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Theorem ustund 22025
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 5675 . . 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 5172 . . . . 5 |- ( ( A u. B ) X. ( A i^i B ) ) = ( ( A X. ( A i^i B ) ) u. ( B X. ( A i^i B ) ) )
5 xpindi 5255 . . . . . . 7 |- ( A X. ( A i^i B ) ) = ( ( A X. A ) i^i ( A X. B ) )
6 inss1 3833 . . . . . . . 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 3614 . . . . . . 7 |- ( ph -> ( ( A X. A ) i^i ( A X. B ) ) C_ V )
95, 8syl5eqss 3649 . . . . . 6 |- ( ph -> ( A X. ( A i^i B ) ) C_ V )
10 xpindi 5255 . . . . . . 7 |- ( B X. ( A i^i B ) ) = ( ( B X. A ) i^i ( B X. B ) )
11 inss2 3834 . . . . . . . 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 3614 . . . . . . 7 |- ( ph -> ( ( B X. A ) i^i ( B X. B ) ) C_ V )
1410, 13syl5eqss 3649 . . . . . 6 |- ( ph -> ( B X. ( A i^i B ) ) C_ V )
159, 14unssd 3789 . . . . 5 |- ( ph -> ( ( A X. ( A i^i B ) ) u. ( B X. ( A i^i B ) ) ) C_ V )
164, 15syl5eqss 3649 . . . 4 |- ( ph -> ( ( A u. B ) X. ( A i^i B ) ) C_ V )
17 coss2 5278 . . . 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 5171 . . . . 5 |- ( ( A i^i B ) X. ( A u. B ) ) = ( ( ( A i^i B ) X. A ) u. ( ( A i^i B ) X. B ) )
20 xpindir 5256 . . . . . . 7 |- ( ( A i^i B ) X. A ) = ( ( A X. A ) i^i ( B X. A ) )
21 inss1 3833 . . . . . . . 8 |- ( ( A X. A ) i^i ( B X. A ) ) C_ ( A X. A )
2221, 7syl5ss 3614 . . . . . . 7 |- ( ph -> ( ( A X. A ) i^i ( B X. A ) ) C_ V )
2320, 22syl5eqss 3649 . . . . . 6 |- ( ph -> ( ( A i^i B ) X. A ) C_ V )
24 xpindir 5256 . . . . . . 7 |- ( ( A i^i B ) X. B ) = ( ( A X. B ) i^i ( B X. B ) )
25 inss2 3834 . . . . . . . 8 |- ( ( A X. B ) i^i ( B X. B ) ) C_ ( B X. B )
2625, 12syl5ss 3614 . . . . . . 7 |- ( ph -> ( ( A X. B ) i^i ( B X. B ) ) C_ V )
2724, 26syl5eqss 3649 . . . . . 6 |- ( ph -> ( ( A i^i B ) X. B ) C_ V )
2823, 27unssd 3789 . . . . 5 |- ( ph -> ( ( ( A i^i B ) X. A ) u. ( ( A i^i B ) X. B ) ) C_ V )
2919, 28syl5eqss 3649 . . . 4 |- ( ph -> ( ( A i^i B ) X. ( A u. B ) ) C_ V )
30 coss1 5277 . . . 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 3613 . 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 3640 1 |- ( ph -> ( ( A u. B ) X. ( A u. B ) ) C_ ( V o. V ) )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   = wceq 1483   =/= wne 2794   u. cun 3572   i^i cin 3573   C_ wss 3574   (/)c0 3915   X. cxp 5112   o. ccom 5118
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-co 5123
This theorem is referenced by: (None)
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