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Theorem ustund 21018
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 5366 . . 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 17 . 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 4879 . . . . 5  |-  ( ( A  u.  B )  X.  ( A  i^i  B ) )  =  ( ( A  X.  ( A  i^i  B ) )  u.  ( B  X.  ( A  i^i  B ) ) )
5 xpindi 4959 . . . . . . 7  |-  ( A  X.  ( A  i^i  B ) )  =  ( ( A  X.  A
)  i^i  ( A  X.  B ) )
6 inss1 3661 . . . . . . . 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 3455 . . . . . . 7  |-  ( ph  ->  ( ( A  X.  A )  i^i  ( A  X.  B ) ) 
C_  V )
95, 8syl5eqss 3488 . . . . . 6  |-  ( ph  ->  ( A  X.  ( A  i^i  B ) ) 
C_  V )
10 xpindi 4959 . . . . . . 7  |-  ( B  X.  ( A  i^i  B ) )  =  ( ( B  X.  A
)  i^i  ( B  X.  B ) )
11 inss2 3662 . . . . . . . 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 3455 . . . . . . 7  |-  ( ph  ->  ( ( B  X.  A )  i^i  ( B  X.  B ) ) 
C_  V )
1410, 13syl5eqss 3488 . . . . . 6  |-  ( ph  ->  ( B  X.  ( A  i^i  B ) ) 
C_  V )
159, 14unssd 3621 . . . . 5  |-  ( ph  ->  ( ( A  X.  ( A  i^i  B ) )  u.  ( B  X.  ( A  i^i  B ) ) )  C_  V )
164, 15syl5eqss 3488 . . . 4  |-  ( ph  ->  ( ( A  u.  B )  X.  ( A  i^i  B ) ) 
C_  V )
17 coss2 4982 . . . 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 17 . . 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 4878 . . . . 5  |-  ( ( A  i^i  B )  X.  ( A  u.  B ) )  =  ( ( ( A  i^i  B )  X.  A )  u.  (
( A  i^i  B
)  X.  B ) )
20 xpindir 4960 . . . . . . 7  |-  ( ( A  i^i  B )  X.  A )  =  ( ( A  X.  A )  i^i  ( B  X.  A ) )
21 inss1 3661 . . . . . . . 8  |-  ( ( A  X.  A )  i^i  ( B  X.  A ) )  C_  ( A  X.  A
)
2221, 7syl5ss 3455 . . . . . . 7  |-  ( ph  ->  ( ( A  X.  A )  i^i  ( B  X.  A ) ) 
C_  V )
2320, 22syl5eqss 3488 . . . . . 6  |-  ( ph  ->  ( ( A  i^i  B )  X.  A ) 
C_  V )
24 xpindir 4960 . . . . . . 7  |-  ( ( A  i^i  B )  X.  B )  =  ( ( A  X.  B )  i^i  ( B  X.  B ) )
25 inss2 3662 . . . . . . . 8  |-  ( ( A  X.  B )  i^i  ( B  X.  B ) )  C_  ( B  X.  B
)
2625, 12syl5ss 3455 . . . . . . 7  |-  ( ph  ->  ( ( A  X.  B )  i^i  ( B  X.  B ) ) 
C_  V )
2724, 26syl5eqss 3488 . . . . . 6  |-  ( ph  ->  ( ( A  i^i  B )  X.  B ) 
C_  V )
2823, 27unssd 3621 . . . . 5  |-  ( ph  ->  ( ( ( A  i^i  B )  X.  A )  u.  (
( A  i^i  B
)  X.  B ) )  C_  V )
2919, 28syl5eqss 3488 . . . 4  |-  ( ph  ->  ( ( A  i^i  B )  X.  ( A  u.  B ) ) 
C_  V )
30 coss1 4981 . . . 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 17 . . 3  |-  ( ph  ->  ( ( ( A  i^i  B )  X.  ( A  u.  B
) )  o.  V
)  C_  ( V  o.  V ) )
3218, 31sstrd 3454 . 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 3479 1  |-  ( ph  ->  ( ( A  u.  B )  X.  ( A  u.  B )
)  C_  ( V  o.  V ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1407    =/= wne 2600    u. cun 3414    i^i cin 3415    C_ wss 3416   (/)c0 3740    X. cxp 4823    o. ccom 4829
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-sep 4519  ax-nul 4527  ax-pr 4632
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3063  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3741  df-if 3888  df-sn 3975  df-pr 3977  df-op 3981  df-br 4398  df-opab 4456  df-xp 4831  df-rel 4832  df-co 4834
This theorem is referenced by: (None)
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