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Theorem cvnbtwn 29990
Description: The covers relation implies no in-betweenness. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)
Assertion
Ref Expression
cvnbtwn ((𝐴C𝐵C𝐶C ) → (𝐴 𝐵 → ¬ (𝐴𝐶𝐶𝐵)))

Proof of Theorem cvnbtwn
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 cvbr 29986 . . . 4 ((𝐴C𝐵C ) → (𝐴 𝐵 ↔ (𝐴𝐵 ∧ ¬ ∃𝑥C (𝐴𝑥𝑥𝐵))))
2 psseq2 4062 . . . . . . . . 9 (𝑥 = 𝐶 → (𝐴𝑥𝐴𝐶))
3 psseq1 4061 . . . . . . . . 9 (𝑥 = 𝐶 → (𝑥𝐵𝐶𝐵))
42, 3anbi12d 630 . . . . . . . 8 (𝑥 = 𝐶 → ((𝐴𝑥𝑥𝐵) ↔ (𝐴𝐶𝐶𝐵)))
54rspcev 3620 . . . . . . 7 ((𝐶C ∧ (𝐴𝐶𝐶𝐵)) → ∃𝑥C (𝐴𝑥𝑥𝐵))
65ex 413 . . . . . 6 (𝐶C → ((𝐴𝐶𝐶𝐵) → ∃𝑥C (𝐴𝑥𝑥𝐵)))
76con3rr3 158 . . . . 5 (¬ ∃𝑥C (𝐴𝑥𝑥𝐵) → (𝐶C → ¬ (𝐴𝐶𝐶𝐵)))
87adantl 482 . . . 4 ((𝐴𝐵 ∧ ¬ ∃𝑥C (𝐴𝑥𝑥𝐵)) → (𝐶C → ¬ (𝐴𝐶𝐶𝐵)))
91, 8syl6bi 254 . . 3 ((𝐴C𝐵C ) → (𝐴 𝐵 → (𝐶C → ¬ (𝐴𝐶𝐶𝐵))))
109com23 86 . 2 ((𝐴C𝐵C ) → (𝐶C → (𝐴 𝐵 → ¬ (𝐴𝐶𝐶𝐵))))
11103impia 1109 1 ((𝐴C𝐵C𝐶C ) → (𝐴 𝐵 → ¬ (𝐴𝐶𝐶𝐵)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  w3a 1079   = wceq 1528  wcel 2105  wrex 3136  wpss 3934   class class class wbr 5057   C cch 28633   ccv 28668
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-opab 5120  df-cv 29983
This theorem is referenced by:  cvnbtwn2  29991  cvnbtwn3  29992  cvnbtwn4  29993  cvntr  29996
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