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Theorem uniinn0 28241
 Description: Sufficient and necessary condition for a union to intersect with a given set. (Contributed by Thierry Arnoux, 27-Jan-2020.)
Assertion
Ref Expression
uniinn0
Distinct variable groups:   ,   ,

Proof of Theorem uniinn0
StepHypRef Expression
1 nne 2647 . . . 4
21ralbii 2823 . . 3
3 ralnex 2834 . . 3
4 unissb 4221 . . . 4
5 disj2 3816 . . . 4
6 disj2 3816 . . . . 5
76ralbii 2823 . . . 4
84, 5, 73bitr4ri 286 . . 3
92, 3, 83bitr3i 283 . 2
109necon1abii 2691 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wb 189   wceq 1452   wne 2641  wral 2756  wrex 2757  cvv 3031   cdif 3387   cin 3389   wss 3390  c0 3722  cuni 4190 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451 This theorem depends on definitions:  df-bi 190  df-an 378  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-v 3033  df-dif 3393  df-in 3397  df-ss 3404  df-nul 3723  df-uni 4191 This theorem is referenced by:  locfinreflem  28741
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