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

Theorem ustund 19818
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 5398 . . 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 4913 . . . . 5  |-  ( ( A  u.  B )  X.  ( A  i^i  B ) )  =  ( ( A  X.  ( A  i^i  B ) )  u.  ( B  X.  ( A  i^i  B ) ) )
5 xpindi 4994 . . . . . . 7  |-  ( A  X.  ( A  i^i  B ) )  =  ( ( A  X.  A
)  i^i  ( A  X.  B ) )
6 inss1 3591 . . . . . . . 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 3388 . . . . . . 7  |-  ( ph  ->  ( ( A  X.  A )  i^i  ( A  X.  B ) ) 
C_  V )
95, 8syl5eqss 3421 . . . . . 6  |-  ( ph  ->  ( A  X.  ( A  i^i  B ) ) 
C_  V )
10 xpindi 4994 . . . . . . 7  |-  ( B  X.  ( A  i^i  B ) )  =  ( ( B  X.  A
)  i^i  ( B  X.  B ) )
11 inss2 3592 . . . . . . . 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 3388 . . . . . . 7  |-  ( ph  ->  ( ( B  X.  A )  i^i  ( B  X.  B ) ) 
C_  V )
1410, 13syl5eqss 3421 . . . . . 6  |-  ( ph  ->  ( B  X.  ( A  i^i  B ) ) 
C_  V )
159, 14unssd 3553 . . . . 5  |-  ( ph  ->  ( ( A  X.  ( A  i^i  B ) )  u.  ( B  X.  ( A  i^i  B ) ) )  C_  V )
164, 15syl5eqss 3421 . . . 4  |-  ( ph  ->  ( ( A  u.  B )  X.  ( A  i^i  B ) ) 
C_  V )
17 coss2 5017 . . . 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 4912 . . . . 5  |-  ( ( A  i^i  B )  X.  ( A  u.  B ) )  =  ( ( ( A  i^i  B )  X.  A )  u.  (
( A  i^i  B
)  X.  B ) )
20 xpindir 4995 . . . . . . 7  |-  ( ( A  i^i  B )  X.  A )  =  ( ( A  X.  A )  i^i  ( B  X.  A ) )
21 inss1 3591 . . . . . . . 8  |-  ( ( A  X.  A )  i^i  ( B  X.  A ) )  C_  ( A  X.  A
)
2221, 7syl5ss 3388 . . . . . . 7  |-  ( ph  ->  ( ( A  X.  A )  i^i  ( B  X.  A ) ) 
C_  V )
2320, 22syl5eqss 3421 . . . . . 6  |-  ( ph  ->  ( ( A  i^i  B )  X.  A ) 
C_  V )
24 xpindir 4995 . . . . . . 7  |-  ( ( A  i^i  B )  X.  B )  =  ( ( A  X.  B )  i^i  ( B  X.  B ) )
25 inss2 3592 . . . . . . . 8  |-  ( ( A  X.  B )  i^i  ( B  X.  B ) )  C_  ( B  X.  B
)
2625, 12syl5ss 3388 . . . . . . 7  |-  ( ph  ->  ( ( A  X.  B )  i^i  ( B  X.  B ) ) 
C_  V )
2724, 26syl5eqss 3421 . . . . . 6  |-  ( ph  ->  ( ( A  i^i  B )  X.  B ) 
C_  V )
2823, 27unssd 3553 . . . . 5  |-  ( ph  ->  ( ( ( A  i^i  B )  X.  A )  u.  (
( A  i^i  B
)  X.  B ) )  C_  V )
2919, 28syl5eqss 3421 . . . 4  |-  ( ph  ->  ( ( A  i^i  B )  X.  ( A  u.  B ) ) 
C_  V )
30 coss1 5016 . . . 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 3387 . 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 3412 1  |-  ( ph  ->  ( ( A  u.  B )  X.  ( A  u.  B )
)  C_  ( V  o.  V ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369    =/= wne 2620    u. cun 3347    i^i cin 3348    C_ wss 3349   (/)c0 3658    X. cxp 4859    o. ccom 4865
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4434  ax-nul 4442  ax-pr 4552
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-rab 2745  df-v 2995  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-if 3813  df-sn 3899  df-pr 3901  df-op 3905  df-br 4314  df-opab 4372  df-xp 4867  df-rel 4868  df-co 4870
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator