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Theorem difin 3168
Description: Difference with intersection. Theorem 33 of [Suppes] p. 29. (Contributed by NM, 31-Mar-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Assertion
Ref Expression
difin 
\  i^i  \

Proof of Theorem difin
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 ax-in2 545 . . . . . . . 8
21expd 245 . . . . . . 7
3 dfnot 1261 . . . . . . 7
42, 3syl6ibr 151 . . . . . 6
54com12 27 . . . . 5
65imdistani 419 . . . 4
7 simpr 103 . . . . . 6
87con3i 561 . . . . 5
98anim2i 324 . . . 4
106, 9impbii 117 . . 3
11 eldif 2921 . . . 4  \  i^i  i^i
12 elin 3120 . . . . . 6  i^i
1312notbii 593 . . . . 5  i^i
1413anbi2i 430 . . . 4  i^i
1511, 14bitri 173 . . 3  \  i^i
16 eldif 2921 . . 3  \
1710, 15, 163bitr4i 201 . 2  \  i^i  \
1817eqriv 2034 1 
\  i^i  \
Colors of variables: wff set class
Syntax hints:   wn 3   wi 4   wa 97   wceq 1242   wfal 1247   wcel 1390    \ cdif 2908    i^i cin 2910
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-fal 1248  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
This theorem is referenced by:  inssddif  3172  symdif1  3196  notrab  3208
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