Theorem cvpss

Description: The covers relation implies proper subset. (Contributed by NM, 10-Jun-2004) (New usage is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		cvbr	$$⊢\left(A\in {\mathbf{C}}_{\mathscr{H}}\wedge B\in {\mathbf{C}}_{\mathscr{H}}\right)\to \left(A{⋖}_{\mathscr{H}}B\leftrightarrow \left(A\subset B\wedge \neg \exists x\in {\mathbf{C}}_{\mathscr{H}}\left(A\subset x\wedge x\subset B\right)\right)\right)$$
2		simpl	$$⊢\left(A\subset B\wedge \neg \exists x\in {\mathbf{C}}_{\mathscr{H}}\left(A\subset x\wedge x\subset B\right)\right)\to A\subset B$$
3	1 2	biimtrdi	$$⊢\left(A\in {\mathbf{C}}_{\mathscr{H}}\wedge B\in {\mathbf{C}}_{\mathscr{H}}\right)\to \left(A{⋖}_{\mathscr{H}}B\to A\subset B\right)$$