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Theorem srefwref 25067
Description: Strong reflexivity implies weak reflexivity. (Strong and weak reflexivity is the difference between a toset and a poset). (Contributed by FL, 29-Dec-2011.)
Assertion
Ref Expression
srefwref  |-  ( A. x  e.  ( dom  R  u.  ran  R ) A. y  e.  ( dom  R  u.  ran  R ) ( x R y  \/  y R x )  ->  A. x  e.  ( dom  R  u.  ran  R ) x R x )
Distinct variable groups:    y, R    x, y
Allowed substitution hint:    R( x)

Proof of Theorem srefwref
StepHypRef Expression
1 breq2 4027 . . . . 5  |-  ( y  =  x  ->  (
x R y  <->  x R x ) )
2 breq1 4026 . . . . 5  |-  ( y  =  x  ->  (
y R x  <->  x R x ) )
31, 2orbi12d 690 . . . 4  |-  ( y  =  x  ->  (
( x R y  \/  y R x )  <->  ( x R x  \/  x R x ) ) )
43rspcva 2882 . . 3  |-  ( ( x  e.  ( dom 
R  u.  ran  R
)  /\  A. y  e.  ( dom  R  u.  ran  R ) ( x R y  \/  y R x ) )  ->  ( x R x  \/  x R x ) )
5 pm4.25 501 . . 3  |-  ( x R x  <->  ( x R x  \/  x R x ) )
64, 5sylibr 203 . 2  |-  ( ( x  e.  ( dom 
R  u.  ran  R
)  /\  A. y  e.  ( dom  R  u.  ran  R ) ( x R y  \/  y R x ) )  ->  x R x )
76ralimiaa 2617 1  |-  ( A. x  e.  ( dom  R  u.  ran  R ) A. y  e.  ( dom  R  u.  ran  R ) ( x R y  \/  y R x )  ->  A. x  e.  ( dom  R  u.  ran  R ) x R x )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 357    /\ wa 358    e. wcel 1684   A.wral 2543    u. cun 3150   class class class wbr 4023   dom cdm 4689   ran crn 4690
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-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
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-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  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
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