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Theorem disjpss 3793
Description: A class is a proper subset of its union with a disjoint nonempty class. (Contributed by NM, 15-Sep-2004.)
Assertion
Ref Expression
disjpss  |-  ( ( ( A  i^i  B
)  =  (/)  /\  B  =/=  (/) )  ->  A  C.  ( A  u.  B
) )

Proof of Theorem disjpss
StepHypRef Expression
1 ssid 3436 . . . . . . . 8  |-  B  C_  B
21biantru 503 . . . . . . 7  |-  ( B 
C_  A  <->  ( B  C_  A  /\  B  C_  B ) )
3 ssin 3634 . . . . . . 7  |-  ( ( B  C_  A  /\  B  C_  B )  <->  B  C_  ( A  i^i  B ) )
42, 3bitri 249 . . . . . 6  |-  ( B 
C_  A  <->  B  C_  ( A  i^i  B ) )
5 sseq2 3439 . . . . . 6  |-  ( ( A  i^i  B )  =  (/)  ->  ( B 
C_  ( A  i^i  B )  <->  B  C_  (/) ) )
64, 5syl5bb 257 . . . . 5  |-  ( ( A  i^i  B )  =  (/)  ->  ( B 
C_  A  <->  B  C_  (/) ) )
7 ss0 3743 . . . . 5  |-  ( B 
C_  (/)  ->  B  =  (/) )
86, 7syl6bi 228 . . . 4  |-  ( ( A  i^i  B )  =  (/)  ->  ( B 
C_  A  ->  B  =  (/) ) )
98necon3ad 2592 . . 3  |-  ( ( A  i^i  B )  =  (/)  ->  ( B  =/=  (/)  ->  -.  B  C_  A ) )
109imp 427 . 2  |-  ( ( ( A  i^i  B
)  =  (/)  /\  B  =/=  (/) )  ->  -.  B  C_  A )
11 nsspssun 3656 . . 3  |-  ( -.  B  C_  A  <->  A  C.  ( B  u.  A )
)
12 uncom 3562 . . . 4  |-  ( B  u.  A )  =  ( A  u.  B
)
1312psseq2i 3508 . . 3  |-  ( A 
C.  ( B  u.  A )  <->  A  C.  ( A  u.  B )
)
1411, 13bitri 249 . 2  |-  ( -.  B  C_  A  <->  A  C.  ( A  u.  B )
)
1510, 14sylib 196 1  |-  ( ( ( A  i^i  B
)  =  (/)  /\  B  =/=  (/) )  ->  A  C.  ( A  u.  B
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 367    = wceq 1399    =/= wne 2577    u. cun 3387    i^i cin 3388    C_ wss 3389    C. wpss 3390   (/)c0 3711
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-v 3036  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712
This theorem is referenced by:  isfin1-3  8679
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