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Theorem cvbr2 22693
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 22692 . 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 633 . . . . . . 7  |-  ( ( ( A  C.  x  /\  x  C_  B )  /\  -.  x  =  B )  <->  ( A  C.  x  /\  (
x  C_  B  /\  -.  x  =  B
) ) )
4 dfpss2 3182 . . . . . . . 8  |-  ( x 
C.  B  <->  ( x  C_  B  /\  -.  x  =  B ) )
54anbi2i 678 . . . . . . 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 2531 . . . 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 2517 . . . 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 678 . 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 1619    e. wcel 1621   A.wral 2509   E.wrex 2510    C_ wss 3078    C. wpss 3079   class class class wbr 3920   CHcch 21339    <oH ccv 21374
This theorem is referenced by:  spansncv2  22703  elat2  22750
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pr 4108
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-rab 2516  df-v 2729  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-sn 3550  df-pr 3551  df-op 3553  df-br 3921  df-opab 3975  df-cv 22689
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