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Theorem cvbr2 22863
Description: Binary relation expressing  B covers  A. Definition of covers in [Kalmbach] p. 15. (Contributed by NM, 9-Jun-2004.) (New usage is discouraged.)
Assertion
Ref Expression
cvbr2  |-  ( ( A  e.  CH  /\  B  e.  CH )  ->  ( A  <oH  B  <->  ( A  C.  B  /\  A. x  e.  CH  ( ( A 
C.  x  /\  x  C_  B )  ->  x  =  B ) ) ) )
Distinct variable groups:    x, A    x, B

Proof of Theorem cvbr2
StepHypRef Expression
1 cvbr 22862 . 2  |-  ( ( A  e.  CH  /\  B  e.  CH )  ->  ( A  <oH  B  <->  ( A  C.  B  /\  -.  E. x  e.  CH  ( A 
C.  x  /\  x  C.  B ) ) ) )
2 iman 413 . . . . . 6  |-  ( ( ( A  C.  x  /\  x  C_  B )  ->  x  =  B )  <->  -.  ( ( A  C.  x  /\  x  C_  B )  /\  -.  x  =  B )
)
3 anass 630 . . . . . . 7  |-  ( ( ( A  C.  x  /\  x  C_  B )  /\  -.  x  =  B )  <->  ( A  C.  x  /\  (
x  C_  B  /\  -.  x  =  B
) ) )
4 dfpss2 3261 . . . . . . . 8  |-  ( x 
C.  B  <->  ( x  C_  B  /\  -.  x  =  B ) )
54anbi2i 675 . . . . . . 7  |-  ( ( A  C.  x  /\  x  C.  B )  <->  ( A  C.  x  /\  (
x  C_  B  /\  -.  x  =  B
) ) )
63, 5bitr4i 243 . . . . . 6  |-  ( ( ( A  C.  x  /\  x  C_  B )  /\  -.  x  =  B )  <->  ( A  C.  x  /\  x  C.  B ) )
72, 6xchbinx 301 . . . . 5  |-  ( ( ( A  C.  x  /\  x  C_  B )  ->  x  =  B )  <->  -.  ( A  C.  x  /\  x  C.  B ) )
87ralbii 2567 . . . 4  |-  ( A. x  e.  CH  ( ( A  C.  x  /\  x  C_  B )  ->  x  =  B )  <->  A. x  e.  CH  -.  ( A  C.  x  /\  x  C.  B ) )
9 ralnex 2553 . . . 4  |-  ( A. x  e.  CH  -.  ( A  C.  x  /\  x  C.  B )  <->  -.  E. x  e.  CH  ( A  C.  x  /\  x  C.  B
) )
108, 9bitri 240 . . 3  |-  ( A. x  e.  CH  ( ( A  C.  x  /\  x  C_  B )  ->  x  =  B )  <->  -. 
E. x  e.  CH  ( A  C.  x  /\  x  C.  B ) )
1110anbi2i 675 . 2  |-  ( ( A  C.  B  /\  A. x  e.  CH  (
( A  C.  x  /\  x  C_  B )  ->  x  =  B ) )  <->  ( A  C.  B  /\  -.  E. x  e.  CH  ( A 
C.  x  /\  x  C.  B ) ) )
121, 11syl6bbr 254 1  |-  ( ( A  e.  CH  /\  B  e.  CH )  ->  ( A  <oH  B  <->  ( A  C.  B  /\  A. x  e.  CH  ( ( A 
C.  x  /\  x  C_  B )  ->  x  =  B ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544    C_ wss 3152    C. wpss 3153   class class class wbr 4023   CHcch 21509    <oH ccv 21544
This theorem is referenced by:  spansncv2  22873  elat2  22920
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-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-cv 22859
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