Theorem cvnbtwn 32579

Description: The covers relation implies no in-betweenness. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)

Assertion

Ref	Expression
cvnbtwn	⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) → (𝐴 ⋖ℋ 𝐵 → ¬ (𝐴 ⊊ 𝐶 ∧ 𝐶 ⊊ 𝐵)))

Proof of Theorem cvnbtwn

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cvbr 32575	. . . 4 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 ⋖ℋ 𝐵 ↔ (𝐴 ⊊ 𝐵 ∧ ¬ ∃𝑥 ∈ Cℋ (𝐴 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝐵))))
2		psseq2 4053	. . . . . . . . 9 ⊢ (𝑥 = 𝐶 → (𝐴 ⊊ 𝑥 ↔ 𝐴 ⊊ 𝐶))
3		psseq1 4052	. . . . . . . . 9 ⊢ (𝑥 = 𝐶 → (𝑥 ⊊ 𝐵 ↔ 𝐶 ⊊ 𝐵))
4	2, 3	anbi12d 643	. . . . . . . 8 ⊢ (𝑥 = 𝐶 → ((𝐴 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝐵) ↔ (𝐴 ⊊ 𝐶 ∧ 𝐶 ⊊ 𝐵)))
5	4	rspcev 3590	. . . . . . 7 ⊢ ((𝐶 ∈ Cℋ ∧ (𝐴 ⊊ 𝐶 ∧ 𝐶 ⊊ 𝐵)) → ∃𝑥 ∈ Cℋ (𝐴 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝐵))
6	5	ex 417	. . . . . 6 ⊢ (𝐶 ∈ Cℋ → ((𝐴 ⊊ 𝐶 ∧ 𝐶 ⊊ 𝐵) → ∃𝑥 ∈ Cℋ (𝐴 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝐵)))
7	6	con3rr3 156	. . . . 5 ⊢ (¬ ∃𝑥 ∈ Cℋ (𝐴 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝐵) → (𝐶 ∈ Cℋ → ¬ (𝐴 ⊊ 𝐶 ∧ 𝐶 ⊊ 𝐵)))
8	7	adantl 486	. . . 4 ⊢ ((𝐴 ⊊ 𝐵 ∧ ¬ ∃𝑥 ∈ Cℋ (𝐴 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝐵)) → (𝐶 ∈ Cℋ → ¬ (𝐴 ⊊ 𝐶 ∧ 𝐶 ⊊ 𝐵)))
9	1, 8	biimtrdi 256	. . 3 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 ⋖ℋ 𝐵 → (𝐶 ∈ Cℋ → ¬ (𝐴 ⊊ 𝐶 ∧ 𝐶 ⊊ 𝐵))))
10	9	com23 87	. 2 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐶 ∈ Cℋ → (𝐴 ⋖ℋ 𝐵 → ¬ (𝐴 ⊊ 𝐶 ∧ 𝐶 ⊊ 𝐵))))
11	10	3impia 1133	1 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) → (𝐴 ⋖ℋ 𝐵 → ¬ (𝐴 ⊊ 𝐶 ∧ 𝐶 ⊊ 𝐵)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149  ∃wrex 3095   ⊊ wpss 3914   class class class wbr 5113   C_ℋ cch 31222   ⋖_ℋ ccv 31257

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5261  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5114  df-opab 5178  df-cv 32572

This theorem is referenced by:  cvnbtwn2  32580  cvnbtwn3  32581  cvnbtwn4  32582  cvntr  32585