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Theorem cvnbtwn 31539
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 31535 . . . 4 ((𝐴C𝐵C ) → (𝐴 𝐵 ↔ (𝐴𝐵 ∧ ¬ ∃𝑥C (𝐴𝑥𝑥𝐵))))
2 psseq2 4089 . . . . . . . . 9 (𝑥 = 𝐶 → (𝐴𝑥𝐴𝐶))
3 psseq1 4088 . . . . . . . . 9 (𝑥 = 𝐶 → (𝑥𝐵𝐶𝐵))
42, 3anbi12d 632 . . . . . . . 8 (𝑥 = 𝐶 → ((𝐴𝑥𝑥𝐵) ↔ (𝐴𝐶𝐶𝐵)))
54rspcev 3613 . . . . . . 7 ((𝐶C ∧ (𝐴𝐶𝐶𝐵)) → ∃𝑥C (𝐴𝑥𝑥𝐵))
65ex 414 . . . . . 6 (𝐶C → ((𝐴𝐶𝐶𝐵) → ∃𝑥C (𝐴𝑥𝑥𝐵)))
76con3rr3 155 . . . . 5 (¬ ∃𝑥C (𝐴𝑥𝑥𝐵) → (𝐶C → ¬ (𝐴𝐶𝐶𝐵)))
87adantl 483 . . . 4 ((𝐴𝐵 ∧ ¬ ∃𝑥C (𝐴𝑥𝑥𝐵)) → (𝐶C → ¬ (𝐴𝐶𝐶𝐵)))
91, 8syl6bi 253 . . 3 ((𝐴C𝐵C ) → (𝐴 𝐵 → (𝐶C → ¬ (𝐴𝐶𝐶𝐵))))
109com23 86 . 2 ((𝐴C𝐵C ) → (𝐶C → (𝐴 𝐵 → ¬ (𝐴𝐶𝐶𝐵))))
11103impia 1118 1 ((𝐴C𝐵C𝐶C ) → (𝐴 𝐵 → ¬ (𝐴𝐶𝐶𝐵)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 397  w3a 1088   = wceq 1542  wcel 2107  wrex 3071  wpss 3950   class class class wbr 5149   C cch 30182   ccv 30217
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-cv 31532
This theorem is referenced by:  cvnbtwn2  31540  cvnbtwn3  31541  cvnbtwn4  31542  cvntr  31545
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