Theorem cvntr

Description: The covers relation is not transitive. (Contributed by NM, 26-Jun-2004) (New usage is discouraged.)

	Ref	Expression
Assertion	cvntr	$$⊢\left(A\in {\mathbf{C}}_{\mathscr{H}}\wedge B\in {\mathbf{C}}_{\mathscr{H}}\wedge C\in {\mathbf{C}}_{\mathscr{H}}\right)\to \left(\left(A{⋖}_{\mathscr{H}}B\wedge B{⋖}_{\mathscr{H}}C\right)\to \neg A{⋖}_{\mathscr{H}}C\right)$$

Proof

Step	Hyp	Ref	Expression
1		cvpss	$$⊢\left(A\in {\mathbf{C}}_{\mathscr{H}}\wedge B\in {\mathbf{C}}_{\mathscr{H}}\right)\to \left(A{⋖}_{\mathscr{H}}B\to A\subset B\right)$$
2	1	3adant3	$$⊢\left(A\in {\mathbf{C}}_{\mathscr{H}}\wedge B\in {\mathbf{C}}_{\mathscr{H}}\wedge C\in {\mathbf{C}}_{\mathscr{H}}\right)\to \left(A{⋖}_{\mathscr{H}}B\to A\subset B\right)$$
3		cvpss	$$⊢\left(B\in {\mathbf{C}}_{\mathscr{H}}\wedge C\in {\mathbf{C}}_{\mathscr{H}}\right)\to \left(B{⋖}_{\mathscr{H}}C\to B\subset C\right)$$
4	3	3adant1	$$⊢\left(A\in {\mathbf{C}}_{\mathscr{H}}\wedge B\in {\mathbf{C}}_{\mathscr{H}}\wedge C\in {\mathbf{C}}_{\mathscr{H}}\right)\to \left(B{⋖}_{\mathscr{H}}C\to B\subset C\right)$$
5		cvnbtwn	$$⊢\left(A\in {\mathbf{C}}_{\mathscr{H}}\wedge C\in {\mathbf{C}}_{\mathscr{H}}\wedge B\in {\mathbf{C}}_{\mathscr{H}}\right)\to \left(A{⋖}_{\mathscr{H}}C\to \neg \left(A\subset B\wedge B\subset C\right)\right)$$
6	5	3com23	$$⊢\left(A\in {\mathbf{C}}_{\mathscr{H}}\wedge B\in {\mathbf{C}}_{\mathscr{H}}\wedge C\in {\mathbf{C}}_{\mathscr{H}}\right)\to \left(A{⋖}_{\mathscr{H}}C\to \neg \left(A\subset B\wedge B\subset C\right)\right)$$
7	6	con2d	$$⊢\left(A\in {\mathbf{C}}_{\mathscr{H}}\wedge B\in {\mathbf{C}}_{\mathscr{H}}\wedge C\in {\mathbf{C}}_{\mathscr{H}}\right)\to \left(\left(A\subset B\wedge B\subset C\right)\to \neg A{⋖}_{\mathscr{H}}C\right)$$
8	2 4 7	syl2and	$$⊢\left(A\in {\mathbf{C}}_{\mathscr{H}}\wedge B\in {\mathbf{C}}_{\mathscr{H}}\wedge C\in {\mathbf{C}}_{\mathscr{H}}\right)\to \left(\left(A{⋖}_{\mathscr{H}}B\wedge B{⋖}_{\mathscr{H}}C\right)\to \neg A{⋖}_{\mathscr{H}}C\right)$$