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Theorem ssin0 37454
Description: If two classes are disjoint, two respective subclasses are disjoint. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Assertion
Ref Expression
ssin0  |-  ( ( ( A  i^i  B
)  =  (/)  /\  C  C_  A  /\  D  C_  B )  ->  ( C  i^i  D )  =  (/) )

Proof of Theorem ssin0
StepHypRef Expression
1 simp2 1031 . . . 4  |-  ( ( ( A  i^i  B
)  =  (/)  /\  C  C_  A  /\  D  C_  B )  ->  C  C_  A )
2 simp3 1032 . . . 4  |-  ( ( ( A  i^i  B
)  =  (/)  /\  C  C_  A  /\  D  C_  B )  ->  D  C_  B )
3 ss2in 3650 . . . 4  |-  ( ( C  C_  A  /\  D  C_  B )  -> 
( C  i^i  D
)  C_  ( A  i^i  B ) )
41, 2, 3syl2anc 673 . . 3  |-  ( ( ( A  i^i  B
)  =  (/)  /\  C  C_  A  /\  D  C_  B )  ->  ( C  i^i  D )  C_  ( A  i^i  B ) )
5 eqimss 3470 . . . 4  |-  ( ( A  i^i  B )  =  (/)  ->  ( A  i^i  B )  C_  (/) )
653ad2ant1 1051 . . 3  |-  ( ( ( A  i^i  B
)  =  (/)  /\  C  C_  A  /\  D  C_  B )  ->  ( A  i^i  B )  C_  (/) )
74, 6sstrd 3428 . 2  |-  ( ( ( A  i^i  B
)  =  (/)  /\  C  C_  A  /\  D  C_  B )  ->  ( C  i^i  D )  C_  (/) )
8 ss0 3768 . 2  |-  ( ( C  i^i  D ) 
C_  (/)  ->  ( C  i^i  D )  =  (/) )
97, 8syl 17 1  |-  ( ( ( A  i^i  B
)  =  (/)  /\  C  C_  A  /\  D  C_  B )  ->  ( C  i^i  D )  =  (/) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 1007    = wceq 1452    i^i cin 3389    C_ wss 3390   (/)c0 3722
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-3an 1009  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-v 3033  df-dif 3393  df-in 3397  df-ss 3404  df-nul 3723
This theorem is referenced by:  sge0resplit  38362
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