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Theorem cvnbtwn 30221
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 30217 . . . 4 ((𝐴C𝐵C ) → (𝐴 𝐵 ↔ (𝐴𝐵 ∧ ¬ ∃𝑥C (𝐴𝑥𝑥𝐵))))
2 psseq2 3979 . . . . . . . . 9 (𝑥 = 𝐶 → (𝐴𝑥𝐴𝐶))
3 psseq1 3978 . . . . . . . . 9 (𝑥 = 𝐶 → (𝑥𝐵𝐶𝐵))
42, 3anbi12d 634 . . . . . . . 8 (𝑥 = 𝐶 → ((𝐴𝑥𝑥𝐵) ↔ (𝐴𝐶𝐶𝐵)))
54rspcev 3526 . . . . . . 7 ((𝐶C ∧ (𝐴𝐶𝐶𝐵)) → ∃𝑥C (𝐴𝑥𝑥𝐵))
65ex 416 . . . . . 6 (𝐶C → ((𝐴𝐶𝐶𝐵) → ∃𝑥C (𝐴𝑥𝑥𝐵)))
76con3rr3 158 . . . . 5 (¬ ∃𝑥C (𝐴𝑥𝑥𝐵) → (𝐶C → ¬ (𝐴𝐶𝐶𝐵)))
87adantl 485 . . . 4 ((𝐴𝐵 ∧ ¬ ∃𝑥C (𝐴𝑥𝑥𝐵)) → (𝐶C → ¬ (𝐴𝐶𝐶𝐵)))
91, 8syl6bi 256 . . 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 399  w3a 1088   = wceq 1542  wcel 2114  wrex 3054  wpss 3844   class class class wbr 5030   C cch 28864   ccv 28899
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pr 5296
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2075  df-clab 2717  df-cleq 2730  df-clel 2811  df-ne 2935  df-ral 3058  df-rex 3059  df-v 3400  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-pss 3862  df-nul 4212  df-if 4415  df-sn 4517  df-pr 4519  df-op 4523  df-br 5031  df-opab 5093  df-cv 30214
This theorem is referenced by:  cvnbtwn2  30222  cvnbtwn3  30223  cvnbtwn4  30224  cvntr  30227
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