Theorem chnle

Description: Equivalent expressions for "not less than" in the Hilbert lattice. (Contributed by NM, 9-Jun-2004) (New usage is discouraged.)

	Ref	Expression
Assertion	chnle	$$⊢\left(A\in {\mathbf{C}}_{\mathscr{H}}\wedge B\in {\mathbf{C}}_{\mathscr{H}}\right)\to \left(\neg B\subseteq A\leftrightarrow A\subset A{\vee }_{\mathscr{H}}B\right)$$

Proof

Step	Hyp	Ref	Expression
1		sseq2	$$⊢A=if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,0_{\mathscr{H}}\right)\to \left(B\subseteq A\leftrightarrow B\subseteq if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,0_{\mathscr{H}}\right)\right)$$
2	1	notbid	$$⊢A=if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,0_{\mathscr{H}}\right)\to \left(\neg B\subseteq A\leftrightarrow \neg B\subseteq if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,0_{\mathscr{H}}\right)\right)$$
3		id	$$⊢A=if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,0_{\mathscr{H}}\right)\to A=if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,0_{\mathscr{H}}\right)$$
4		oveq1	$$⊢A=if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,0_{\mathscr{H}}\right)\to A{\vee }_{\mathscr{H}}B=if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,0_{\mathscr{H}}\right){\vee }_{\mathscr{H}}B$$
5	3 4	psseq12d	$$⊢A=if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,0_{\mathscr{H}}\right)\to \left(A\subset A{\vee }_{\mathscr{H}}B\leftrightarrow if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,0_{\mathscr{H}}\right)\subset if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,0_{\mathscr{H}}\right){\vee }_{\mathscr{H}}B\right)$$
6	2 5	bibi12d	$$⊢A=if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,0_{\mathscr{H}}\right)\to \left(\left(\neg B\subseteq A\leftrightarrow A\subset A{\vee }_{\mathscr{H}}B\right)\leftrightarrow \left(\neg B\subseteq if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,0_{\mathscr{H}}\right)\leftrightarrow if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,0_{\mathscr{H}}\right)\subset if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,0_{\mathscr{H}}\right){\vee }_{\mathscr{H}}B\right)\right)$$
7		sseq1	$$⊢B=if\left(B\in {\mathbf{C}}_{\mathscr{H}},B,0_{\mathscr{H}}\right)\to \left(B\subseteq if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,0_{\mathscr{H}}\right)\leftrightarrow if\left(B\in {\mathbf{C}}_{\mathscr{H}},B,0_{\mathscr{H}}\right)\subseteq if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,0_{\mathscr{H}}\right)\right)$$
8	7	notbid	$$⊢B=if\left(B\in {\mathbf{C}}_{\mathscr{H}},B,0_{\mathscr{H}}\right)\to \left(\neg B\subseteq if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,0_{\mathscr{H}}\right)\leftrightarrow \neg if\left(B\in {\mathbf{C}}_{\mathscr{H}},B,0_{\mathscr{H}}\right)\subseteq if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,0_{\mathscr{H}}\right)\right)$$
9		oveq2	$$⊢B=if\left(B\in {\mathbf{C}}_{\mathscr{H}},B,0_{\mathscr{H}}\right)\to if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,0_{\mathscr{H}}\right){\vee }_{\mathscr{H}}B=if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,0_{\mathscr{H}}\right){\vee }_{\mathscr{H}}if\left(B\in {\mathbf{C}}_{\mathscr{H}},B,0_{\mathscr{H}}\right)$$
10	9	psseq2d	$$⊢B=if\left(B\in {\mathbf{C}}_{\mathscr{H}},B,0_{\mathscr{H}}\right)\to \left(if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,0_{\mathscr{H}}\right)\subset if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,0_{\mathscr{H}}\right){\vee }_{\mathscr{H}}B\leftrightarrow if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,0_{\mathscr{H}}\right)\subset if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,0_{\mathscr{H}}\right){\vee }_{\mathscr{H}}if\left(B\in {\mathbf{C}}_{\mathscr{H}},B,0_{\mathscr{H}}\right)\right)$$
11	8 10	bibi12d	$$⊢B=if\left(B\in {\mathbf{C}}_{\mathscr{H}},B,0_{\mathscr{H}}\right)\to \left(\left(\neg B\subseteq if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,0_{\mathscr{H}}\right)\leftrightarrow if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,0_{\mathscr{H}}\right)\subset if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,0_{\mathscr{H}}\right){\vee }_{\mathscr{H}}B\right)\leftrightarrow \left(\neg if\left(B\in {\mathbf{C}}_{\mathscr{H}},B,0_{\mathscr{H}}\right)\subseteq if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,0_{\mathscr{H}}\right)\leftrightarrow if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,0_{\mathscr{H}}\right)\subset if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,0_{\mathscr{H}}\right){\vee }_{\mathscr{H}}if\left(B\in {\mathbf{C}}_{\mathscr{H}},B,0_{\mathscr{H}}\right)\right)\right)$$
12		h0elch	$$⊢0_{\mathscr{H}}\in {\mathbf{C}}_{\mathscr{H}}$$
13	12	elimel	$$⊢if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,0_{\mathscr{H}}\right)\in {\mathbf{C}}_{\mathscr{H}}$$
14	12	elimel	$$⊢if\left(B\in {\mathbf{C}}_{\mathscr{H}},B,0_{\mathscr{H}}\right)\in {\mathbf{C}}_{\mathscr{H}}$$
15	13 14	chnlei	$$⊢\neg if\left(B\in {\mathbf{C}}_{\mathscr{H}},B,0_{\mathscr{H}}\right)\subseteq if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,0_{\mathscr{H}}\right)\leftrightarrow if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,0_{\mathscr{H}}\right)\subset if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,0_{\mathscr{H}}\right){\vee }_{\mathscr{H}}if\left(B\in {\mathbf{C}}_{\mathscr{H}},B,0_{\mathscr{H}}\right)$$
16	6 11 15	dedth2h	$$⊢\left(A\in {\mathbf{C}}_{\mathscr{H}}\wedge B\in {\mathbf{C}}_{\mathscr{H}}\right)\to \left(\neg B\subseteq A\leftrightarrow A\subset A{\vee }_{\mathscr{H}}B\right)$$