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Theorem disjpss 3678
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 3367 . . . . . . . 8  |-  B  C_  B
21biantru 492 . . . . . . 7  |-  ( B 
C_  A  <->  ( B  C_  A  /\  B  C_  B ) )
3 ssin 3563 . . . . . . 7  |-  ( ( B  C_  A  /\  B  C_  B )  <->  B  C_  ( A  i^i  B ) )
42, 3bitri 241 . . . . . 6  |-  ( B 
C_  A  <->  B  C_  ( A  i^i  B ) )
5 sseq2 3370 . . . . . 6  |-  ( ( A  i^i  B )  =  (/)  ->  ( B 
C_  ( A  i^i  B )  <->  B  C_  (/) ) )
64, 5syl5bb 249 . . . . 5  |-  ( ( A  i^i  B )  =  (/)  ->  ( B 
C_  A  <->  B  C_  (/) ) )
7 ss0 3658 . . . . 5  |-  ( B 
C_  (/)  ->  B  =  (/) )
86, 7syl6bi 220 . . . 4  |-  ( ( A  i^i  B )  =  (/)  ->  ( B 
C_  A  ->  B  =  (/) ) )
98necon3ad 2637 . . 3  |-  ( ( A  i^i  B )  =  (/)  ->  ( B  =/=  (/)  ->  -.  B  C_  A ) )
109imp 419 . 2  |-  ( ( ( A  i^i  B
)  =  (/)  /\  B  =/=  (/) )  ->  -.  B  C_  A )
11 nsspssun 3574 . . 3  |-  ( -.  B  C_  A  <->  A  C.  ( B  u.  A
) )
12 uncom 3491 . . . 4  |-  ( B  u.  A )  =  ( A  u.  B
)
1312psseq2i 3437 . . 3  |-  ( A 
C.  ( B  u.  A )  <->  A  C.  ( A  u.  B
) )
1411, 13bitri 241 . 2  |-  ( -.  B  C_  A  <->  A  C.  ( A  u.  B
) )
1510, 14sylib 189 1  |-  ( ( ( A  i^i  B
)  =  (/)  /\  B  =/=  (/) )  ->  A  C.  ( A  u.  B
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    = wceq 1652    =/= wne 2599    u. cun 3318    i^i cin 3319    C_ wss 3320    C. wpss 3321   (/)c0 3628
This theorem is referenced by:  isfin1-3  8266
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629
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