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Theorem inposetlem 25379
Description: Lemma for inposet 25381. 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 3212 . 2  |-  ( x  =  A  ->  (
x  C_  y  <->  A  C_  y
) )
4 sseq2 3213 . 2  |-  ( y  =  B  ->  ( A  C_  y  <->  A  C_  B
) )
5 inposetlem.3 . 2  |-  C  =  { <. x ,  y
>.  |  x  C_  y }
61, 2, 3, 4, 5brab 4303 1  |-  ( A C B  <->  A  C_  B
)
Colors of variables: wff set class
Syntax hints:    <-> wb 176    = wceq 1632    e. wcel 1696   _Vcvv 2801    C_ wss 3165   class class class wbr 4039   {copab 4092
This theorem is referenced by:  inposet  25381  definc  25382
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-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
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