Mathbox for Frédéric Liné < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  definc Unicode version

Theorem definc 25382
 Description: Definition of the inclusion. (Contributed by FL, 6-Sep-2009.)
Hypotheses
Ref Expression
definc.1
definc.2
definc.3
Assertion
Ref Expression
definc
Distinct variable groups:   ,,   ,,
Allowed substitution hints:   (,)   (,)   (,)   (,)   (,)

Proof of Theorem definc
StepHypRef Expression
1 brin 4086 . 2
2 definc.1 . . . . 5
32elexi 2810 . . . 4
4 definc.2 . . . . 5
54elexi 2810 . . . 4
6 definc.3 . . . 4
73, 5, 6inposetlem 25379 . . 3
8 brxp 4736 . . 3
97, 8anbi12i 678 . 2
10 simprl 732 . . . 4
11 simprr 733 . . . 4
12 simpl 443 . . . 4
1310, 11, 123jca 1132 . . 3
14 simp3 957 . . . 4
15 3simpa 952 . . . 4
1614, 15jca 518 . . 3
1713, 16impbii 180 . 2
181, 9, 173bitri 262 1
 Colors of variables: wff set class Syntax hints:   wb 176   wa 358   w3a 934   wceq 1632   wcel 1696   cin 3164   wss 3165   class class class wbr 4039  copab 4092   cxp 4703 This theorem is referenced by:  toplat  25393 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-xp 4711
 Copyright terms: Public domain W3C validator