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Theorem inssdif0im 3285
Description: Intersection, subclass, and difference relationship. In classical logic the converse would also hold. (Contributed by Jim Kingdon, 3-Aug-2018.)
Assertion
Ref Expression
inssdif0im  i^i 
C_  C  i^i  \  C  (/)

Proof of Theorem inssdif0im
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 elin 3120 . . . . . 6  i^i
21imbi1i 227 . . . . 5  i^i  C  C
3 imanim 784 . . . . 5  C  C
42, 3sylbi 114 . . . 4  i^i  C  C
5 eldif 2921 . . . . . 6  \  C  C
65anbi2i 430 . . . . 5  \  C  C
7 elin 3120 . . . . 5  i^i  \  C  \  C
8 anass 381 . . . . 5  C  C
96, 7, 83bitr4ri 202 . . . 4  C  i^i  \  C
104, 9sylnib 600 . . 3  i^i  C  i^i 
\  C
1110alimi 1341 . 2  i^i  C  i^i  \  C
12 dfss2 2928 . 2  i^i 
C_  C  i^i  C
13 eq0 3233 . 2  i^i 
\  C  (/)  i^i  \  C
1411, 12, 133imtr4i 190 1  i^i 
C_  C  i^i  \  C  (/)
Colors of variables: wff set class
Syntax hints:   wn 3   wi 4   wa 97  wal 1240   wceq 1242   wcel 1390    \ cdif 2908    i^i cin 2910    C_ wss 2911   (/)c0 3218
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-v 2553  df-dif 2914  df-in 2918  df-ss 2925  df-nul 3219
This theorem is referenced by:  disjdif  3290
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