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Theorem disjpss 3505
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 3197 . . . . . . . 8  |-  B  C_  B
21biantru 491 . . . . . . 7  |-  ( B 
C_  A  <->  ( B  C_  A  /\  B  C_  B ) )
3 ssin 3391 . . . . . . 7  |-  ( ( B  C_  A  /\  B  C_  B )  <->  B  C_  ( A  i^i  B ) )
42, 3bitri 240 . . . . . 6  |-  ( B 
C_  A  <->  B  C_  ( A  i^i  B ) )
5 sseq2 3200 . . . . . 6  |-  ( ( A  i^i  B )  =  (/)  ->  ( B 
C_  ( A  i^i  B )  <->  B  C_  (/) ) )
64, 5syl5bb 248 . . . . 5  |-  ( ( A  i^i  B )  =  (/)  ->  ( B 
C_  A  <->  B  C_  (/) ) )
7 ss0 3485 . . . . 5  |-  ( B 
C_  (/)  ->  B  =  (/) )
86, 7syl6bi 219 . . . 4  |-  ( ( A  i^i  B )  =  (/)  ->  ( B 
C_  A  ->  B  =  (/) ) )
98necon3ad 2482 . . 3  |-  ( ( A  i^i  B )  =  (/)  ->  ( B  =/=  (/)  ->  -.  B  C_  A ) )
109imp 418 . 2  |-  ( ( ( A  i^i  B
)  =  (/)  /\  B  =/=  (/) )  ->  -.  B  C_  A )
11 nsspssun 3402 . . 3  |-  ( -.  B  C_  A  <->  A  C.  ( B  u.  A
) )
12 uncom 3319 . . . 4  |-  ( B  u.  A )  =  ( A  u.  B
)
1312psseq2i 3266 . . 3  |-  ( A 
C.  ( B  u.  A )  <->  A  C.  ( A  u.  B
) )
1411, 13bitri 240 . 2  |-  ( -.  B  C_  A  <->  A  C.  ( A  u.  B
) )
1510, 14sylib 188 1  |-  ( ( ( A  i^i  B
)  =  (/)  /\  B  =/=  (/) )  ->  A  C.  ( A  u.  B
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1623    =/= wne 2446    u. cun 3150    i^i cin 3151    C_ wss 3152    C. wpss 3153   (/)c0 3455
This theorem is referenced by:  isfin1-3  8012
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456
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