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Theorem ssin 3159
 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 3126 . . . . 5
21imbi2i 215 . . . 4
32albii 1359 . . 3
4 jcab 535 . . . 4
54albii 1359 . . 3
6 19.26 1370 . . 3
73, 5, 63bitrri 196 . 2
8 dfss2 2934 . . 3
9 dfss2 2934 . . 3
108, 9anbi12i 433 . 2
11 dfss2 2934 . 2
127, 10, 113bitr4i 201 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 97   wb 98  wal 1241   wcel 1393   cin 2916   wss 2917 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-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-in 2924  df-ss 2931 This theorem is referenced by:  ssini  3160  ssind  3161  uneqin  3188  disjpss  3278  trin  3864  pwin  4019  peano5  4321  fin  5076
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