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Theorem cvbr 23773
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 2495 . . . . 5  |-  ( y  =  A  ->  (
y  e.  CH  <->  A  e.  CH ) )
21anbi1d 686 . . . 4  |-  ( y  =  A  ->  (
( y  e.  CH  /\  z  e.  CH )  <->  ( A  e.  CH  /\  z  e.  CH )
) )
3 psseq1 3426 . . . . 5  |-  ( y  =  A  ->  (
y  C.  z  <->  A  C.  z ) )
4 psseq1 3426 . . . . . . . 8  |-  ( y  =  A  ->  (
y  C.  x  <->  A  C.  x ) )
54anbi1d 686 . . . . . . 7  |-  ( y  =  A  ->  (
( y  C.  x  /\  x  C.  z )  <-> 
( A  C.  x  /\  x  C.  z ) ) )
65rexbidv 2718 . . . . . 6  |-  ( y  =  A  ->  ( E. x  e.  CH  (
y  C.  x  /\  x  C.  z )  <->  E. x  e.  CH  ( A  C.  x  /\  x  C.  z
) ) )
76notbid 286 . . . . 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 692 . . . 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 692 . . 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 2495 . . . . 5  |-  ( z  =  B  ->  (
z  e.  CH  <->  B  e.  CH ) )
1110anbi2d 685 . . . 4  |-  ( z  =  B  ->  (
( A  e.  CH  /\  z  e.  CH )  <->  ( A  e.  CH  /\  B  e.  CH )
) )
12 psseq2 3427 . . . . 5  |-  ( z  =  B  ->  ( A  C.  z  <->  A  C.  B ) )
13 psseq2 3427 . . . . . . . 8  |-  ( z  =  B  ->  (
x  C.  z  <->  x  C.  B ) )
1413anbi2d 685 . . . . . . 7  |-  ( z  =  B  ->  (
( A  C.  x  /\  x  C.  z )  <-> 
( A  C.  x  /\  x  C.  B ) ) )
1514rexbidv 2718 . . . . . 6  |-  ( z  =  B  ->  ( E. x  e.  CH  ( A  C.  x  /\  x  C.  z )  <->  E. x  e.  CH  ( A  C.  x  /\  x  C.  B
) ) )
1615notbid 286 . . . . 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 692 . . . 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 692 . . 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 23770 . . 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 4466 . 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 851 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 177    /\ wa 359    = wceq 1652    e. wcel 1725   E.wrex 2698    C. wpss 3313   class class class wbr 4204   CHcch 22420    <oH ccv 22455
This theorem is referenced by:  cvbr2  23774  cvcon3  23775  cvpss  23776  cvnbtwn  23777
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-cv 23770
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