Theorem cvnbtwnt 5718

Description: The covering relation implies no in-betweenness.

Assertion

Ref	Expression
cvnbtwnt	⊢ ((A ∈ Cℋ ∧ B ∈ Cℋ ∧ C ∈ Cℋ ) → (A ⋖ B → ¬ (A ⊂ C ∧ C ⊂ B)))

Proof of Theorem cvnbtwnt

Step	Hyp	Ref	Expression
1		cvbrt 5714	. . . . 5 ⊢ ((A ∈ Cℋ ∧ B ∈ Cℋ ) → (A ⋖ B ↔ (A ⊂ B ∧ ¬ ∃x ∈ Cℋ (A ⊂ x ∧ x ⊂ B))))
2		psseq2 1560	. . . . . . . . . . 11 ⊢ (x = C → (A ⊂ x ↔ A ⊂ C))
3		psseq1 1559	. . . . . . . . . . 11 ⊢ (x = C → (x ⊂ B ↔ C ⊂ B))
4	2, 3	anbi12d 476	. . . . . . . . . 10 ⊢ (x = C → ((A ⊂ x ∧ x ⊂ B) ↔ (A ⊂ C ∧ C ⊂ B)))
5	4	rcla4ev 1403	. . . . . . . . 9 ⊢ ((C ∈ Cℋ ∧ (A ⊂ C ∧ C ⊂ B)) → ∃x ∈ Cℋ (A ⊂ x ∧ x ⊂ B))
6	5	exp 291	. . . . . . . 8 ⊢ (C ∈ Cℋ → ((A ⊂ C ∧ C ⊂ B) → ∃x ∈ Cℋ (A ⊂ x ∧ x ⊂ B)))
7	6	con3d 87	. . . . . . 7 ⊢ (C ∈ Cℋ → (¬ ∃x ∈ Cℋ (A ⊂ x ∧ x ⊂ B) → ¬ (A ⊂ C ∧ C ⊂ B)))
8	7	com12 13	. . . . . 6 ⊢ (¬ ∃x ∈ Cℋ (A ⊂ x ∧ x ⊂ B) → (C ∈ Cℋ → ¬ (A ⊂ C ∧ C ⊂ B)))
9	8	adantl 305	. . . . 5 ⊢ ((A ⊂ B ∧ ¬ ∃x ∈ Cℋ (A ⊂ x ∧ x ⊂ B)) → (C ∈ Cℋ → ¬ (A ⊂ C ∧ C ⊂ B)))
10	1, 9	syl6bi 187	. . . 4 ⊢ ((A ∈ Cℋ ∧ B ∈ Cℋ ) → (A ⋖ B → (C ∈ Cℋ → ¬ (A ⊂ C ∧ C ⊂ B))))
11	10	com23 32	. . 3 ⊢ ((A ∈ Cℋ ∧ B ∈ Cℋ ) → (C ∈ Cℋ → (A ⋖ B → ¬ (A ⊂ C ∧ C ⊂ B))))
12	11	imp 277	. 2 ⊢ (((A ∈ Cℋ ∧ B ∈ Cℋ ) ∧ C ∈ Cℋ ) → (A ⋖ B → ¬ (A ⊂ C ∧ C ⊂ B)))
13	12	3impa 609	1 ⊢ ((A ∈ Cℋ ∧ B ∈ Cℋ ∧ C ∈ Cℋ ) → (A ⋖ B → ¬ (A ⊂ C ∧ C ⊂ B)))

Syntax hints:  ¬ wn 1   → wi 2   ∧ wa 196   ∧ w3a 581   = wceq 1091   ∈ wcel 1092  ∃wrex 1202   ⊂ wpss 1488   class class class wbr 2054   C_ℋ cch 4968   ⋖ ccv 4981

This theorem is referenced by:  cvnbtwn2t 5719  cvnbtwn3t 5720  cvnbtwn4t 5721  cvntrt 5724

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3an 583  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ne 1192  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-pss 1494  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-opab 2098  df-cv 5712

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