_  ~P X )  ->  ( o  C_  U. O  ->  ( ( int `  J ) `  o )  C_  (
( int `  J
) `  U. O ) ) )
71, 6syl5 32 . . 3  |-  ( ( J  e.  Top  /\  O  C_  ~P X )  ->  ( o  e.  O  ->  ( ( int `  J ) `  o )  C_  (
( int `  J
) `  U. O ) ) )
87ralrimiv 2869 . 2  |-  ( ( J  e.  Top  /\  O  C_  ~P X )  ->  A. o  e.  O  ( ( int `  J
) `  o )  C_  ( ( int `  J
) `  U. O ) )
9 iunss 4373 . 2  |-  ( U_ o  e.  O  (
( int `  J
) `  o )  C_  ( ( int `  J
) `  U. O )  <->  A. o  e.  O  ( ( int `  J
) `  o )  C_  ( ( int `  J
) `  U. O ) )
108, 9sylibr 212 1  |-  ( ( J  e.  Top  /\  O  C_  ~P X )  ->  U_ o  e.  O  ( ( int `  J
) `  o )  C_  ( ( int `  J
) `  U. O ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1395    e. wcel 1819   A.wral 2807    C_ wss 3471   ~Pcpw 4015   U.cuni 4251   U_ciun 4332   ` cfv 5594   Topctop 19521   intcnt 19645
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-iin 4335  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-top 19526  df-cld 19647  df-ntr 19648  df-cls 19649
This theorem is referenced by: (None)
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