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Theorem ssin 3195
 Description: Subclass of intersection. Theorem 2.8(vii) of [Monk1] p. 26. (Contributed by NM, 15-Jun-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Assertion
Ref Expression
ssin

Proof of Theorem ssin
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 elin 3156 . . . . 5
21imbi2i 224 . . . 4
32albii 1400 . . 3
4 jcab 568 . . . 4
54albii 1400 . . 3
6 19.26 1411 . . 3
73, 5, 63bitrri 205 . 2
8 dfss2 2989 . . 3
9 dfss2 2989 . . 3
108, 9anbi12i 448 . 2
11 dfss2 2989 . 2
127, 10, 113bitr4i 210 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 102   wb 103  wal 1283   wcel 1434   cin 2973   wss 2974 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064 This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-in 2980  df-ss 2987 This theorem is referenced by:  ssini  3196  ssind  3197  uneqin  3222  trin  3893  pwin  4045  peano5  4347  fin  5107
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