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Theorem cvbr 22862
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
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq1 2343 . . . . 5  |-  ( y  =  A  ->  (
y  e.  CH  <->  A  e.  CH ) )
21anbi1d 685 . . . 4  |-  ( y  =  A  ->  (
( y  e.  CH  /\  z  e.  CH )  <->  ( A  e.  CH  /\  z  e.  CH )
) )
3 psseq1 3263 . . . . 5  |-  ( y  =  A  ->  (
y  C.  z  <->  A  C.  z ) )
4 psseq1 3263 . . . . . . . 8  |-  ( y  =  A  ->  (
y  C.  x  <->  A  C.  x ) )
54anbi1d 685 . . . . . . 7  |-  ( y  =  A  ->  (
( y  C.  x  /\  x  C.  z )  <-> 
( A  C.  x  /\  x  C.  z ) ) )
65rexbidv 2564 . . . . . 6  |-  ( y  =  A  ->  ( E. x  e.  CH  (
y  C.  x  /\  x  C.  z )  <->  E. x  e.  CH  ( A  C.  x  /\  x  C.  z
) ) )
76notbid 285 . . . . 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 691 . . . 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 691 . . 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 2343 . . . . 5  |-  ( z  =  B  ->  (
z  e.  CH  <->  B  e.  CH ) )
1110anbi2d 684 . . . 4  |-  ( z  =  B  ->  (
( A  e.  CH  /\  z  e.  CH )  <->  ( A  e.  CH  /\  B  e.  CH )
) )
12 psseq2 3264 . . . . 5  |-  ( z  =  B  ->  ( A  C.  z  <->  A  C.  B ) )
13 psseq2 3264 . . . . . . . 8  |-  ( z  =  B  ->  (
x  C.  z  <->  x  C.  B ) )
1413anbi2d 684 . . . . . . 7  |-  ( z  =  B  ->  (
( A  C.  x  /\  x  C.  z )  <-> 
( A  C.  x  /\  x  C.  B ) ) )
1514rexbidv 2564 . . . . . 6  |-  ( z  =  B  ->  ( E. x  e.  CH  ( A  C.  x  /\  x  C.  z )  <->  E. x  e.  CH  ( A  C.  x  /\  x  C.  B
) ) )
1615notbid 285 . . . . 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 691 . . . 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 691 . . 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 22859 . . 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 4284 . 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 850 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 3    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   E.wrex 2544    C. wpss 3153   class class class wbr 4023   CHcch 21509    <oH ccv 21544
This theorem is referenced by:  cvbr2  22863  cvcon3  22864  cvpss  22865  cvnbtwn  22866
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-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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