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Theorem difin2 3193
Description: Represent a set difference as an intersection with a larger difference. (Contributed by Jeff Madsen, 2-Sep-2009.)
Assertion
Ref Expression
difin2 
C_  C  \  C  \  i^i

Proof of Theorem difin2
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 ssel 2933 . . . . 5 
C_  C  C
21pm4.71d 373 . . . 4 
C_  C  C
32anbi1d 438 . . 3 
C_  C  C
4 eldif 2921 . . 3  \
5 elin 3120 . . . 4  C 
\  i^i  C  \
6 eldif 2921 . . . . 5  C  \  C
76anbi1i 431 . . . 4  C 
\  C
8 ancom 253 . . . . 5  C  C
9 anass 381 . . . . 5  C  C
108, 9bitr4i 176 . . . 4  C  C
115, 7, 103bitri 195 . . 3  C 
\  i^i  C
123, 4, 113bitr4g 212 . 2 
C_  C  \  C  \  i^i
1312eqrdv 2035 1 
C_  C  \  C  \  i^i
Colors of variables: wff set class
Syntax hints:   wn 3   wi 4   wa 97   wceq 1242   wcel 1390    \ cdif 2908    i^i cin 2910    C_ wss 2911
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-dif 2914  df-in 2918  df-ss 2925
This theorem is referenced by: (None)
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