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Theorem ustund 21018
 Description: If two intersecting sets and are both small in , their union is small in . 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
ustund.2
ustund.3
Assertion
Ref Expression
ustund

Proof of Theorem ustund
StepHypRef Expression
1 ustund.3 . . 3
2 xpco 5366 . . 3
31, 2syl 17 . 2
4 xpundir 4879 . . . . 5
5 xpindi 4959 . . . . . . 7
6 inss1 3661 . . . . . . . 8
7 ustund.1 . . . . . . . 8
86, 7syl5ss 3455 . . . . . . 7
95, 8syl5eqss 3488 . . . . . 6
10 xpindi 4959 . . . . . . 7
11 inss2 3662 . . . . . . . 8
12 ustund.2 . . . . . . . 8
1311, 12syl5ss 3455 . . . . . . 7
1410, 13syl5eqss 3488 . . . . . 6
159, 14unssd 3621 . . . . 5
164, 15syl5eqss 3488 . . . 4
17 coss2 4982 . . . 4
1816, 17syl 17 . . 3
19 xpundi 4878 . . . . 5
20 xpindir 4960 . . . . . . 7
21 inss1 3661 . . . . . . . 8
2221, 7syl5ss 3455 . . . . . . 7
2320, 22syl5eqss 3488 . . . . . 6
24 xpindir 4960 . . . . . . 7
25 inss2 3662 . . . . . . . 8
2625, 12syl5ss 3455 . . . . . . 7
2724, 26syl5eqss 3488 . . . . . 6
2823, 27unssd 3621 . . . . 5
2919, 28syl5eqss 3488 . . . 4
30 coss1 4981 . . . 4
3129, 30syl 17 . . 3
3218, 31sstrd 3454 . 2
333, 32eqsstr3d 3479 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wceq 1407   wne 2600   cun 3414   cin 3415   wss 3416  c0 3740   cxp 4823   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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