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Theorem cvbr2 23739
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 23738 . 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 414 . . . . . 6  |-  ( ( ( A  C.  x  /\  x  C_  B )  ->  x  =  B )  <->  -.  ( ( A  C.  x  /\  x  C_  B )  /\  -.  x  =  B )
)
3 anass 631 . . . . . . 7  |-  ( ( ( A  C.  x  /\  x  C_  B )  /\  -.  x  =  B )  <->  ( A  C.  x  /\  (
x  C_  B  /\  -.  x  =  B
) ) )
4 dfpss2 3392 . . . . . . . 8  |-  ( x 
C.  B  <->  ( x  C_  B  /\  -.  x  =  B ) )
54anbi2i 676 . . . . . . 7  |-  ( ( A  C.  x  /\  x  C.  B )  <->  ( A  C.  x  /\  (
x  C_  B  /\  -.  x  =  B
) ) )
63, 5bitr4i 244 . . . . . 6  |-  ( ( ( A  C.  x  /\  x  C_  B )  /\  -.  x  =  B )  <->  ( A  C.  x  /\  x  C.  B ) )
72, 6xchbinx 302 . . . . 5  |-  ( ( ( A  C.  x  /\  x  C_  B )  ->  x  =  B )  <->  -.  ( A  C.  x  /\  x  C.  B ) )
87ralbii 2690 . . . 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 2676 . . . 4  |-  ( A. x  e.  CH  -.  ( A  C.  x  /\  x  C.  B )  <->  -.  E. x  e.  CH  ( A  C.  x  /\  x  C.  B
) )
108, 9bitri 241 . . 3  |-  ( A. x  e.  CH  ( ( A  C.  x  /\  x  C_  B )  ->  x  =  B )  <->  -. 
E. x  e.  CH  ( A  C.  x  /\  x  C.  B ) )
1110anbi2i 676 . 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 255 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 3    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721   A.wral 2666   E.wrex 2667    C_ wss 3280    C. wpss 3281   class class class wbr 4172   CHcch 22385    <oH ccv 22420
This theorem is referenced by:  spansncv2  23749  elat2  23796
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-br 4173  df-opab 4227  df-cv 23735
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