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Theorem difdif2ss 3188
Description: Set difference with a set difference. In classical logic this would be equality rather than subset. (Contributed by Jim Kingdon, 27-Jul-2018.)
Assertion
Ref Expression
difdif2ss  \  u.  i^i  C  C_  \  \  C

Proof of Theorem difdif2ss
StepHypRef Expression
1 inssdif 3167 . . . 4  i^i  C  C_  \  _V  \  C
2 unss2 3108 . . . 4  i^i  C 
C_  \  _V  \  C  \  u.  i^i  C  C_  \  u.  \  _V  \  C
31, 2ax-mp 7 . . 3  \  u.  i^i  C  C_  \  u.  \  _V  \  C
4 difindiss 3185 . . 3  \  u.  \  _V  \  C  C_  \  i^i  _V  \  C
53, 4sstri 2948 . 2  \  u.  i^i  C  C_  \  i^i  _V  \  C
6 invdif 3173 . . . 4  i^i  _V  \  C  \  C
76eqcomi 2041 . . 3 
\  C  i^i  _V  \  C
87difeq2i 3053 . 2 
\  \  C  \  i^i  _V  \  C
95, 8sseqtr4i 2972 1  \  u.  i^i  C  C_  \  \  C
Colors of variables: wff set class
Syntax hints:   _Vcvv 2551    \ cdif 2908    u. cun 2909    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-bndl 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-ral 2305  df-rab 2309  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925
This theorem is referenced by: (None)
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