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Theorem cvpss 23741
Description: The covers relation implies proper subset. (Contributed by NM, 10-Jun-2004.) (New usage is discouraged.)
Assertion
Ref Expression
cvpss  |-  ( ( A  e.  CH  /\  B  e.  CH )  ->  ( A  <oH  B  ->  A  C.  B ) )

Proof of Theorem cvpss
Dummy variable  x is distinct from all other variables.
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 simpl 444 . 2  |-  ( ( A  C.  B  /\  -.  E. x  e.  CH  ( A  C.  x  /\  x  C.  B ) )  ->  A  C.  B
)
31, 2syl6bi 220 1  |-  ( ( A  e.  CH  /\  B  e.  CH )  ->  ( A  <oH  B  ->  A  C.  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    e. wcel 1721   E.wrex 2667    C. wpss 3281   class class class wbr 4172   CHcch 22385    <oH ccv 22420
This theorem is referenced by:  cvnsym  23746  cvntr  23748  atcveq0  23804  chcv1  23811  cvati  23822  cvbr4i  23823  cvexchlem  23824  atexch  23837  atcvat2i  23843
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-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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