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Theorem inposetlem 25276
Description: Lemma for inposet 25278. Definition of inclusion of sets using a class. (Contributed by FL, 22-Sep-2008.)
Hypotheses
Ref Expression
inposetlem.1  |-  A  e. 
_V
inposetlem.2  |-  B  e. 
_V
inposetlem.3  |-  C  =  { <. x ,  y
>.  |  x  C_  y }
Assertion
Ref Expression
inposetlem  |-  ( A C B  <->  A  C_  B
)
Distinct variable groups:    x, A, y    x, B, y
Allowed substitution hints:    C( x, y)

Proof of Theorem inposetlem
StepHypRef Expression
1 inposetlem.1 . 2  |-  A  e. 
_V
2 inposetlem.2 . 2  |-  B  e. 
_V
3 sseq1 3199 . 2  |-  ( x  =  A  ->  (
x  C_  y  <->  A  C_  y
) )
4 sseq2 3200 . 2  |-  ( y  =  B  ->  ( A  C_  y  <->  A  C_  B
) )
5 inposetlem.3 . 2  |-  C  =  { <. x ,  y
>.  |  x  C_  y }
61, 2, 3, 4, 5brab 4287 1  |-  ( A C B  <->  A  C_  B
)
Colors of variables: wff set class
Syntax hints:    <-> wb 176    = wceq 1623    e. wcel 1684   _Vcvv 2788    C_ wss 3152   class class class wbr 4023   {copab 4076
This theorem is referenced by:  inposet  25278  definc  25279
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078
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