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Theorem disjss3 4455
Description: Expand a disjoint collection with any number of empty sets. (Contributed by Mario Carneiro, 15-Nov-2016.)
Assertion
Ref Expression
disjss3  |-  ( ( A  C_  B  /\  A. x  e.  ( B 
\  A ) C  =  (/) )  ->  (Disj  x  e.  A  C  <-> Disj  x  e.  B  C ) )
Distinct variable groups:    x, A    x, B
Allowed substitution hint:    C( x)

Proof of Theorem disjss3
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 df-ral 2812 . . . . . . 7  |-  ( A. x  e.  ( B  \  A ) C  =  (/) 
<-> 
A. x ( x  e.  ( B  \  A )  ->  C  =  (/) ) )
2 simp3r 1025 . . . . . . . . . . . 12  |-  ( ( A  C_  B  /\  ( x  e.  ( B  \  A )  ->  C  =  (/) )  /\  ( x  e.  B  /\  y  e.  C
) )  ->  y  e.  C )
3 n0i 3798 . . . . . . . . . . . 12  |-  ( y  e.  C  ->  -.  C  =  (/) )
42, 3syl 16 . . . . . . . . . . 11  |-  ( ( A  C_  B  /\  ( x  e.  ( B  \  A )  ->  C  =  (/) )  /\  ( x  e.  B  /\  y  e.  C
) )  ->  -.  C  =  (/) )
5 simp3l 1024 . . . . . . . . . . . 12  |-  ( ( A  C_  B  /\  ( x  e.  ( B  \  A )  ->  C  =  (/) )  /\  ( x  e.  B  /\  y  e.  C
) )  ->  x  e.  B )
6 eldif 3481 . . . . . . . . . . . . 13  |-  ( x  e.  ( B  \  A )  <->  ( x  e.  B  /\  -.  x  e.  A ) )
7 simp2 997 . . . . . . . . . . . . 13  |-  ( ( A  C_  B  /\  ( x  e.  ( B  \  A )  ->  C  =  (/) )  /\  ( x  e.  B  /\  y  e.  C
) )  ->  (
x  e.  ( B 
\  A )  ->  C  =  (/) ) )
86, 7syl5bir 218 . . . . . . . . . . . 12  |-  ( ( A  C_  B  /\  ( x  e.  ( B  \  A )  ->  C  =  (/) )  /\  ( x  e.  B  /\  y  e.  C
) )  ->  (
( x  e.  B  /\  -.  x  e.  A
)  ->  C  =  (/) ) )
95, 8mpand 675 . . . . . . . . . . 11  |-  ( ( A  C_  B  /\  ( x  e.  ( B  \  A )  ->  C  =  (/) )  /\  ( x  e.  B  /\  y  e.  C
) )  ->  ( -.  x  e.  A  ->  C  =  (/) ) )
104, 9mt3d 125 . . . . . . . . . 10  |-  ( ( A  C_  B  /\  ( x  e.  ( B  \  A )  ->  C  =  (/) )  /\  ( x  e.  B  /\  y  e.  C
) )  ->  x  e.  A )
1110, 2jca 532 . . . . . . . . 9  |-  ( ( A  C_  B  /\  ( x  e.  ( B  \  A )  ->  C  =  (/) )  /\  ( x  e.  B  /\  y  e.  C
) )  ->  (
x  e.  A  /\  y  e.  C )
)
12113exp 1195 . . . . . . . 8  |-  ( A 
C_  B  ->  (
( x  e.  ( B  \  A )  ->  C  =  (/) )  ->  ( ( x  e.  B  /\  y  e.  C )  ->  (
x  e.  A  /\  y  e.  C )
) ) )
1312alimdv 1710 . . . . . . 7  |-  ( A 
C_  B  ->  ( A. x ( x  e.  ( B  \  A
)  ->  C  =  (/) )  ->  A. x
( ( x  e.  B  /\  y  e.  C )  ->  (
x  e.  A  /\  y  e.  C )
) ) )
141, 13syl5bi 217 . . . . . 6  |-  ( A 
C_  B  ->  ( A. x  e.  ( B  \  A ) C  =  (/)  ->  A. x
( ( x  e.  B  /\  y  e.  C )  ->  (
x  e.  A  /\  y  e.  C )
) ) )
1514imp 429 . . . . 5  |-  ( ( A  C_  B  /\  A. x  e.  ( B 
\  A ) C  =  (/) )  ->  A. x
( ( x  e.  B  /\  y  e.  C )  ->  (
x  e.  A  /\  y  e.  C )
) )
16 moim 2340 . . . . 5  |-  ( A. x ( ( x  e.  B  /\  y  e.  C )  ->  (
x  e.  A  /\  y  e.  C )
)  ->  ( E* x ( x  e.  A  /\  y  e.  C )  ->  E* x ( x  e.  B  /\  y  e.  C ) ) )
1715, 16syl 16 . . . 4  |-  ( ( A  C_  B  /\  A. x  e.  ( B 
\  A ) C  =  (/) )  ->  ( E* x ( x  e.  A  /\  y  e.  C )  ->  E* x ( x  e.  B  /\  y  e.  C ) ) )
1817alimdv 1710 . . 3  |-  ( ( A  C_  B  /\  A. x  e.  ( B 
\  A ) C  =  (/) )  ->  ( A. y E* x ( x  e.  A  /\  y  e.  C )  ->  A. y E* x
( x  e.  B  /\  y  e.  C
) ) )
19 dfdisj2 4429 . . 3  |-  (Disj  x  e.  A  C  <->  A. y E* x ( x  e.  A  /\  y  e.  C ) )
20 dfdisj2 4429 . . 3  |-  (Disj  x  e.  B  C  <->  A. y E* x ( x  e.  B  /\  y  e.  C ) )
2118, 19, 203imtr4g 270 . 2  |-  ( ( A  C_  B  /\  A. x  e.  ( B 
\  A ) C  =  (/) )  ->  (Disj  x  e.  A  C  -> Disj  x  e.  B  C ) )
22 disjss1 4433 . . 3  |-  ( A 
C_  B  ->  (Disj  x  e.  B  C  -> Disj  x  e.  A  C ) )
2322adantr 465 . 2  |-  ( ( A  C_  B  /\  A. x  e.  ( B 
\  A ) C  =  (/) )  ->  (Disj  x  e.  B  C  -> Disj  x  e.  A  C ) )
2421, 23impbid 191 1  |-  ( ( A  C_  B  /\  A. x  e.  ( B 
\  A ) C  =  (/) )  ->  (Disj  x  e.  A  C  <-> Disj  x  e.  B  C ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973   A.wal 1393    = wceq 1395    e. wcel 1819   E*wmo 2284   A.wral 2807    \ cdif 3468    C_ wss 3471   (/)c0 3793  Disj wdisj 4427
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ral 2812  df-rmo 2815  df-v 3111  df-dif 3474  df-in 3478  df-ss 3485  df-nul 3794  df-disj 4428
This theorem is referenced by: (None)
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