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Theorem cvnbtwn 31806
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 31802 . . . 4 ((𝐴C𝐵C ) → (𝐴 𝐵 ↔ (𝐴𝐵 ∧ ¬ ∃𝑥C (𝐴𝑥𝑥𝐵))))
2 psseq2 4087 . . . . . . . . 9 (𝑥 = 𝐶 → (𝐴𝑥𝐴𝐶))
3 psseq1 4086 . . . . . . . . 9 (𝑥 = 𝐶 → (𝑥𝐵𝐶𝐵))
42, 3anbi12d 629 . . . . . . . 8 (𝑥 = 𝐶 → ((𝐴𝑥𝑥𝐵) ↔ (𝐴𝐶𝐶𝐵)))
54rspcev 3611 . . . . . . 7 ((𝐶C ∧ (𝐴𝐶𝐶𝐵)) → ∃𝑥C (𝐴𝑥𝑥𝐵))
65ex 411 . . . . . 6 (𝐶C → ((𝐴𝐶𝐶𝐵) → ∃𝑥C (𝐴𝑥𝑥𝐵)))
76con3rr3 155 . . . . 5 (¬ ∃𝑥C (𝐴𝑥𝑥𝐵) → (𝐶C → ¬ (𝐴𝐶𝐶𝐵)))
87adantl 480 . . . 4 ((𝐴𝐵 ∧ ¬ ∃𝑥C (𝐴𝑥𝑥𝐵)) → (𝐶C → ¬ (𝐴𝐶𝐶𝐵)))
91, 8syl6bi 252 . . 3 ((𝐴C𝐵C ) → (𝐴 𝐵 → (𝐶C → ¬ (𝐴𝐶𝐶𝐵))))
109com23 86 . 2 ((𝐴C𝐵C ) → (𝐶C → (𝐴 𝐵 → ¬ (𝐴𝐶𝐶𝐵))))
11103impia 1115 1 ((𝐴C𝐵C𝐶C ) → (𝐴 𝐵 → ¬ (𝐴𝐶𝐶𝐵)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 394  w3a 1085   = wceq 1539  wcel 2104  wrex 3068  wpss 3948   class class class wbr 5147   C cch 30449   ccv 30484
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-br 5148  df-opab 5210  df-cv 31799
This theorem is referenced by:  cvnbtwn2  31807  cvnbtwn3  31808  cvnbtwn4  31809  cvntr  31812
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