Theorem cvnbtwn 32375

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 32371	. . . 4 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 ⋖ℋ 𝐵 ↔ (𝐴 ⊊ 𝐵 ∧ ¬ ∃𝑥 ∈ Cℋ (𝐴 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝐵))))
2		psseq2 4022	. . . . . . . . 9 ⊢ (𝑥 = 𝐶 → (𝐴 ⊊ 𝑥 ↔ 𝐴 ⊊ 𝐶))
3		psseq1 4021	. . . . . . . . 9 ⊢ (𝑥 = 𝐶 → (𝑥 ⊊ 𝐵 ↔ 𝐶 ⊊ 𝐵))
4	2, 3	anbi12d 638	. . . . . . . 8 ⊢ (𝑥 = 𝐶 → ((𝐴 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝐵) ↔ (𝐴 ⊊ 𝐶 ∧ 𝐶 ⊊ 𝐵)))
5	4	rspcev 3560	. . . . . . 7 ⊢ ((𝐶 ∈ Cℋ ∧ (𝐴 ⊊ 𝐶 ∧ 𝐶 ⊊ 𝐵)) → ∃𝑥 ∈ Cℋ (𝐴 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝐵))
6	5	ex 413	. . . . . 6 ⊢ (𝐶 ∈ Cℋ → ((𝐴 ⊊ 𝐶 ∧ 𝐶 ⊊ 𝐵) → ∃𝑥 ∈ Cℋ (𝐴 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝐵)))
7	6	con3rr3 155	. . . . 5 ⊢ (¬ ∃𝑥 ∈ Cℋ (𝐴 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝐵) → (𝐶 ∈ Cℋ → ¬ (𝐴 ⊊ 𝐶 ∧ 𝐶 ⊊ 𝐵)))
8	7	adantl 482	. . . 4 ⊢ ((𝐴 ⊊ 𝐵 ∧ ¬ ∃𝑥 ∈ Cℋ (𝐴 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝐵)) → (𝐶 ∈ Cℋ → ¬ (𝐴 ⊊ 𝐶 ∧ 𝐶 ⊊ 𝐵)))
9	1, 8	biimtrdi 254	. . 3 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 ⋖ℋ 𝐵 → (𝐶 ∈ Cℋ → ¬ (𝐴 ⊊ 𝐶 ∧ 𝐶 ⊊ 𝐵))))
10	9	com23 86	. 2 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐶 ∈ Cℋ → (𝐴 ⋖ℋ 𝐵 → ¬ (𝐴 ⊊ 𝐶 ∧ 𝐶 ⊊ 𝐵))))
11	10	3impia 1123	1 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) → (𝐴 ⋖ℋ 𝐵 → ¬ (𝐴 ⊊ 𝐶 ∧ 𝐶 ⊊ 𝐵)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119  ∃wrex 3063   ⊊ wpss 3884   class class class wbr 5072   C_ℋ cch 31018   ⋖_ℋ ccv 31053

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 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711  ax-sep 5218  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-br 5073  df-opab 5135  df-cv 32368

This theorem is referenced by:  cvnbtwn2  32376  cvnbtwn3  32377  cvnbtwn4  32378  cvntr  32381