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Theorem cvbr2 22855
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 22854 . 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 415 . . . . . 6  |-  ( ( ( A  C.  x  /\  x  C_  B )  ->  x  =  B )  <->  -.  ( ( A  C.  x  /\  x  C_  B )  /\  -.  x  =  B )
)
3 anass 632 . . . . . . 7  |-  ( ( ( A  C.  x  /\  x  C_  B )  /\  -.  x  =  B )  <->  ( A  C.  x  /\  (
x  C_  B  /\  -.  x  =  B
) ) )
4 dfpss2 3262 . . . . . . . 8  |-  ( x 
C.  B  <->  ( x  C_  B  /\  -.  x  =  B ) )
54anbi2i 677 . . . . . . 7  |-  ( ( A  C.  x  /\  x  C.  B )  <->  ( A  C.  x  /\  (
x  C_  B  /\  -.  x  =  B
) ) )
63, 5bitr4i 245 . . . . . 6  |-  ( ( ( A  C.  x  /\  x  C_  B )  /\  -.  x  =  B )  <->  ( A  C.  x  /\  x  C.  B ) )
72, 6xchbinx 303 . . . . 5  |-  ( ( ( A  C.  x  /\  x  C_  B )  ->  x  =  B )  <->  -.  ( A  C.  x  /\  x  C.  B ) )
87ralbii 2568 . . . 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 2554 . . . 4  |-  ( A. x  e.  CH  -.  ( A  C.  x  /\  x  C.  B )  <->  -.  E. x  e.  CH  ( A  C.  x  /\  x  C.  B
) )
108, 9bitri 242 . . 3  |-  ( A. x  e.  CH  ( ( A  C.  x  /\  x  C_  B )  ->  x  =  B )  <->  -. 
E. x  e.  CH  ( A  C.  x  /\  x  C.  B ) )
1110anbi2i 677 . 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 256 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 5    -> wi 6    <-> wb 178    /\ wa 360    = wceq 1624    e. wcel 1685   A.wral 2544   E.wrex 2545    C_ wss 3153    C. wpss 3154   class class class wbr 4024   CHcch 21501    <oH ccv 21536
This theorem is referenced by:  spansncv2  22865  elat2  22912
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-op 3650  df-br 4025  df-opab 4079  df-cv 22851
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