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Theorem cvbr 22692
Description: Binary relation expressing  B covers  A, which means that  B is larger than  A and there is nothing in between. Definition 3.2.18 of [PtakPulmannova] p. 68. (Contributed by NM, 4-Jun-2004.) (New usage is discouraged.)
Assertion
Ref Expression
cvbr  |-  ( ( A  e.  CH  /\  B  e.  CH )  ->  ( A  <oH  B  <->  ( A  C.  B  /\  -.  E. x  e.  CH  ( A 
C.  x  /\  x  C.  B ) ) ) )
Distinct variable groups:    x, A    x, B

Proof of Theorem cvbr
StepHypRef Expression
1 eleq1 2313 . . . . 5  |-  ( y  =  A  ->  (
y  e.  CH  <->  A  e.  CH ) )
21anbi1d 688 . . . 4  |-  ( y  =  A  ->  (
( y  e.  CH  /\  z  e.  CH )  <->  ( A  e.  CH  /\  z  e.  CH )
) )
3 psseq1 3184 . . . . 5  |-  ( y  =  A  ->  (
y  C.  z  <->  A  C.  z ) )
4 psseq1 3184 . . . . . . . 8  |-  ( y  =  A  ->  (
y  C.  x  <->  A  C.  x ) )
54anbi1d 688 . . . . . . 7  |-  ( y  =  A  ->  (
( y  C.  x  /\  x  C.  z )  <-> 
( A  C.  x  /\  x  C.  z ) ) )
65rexbidv 2528 . . . . . 6  |-  ( y  =  A  ->  ( E. x  e.  CH  (
y  C.  x  /\  x  C.  z )  <->  E. x  e.  CH  ( A  C.  x  /\  x  C.  z
) ) )
76notbid 287 . . . . 5  |-  ( y  =  A  ->  ( -.  E. x  e.  CH  ( y  C.  x  /\  x  C.  z )  <->  -.  E. x  e.  CH  ( A  C.  x  /\  x  C.  z ) ) )
83, 7anbi12d 694 . . . 4  |-  ( y  =  A  ->  (
( y  C.  z  /\  -.  E. x  e. 
CH  ( y  C.  x  /\  x  C.  z
) )  <->  ( A  C.  z  /\  -.  E. x  e.  CH  ( A 
C.  x  /\  x  C.  z ) ) ) )
92, 8anbi12d 694 . . 3  |-  ( y  =  A  ->  (
( ( y  e. 
CH  /\  z  e.  CH )  /\  ( y 
C.  z  /\  -.  E. x  e.  CH  (
y  C.  x  /\  x  C.  z ) ) )  <->  ( ( A  e.  CH  /\  z  e.  CH )  /\  ( A  C.  z  /\  -.  E. x  e.  CH  ( A  C.  x  /\  x  C.  z ) ) ) ) )
10 eleq1 2313 . . . . 5  |-  ( z  =  B  ->  (
z  e.  CH  <->  B  e.  CH ) )
1110anbi2d 687 . . . 4  |-  ( z  =  B  ->  (
( A  e.  CH  /\  z  e.  CH )  <->  ( A  e.  CH  /\  B  e.  CH )
) )
12 psseq2 3185 . . . . 5  |-  ( z  =  B  ->  ( A  C.  z  <->  A  C.  B ) )
13 psseq2 3185 . . . . . . . 8  |-  ( z  =  B  ->  (
x  C.  z  <->  x  C.  B ) )
1413anbi2d 687 . . . . . . 7  |-  ( z  =  B  ->  (
( A  C.  x  /\  x  C.  z )  <-> 
( A  C.  x  /\  x  C.  B ) ) )
1514rexbidv 2528 . . . . . 6  |-  ( z  =  B  ->  ( E. x  e.  CH  ( A  C.  x  /\  x  C.  z )  <->  E. x  e.  CH  ( A  C.  x  /\  x  C.  B
) ) )
1615notbid 287 . . . . 5  |-  ( z  =  B  ->  ( -.  E. x  e.  CH  ( A  C.  x  /\  x  C.  z )  <->  -.  E. x  e.  CH  ( A  C.  x  /\  x  C.  B
) ) )
1712, 16anbi12d 694 . . . 4  |-  ( z  =  B  ->  (
( A  C.  z  /\  -.  E. x  e. 
CH  ( A  C.  x  /\  x  C.  z
) )  <->  ( A  C.  B  /\  -.  E. x  e.  CH  ( A 
C.  x  /\  x  C.  B ) ) ) )
1811, 17anbi12d 694 . . 3  |-  ( z  =  B  ->  (
( ( A  e. 
CH  /\  z  e.  CH )  /\  ( A 
C.  z  /\  -.  E. x  e.  CH  ( A  C.  x  /\  x  C.  z ) ) )  <-> 
( ( A  e. 
CH  /\  B  e.  CH )  /\  ( A 
C.  B  /\  -.  E. x  e.  CH  ( A  C.  x  /\  x  C.  B ) ) ) ) )
19 df-cv 22689 . . 3  |-  <oH  =  { <. y ,  z >.  |  ( ( y  e.  CH  /\  z  e.  CH )  /\  (
y  C.  z  /\  -.  E. x  e.  CH  ( y  C.  x  /\  x  C.  z ) ) ) }
209, 18, 19brabg 4177 . 2  |-  ( ( A  e.  CH  /\  B  e.  CH )  ->  ( A  <oH  B  <->  ( ( A  e.  CH  /\  B  e.  CH )  /\  ( A  C.  B  /\  -.  E. x  e.  CH  ( A  C.  x  /\  x  C.  B ) ) ) ) )
2120bianabs 855 1  |-  ( ( A  e.  CH  /\  B  e.  CH )  ->  ( A  <oH  B  <->  ( A  C.  B  /\  -.  E. x  e.  CH  ( A 
C.  x  /\  x  C.  B ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    <-> wb 178    /\ wa 360    = wceq 1619    e. wcel 1621   E.wrex 2510    C. wpss 3079   class class class wbr 3920   CHcch 21339    <oH ccv 21374
This theorem is referenced by:  cvbr2  22693  cvcon3  22694  cvpss  22695  cvnbtwn  22696
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-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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