Theorem cvbr2

Description: Binary relation expressing B covers A . Definition of covers in Kalmbach p. 15. (Contributed by NM, 9-Jun-2004) (New usage is discouraged.)

	Ref	Expression
Assertion	cvbr2	$$⊢\left(A\in {\mathbf{C}}_{\mathscr{H}}\wedge B\in {\mathbf{C}}_{\mathscr{H}}\right)\to \left(A{⋖}_{\mathscr{H}}B\leftrightarrow \left(A\subset B\wedge \forall x\in {\mathbf{C}}_{\mathscr{H}}\left(\left(A\subset x\wedge x\subseteq B\right)\to x=B\right)\right)\right)$$

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		iman	$$⊢\left(\left(A\subset x\wedge x\subseteq B\right)\to x=B\right)\leftrightarrow \neg \left(\left(A\subset x\wedge x\subseteq B\right)\wedge \neg x=B\right)$$
3		anass	$$⊢\left(\left(A\subset x\wedge x\subseteq B\right)\wedge \neg x=B\right)\leftrightarrow \left(A\subset x\wedge \left(x\subseteq B\wedge \neg x=B\right)\right)$$
4		dfpss2	$$⊢x\subset B\leftrightarrow \left(x\subseteq B\wedge \neg x=B\right)$$
5	4	anbi2i	$$⊢\left(A\subset x\wedge x\subset B\right)\leftrightarrow \left(A\subset x\wedge \left(x\subseteq B\wedge \neg x=B\right)\right)$$
6	3 5	bitr4i	$$⊢\left(\left(A\subset x\wedge x\subseteq B\right)\wedge \neg x=B\right)\leftrightarrow \left(A\subset x\wedge x\subset B\right)$$
7	2 6	xchbinx	$$⊢\left(\left(A\subset x\wedge x\subseteq B\right)\to x=B\right)\leftrightarrow \neg \left(A\subset x\wedge x\subset B\right)$$
8	7	ralbii	$$⊢\forall x\in {\mathbf{C}}_{\mathscr{H}}\left(\left(A\subset x\wedge x\subseteq B\right)\to x=B\right)\leftrightarrow \forall x\in {\mathbf{C}}_{\mathscr{H}}\neg \left(A\subset x\wedge x\subset B\right)$$
9		ralnex	$$⊢\forall x\in {\mathbf{C}}_{\mathscr{H}}\neg \left(A\subset x\wedge x\subset B\right)\leftrightarrow \neg \exists x\in {\mathbf{C}}_{\mathscr{H}}\left(A\subset x\wedge x\subset B\right)$$
10	8 9	bitri	$$⊢\forall x\in {\mathbf{C}}_{\mathscr{H}}\left(\left(A\subset x\wedge x\subseteq B\right)\to x=B\right)\leftrightarrow \neg \exists x\in {\mathbf{C}}_{\mathscr{H}}\left(A\subset x\wedge x\subset B\right)$$
11	10	anbi2i	$$⊢\left(A\subset B\wedge \forall x\in {\mathbf{C}}_{\mathscr{H}}\left(\left(A\subset x\wedge x\subseteq B\right)\to x=B\right)\right)\leftrightarrow \left(A\subset B\wedge \neg \exists x\in {\mathbf{C}}_{\mathscr{H}}\left(A\subset x\wedge x\subset B\right)\right)$$
12	1 11	bitr4di	$$⊢\left(A\in {\mathbf{C}}_{\mathscr{H}}\wedge B\in {\mathbf{C}}_{\mathscr{H}}\right)\to \left(A{⋖}_{\mathscr{H}}B\leftrightarrow \left(A\subset B\wedge \forall x\in {\mathbf{C}}_{\mathscr{H}}\left(\left(A\subset x\wedge x\subseteq B\right)\to x=B\right)\right)\right)$$