MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ustund Structured version   Unicode version

Theorem ustund 20487
Description: If two intersecting sets  A and  B are both small in  V, their union is small in  ( V ^ 2 ). Proposition 1 of [BourbakiTop1] p. II.12. This proposition actually does not require any axiom of the definition of uniform structures. (Contributed by Thierry Arnoux, 17-Nov-2017.)
Hypotheses
Ref Expression
ustund.1  |-  ( ph  ->  ( A  X.  A
)  C_  V )
ustund.2  |-  ( ph  ->  ( B  X.  B
)  C_  V )
ustund.3  |-  ( ph  ->  ( A  i^i  B
)  =/=  (/) )
Assertion
Ref Expression
ustund  |-  ( ph  ->  ( ( A  u.  B )  X.  ( A  u.  B )
)  C_  ( V  o.  V ) )

Proof of Theorem ustund
StepHypRef Expression
1 ustund.3 . . 3  |-  ( ph  ->  ( A  i^i  B
)  =/=  (/) )
2 xpco 5547 . . 3  |-  ( ( A  i^i  B )  =/=  (/)  ->  ( (
( A  i^i  B
)  X.  ( A  u.  B ) )  o.  ( ( A  u.  B )  X.  ( A  i^i  B
) ) )  =  ( ( A  u.  B )  X.  ( A  u.  B )
) )
31, 2syl 16 . 2  |-  ( ph  ->  ( ( ( A  i^i  B )  X.  ( A  u.  B
) )  o.  (
( A  u.  B
)  X.  ( A  i^i  B ) ) )  =  ( ( A  u.  B )  X.  ( A  u.  B ) ) )
4 xpundir 5053 . . . . 5  |-  ( ( A  u.  B )  X.  ( A  i^i  B ) )  =  ( ( A  X.  ( A  i^i  B ) )  u.  ( B  X.  ( A  i^i  B ) ) )
5 xpindi 5136 . . . . . . 7  |-  ( A  X.  ( A  i^i  B ) )  =  ( ( A  X.  A
)  i^i  ( A  X.  B ) )
6 inss1 3718 . . . . . . . 8  |-  ( ( A  X.  A )  i^i  ( A  X.  B ) )  C_  ( A  X.  A
)
7 ustund.1 . . . . . . . 8  |-  ( ph  ->  ( A  X.  A
)  C_  V )
86, 7syl5ss 3515 . . . . . . 7  |-  ( ph  ->  ( ( A  X.  A )  i^i  ( A  X.  B ) ) 
C_  V )
95, 8syl5eqss 3548 . . . . . 6  |-  ( ph  ->  ( A  X.  ( A  i^i  B ) ) 
C_  V )
10 xpindi 5136 . . . . . . 7  |-  ( B  X.  ( A  i^i  B ) )  =  ( ( B  X.  A
)  i^i  ( B  X.  B ) )
11 inss2 3719 . . . . . . . 8  |-  ( ( B  X.  A )  i^i  ( B  X.  B ) )  C_  ( B  X.  B
)
12 ustund.2 . . . . . . . 8  |-  ( ph  ->  ( B  X.  B
)  C_  V )
1311, 12syl5ss 3515 . . . . . . 7  |-  ( ph  ->  ( ( B  X.  A )  i^i  ( B  X.  B ) ) 
C_  V )
1410, 13syl5eqss 3548 . . . . . 6  |-  ( ph  ->  ( B  X.  ( A  i^i  B ) ) 
C_  V )
159, 14unssd 3680 . . . . 5  |-  ( ph  ->  ( ( A  X.  ( A  i^i  B ) )  u.  ( B  X.  ( A  i^i  B ) ) )  C_  V )
164, 15syl5eqss 3548 . . . 4  |-  ( ph  ->  ( ( A  u.  B )  X.  ( A  i^i  B ) ) 
C_  V )
17 coss2 5159 . . . 4  |-  ( ( ( A  u.  B
)  X.  ( A  i^i  B ) ) 
C_  V  ->  (
( ( A  i^i  B )  X.  ( A  u.  B ) )  o.  ( ( A  u.  B )  X.  ( A  i^i  B
) ) )  C_  ( ( ( A  i^i  B )  X.  ( A  u.  B
) )  o.  V
) )
1816, 17syl 16 . . 3  |-  ( ph  ->  ( ( ( A  i^i  B )  X.  ( A  u.  B
) )  o.  (
( A  u.  B
)  X.  ( A  i^i  B ) ) )  C_  ( (
( A  i^i  B
)  X.  ( A  u.  B ) )  o.  V ) )
19 xpundi 5052 . . . . 5  |-  ( ( A  i^i  B )  X.  ( A  u.  B ) )  =  ( ( ( A  i^i  B )  X.  A )  u.  (
( A  i^i  B
)  X.  B ) )
20 xpindir 5137 . . . . . . 7  |-  ( ( A  i^i  B )  X.  A )  =  ( ( A  X.  A )  i^i  ( B  X.  A ) )
21 inss1 3718 . . . . . . . 8  |-  ( ( A  X.  A )  i^i  ( B  X.  A ) )  C_  ( A  X.  A
)
2221, 7syl5ss 3515 . . . . . . 7  |-  ( ph  ->  ( ( A  X.  A )  i^i  ( B  X.  A ) ) 
C_  V )
2320, 22syl5eqss 3548 . . . . . 6  |-  ( ph  ->  ( ( A  i^i  B )  X.  A ) 
C_  V )
24 xpindir 5137 . . . . . . 7  |-  ( ( A  i^i  B )  X.  B )  =  ( ( A  X.  B )  i^i  ( B  X.  B ) )
25 inss2 3719 . . . . . . . 8  |-  ( ( A  X.  B )  i^i  ( B  X.  B ) )  C_  ( B  X.  B
)
2625, 12syl5ss 3515 . . . . . . 7  |-  ( ph  ->  ( ( A  X.  B )  i^i  ( B  X.  B ) ) 
C_  V )
2724, 26syl5eqss 3548 . . . . . 6  |-  ( ph  ->  ( ( A  i^i  B )  X.  B ) 
C_  V )
2823, 27unssd 3680 . . . . 5  |-  ( ph  ->  ( ( ( A  i^i  B )  X.  A )  u.  (
( A  i^i  B
)  X.  B ) )  C_  V )
2919, 28syl5eqss 3548 . . . 4  |-  ( ph  ->  ( ( A  i^i  B )  X.  ( A  u.  B ) ) 
C_  V )
30 coss1 5158 . . . 4  |-  ( ( ( A  i^i  B
)  X.  ( A  u.  B ) ) 
C_  V  ->  (
( ( A  i^i  B )  X.  ( A  u.  B ) )  o.  V )  C_  ( V  o.  V
) )
3129, 30syl 16 . . 3  |-  ( ph  ->  ( ( ( A  i^i  B )  X.  ( A  u.  B
) )  o.  V
)  C_  ( V  o.  V ) )
3218, 31sstrd 3514 . 2  |-  ( ph  ->  ( ( ( A  i^i  B )  X.  ( A  u.  B
) )  o.  (
( A  u.  B
)  X.  ( A  i^i  B ) ) )  C_  ( V  o.  V ) )
333, 32eqsstr3d 3539 1  |-  ( ph  ->  ( ( A  u.  B )  X.  ( A  u.  B )
)  C_  ( V  o.  V ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    =/= wne 2662    u. cun 3474    i^i cin 3475    C_ wss 3476   (/)c0 3785    X. cxp 4997    o. ccom 5003
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-xp 5005  df-rel 5006  df-co 5008
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator