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Theorem cvnbtwn3 29147
Description: The covers relation implies no in-betweenness. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)
Assertion
Ref Expression
cvnbtwn3 |- ( ( A e. CH /\ B e. CH /\ C e. CH ) -> ( A <oH B -> ( ( A C_ C /\ C C. B ) -> C = A ) ) )
Proof of Theorem cvnbtwn3
StepHypRef Expression
1 cvnbtwn 29145 . 2 |- ( ( A e. CH /\ B e. CH /\ C e. CH ) -> ( A <oH B -> -. ( A C. C /\ C C. B ) ) )
2 iman 440 . . 3 |- ( ( ( A C_ C /\ C C. B ) -> A = C ) <-> -. ( ( A C_ C /\ C C. B ) /\ -. A = C ) )
3 eqcom 2629 . . . 4 |- ( C = A <-> A = C )
43imbi2i 326 . . 3 |- ( ( ( A C_ C /\ C C. B ) -> C = A ) <-> ( ( A C_ C /\ C C. B ) -> A = C ) )
5 dfpss2 3692 . . . . . 6 |- ( A C. C <-> ( A C_ C /\ -. A = C ) )
65anbi1i 731 . . . . 5 |- ( ( A C. C /\ C C. B ) <-> ( ( A C_ C /\ -. A = C ) /\ C C. B ) )
7 an32 839 . . . . 5 |- ( ( ( A C_ C /\ -. A = C ) /\ C C. B ) <-> ( ( A C_ C /\ C C. B ) /\ -. A = C ) )
86, 7bitri 264 . . . 4 |- ( ( A C. C /\ C C. B ) <-> ( ( A C_ C /\ C C. B ) /\ -. A = C ) )
98notbii 310 . . 3 |- ( -. ( A C. C /\ C C. B ) <-> -. ( ( A C_ C /\ C C. B ) /\ -. A = C ) )
102, 4, 93bitr4ri 293 . 2 |- ( -. ( A C. C /\ C C. B ) <-> ( ( A C_ C /\ C C. B ) -> C = A ) )
111, 10syl6ib 241 1 |- ( ( A e. CH /\ B e. CH /\ C e. CH ) -> ( A <oH B -> ( ( A C_ C /\ C C. B ) -> C = A ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   C_ wss 3574   C. wpss 3575   class class class wbr 4653   CHcch 27786   <oH ccv 27821
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-cv 29138
This theorem is referenced by:  atcveq0  29207  atcvatlem  29244
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