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Theorem cvnbtwn2 30649
Description: The covers relation implies no in-betweenness. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)
Assertion
Ref Expression
cvnbtwn2 ((𝐴C𝐵C𝐶C ) → (𝐴 𝐵 → ((𝐴𝐶𝐶𝐵) → 𝐶 = 𝐵)))

Proof of Theorem cvnbtwn2
StepHypRef Expression
1 cvnbtwn 30648 . 2 ((𝐴C𝐵C𝐶C ) → (𝐴 𝐵 → ¬ (𝐴𝐶𝐶𝐵)))
2 iman 402 . . 3 (((𝐴𝐶𝐶𝐵) → 𝐶 = 𝐵) ↔ ¬ ((𝐴𝐶𝐶𝐵) ∧ ¬ 𝐶 = 𝐵))
3 anass 469 . . . . 5 (((𝐴𝐶𝐶𝐵) ∧ ¬ 𝐶 = 𝐵) ↔ (𝐴𝐶 ∧ (𝐶𝐵 ∧ ¬ 𝐶 = 𝐵)))
4 dfpss2 4020 . . . . . 6 (𝐶𝐵 ↔ (𝐶𝐵 ∧ ¬ 𝐶 = 𝐵))
54anbi2i 623 . . . . 5 ((𝐴𝐶𝐶𝐵) ↔ (𝐴𝐶 ∧ (𝐶𝐵 ∧ ¬ 𝐶 = 𝐵)))
63, 5bitr4i 277 . . . 4 (((𝐴𝐶𝐶𝐵) ∧ ¬ 𝐶 = 𝐵) ↔ (𝐴𝐶𝐶𝐵))
76notbii 320 . . 3 (¬ ((𝐴𝐶𝐶𝐵) ∧ ¬ 𝐶 = 𝐵) ↔ ¬ (𝐴𝐶𝐶𝐵))
82, 7bitr2i 275 . 2 (¬ (𝐴𝐶𝐶𝐵) ↔ ((𝐴𝐶𝐶𝐵) → 𝐶 = 𝐵))
91, 8syl6ib 250 1 ((𝐴C𝐵C𝐶C ) → (𝐴 𝐵 → ((𝐴𝐶𝐶𝐵) → 𝐶 = 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  w3a 1086   = wceq 1539  wcel 2106  wss 3887  wpss 3888   class class class wbr 5074   C cch 29291   ccv 29326
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-cv 30641
This theorem is referenced by:  cvati  30728  cvexchlem  30730  atexch  30743
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