Theorem atss

Description: A lattice element smaller than an atom is either the atom or zero. (Contributed by NM, 25-Jun-2004) (New usage is discouraged.)

	Ref	Expression
Assertion	atss	$$⊢\left(A\in {\mathbf{C}}_{\mathscr{H}}\wedge B\in \mathrm{HAtoms}\right)\to \left(A\subseteq B\to \left(A=B\vee A=0_{\mathscr{H}}\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		elat2	$$⊢B\in \mathrm{HAtoms}\leftrightarrow \left(B\in {\mathbf{C}}_{\mathscr{H}}\wedge \left(B\ne 0_{\mathscr{H}}\wedge \forall x\in {\mathbf{C}}_{\mathscr{H}}\left(x\subseteq B\to \left(x=B\vee x=0_{\mathscr{H}}\right)\right)\right)\right)$$
2		sseq1	$$⊢x=A\to \left(x\subseteq B\leftrightarrow A\subseteq B\right)$$
3		eqeq1	$$⊢x=A\to \left(x=B\leftrightarrow A=B\right)$$
4		eqeq1	$$⊢x=A\to \left(x=0_{\mathscr{H}}\leftrightarrow A=0_{\mathscr{H}}\right)$$
5	3 4	orbi12d	$$⊢x=A\to \left(\left(x=B\vee x=0_{\mathscr{H}}\right)\leftrightarrow \left(A=B\vee A=0_{\mathscr{H}}\right)\right)$$
6	2 5	imbi12d	$$⊢x=A\to \left(\left(x\subseteq B\to \left(x=B\vee x=0_{\mathscr{H}}\right)\right)\leftrightarrow \left(A\subseteq B\to \left(A=B\vee A=0_{\mathscr{H}}\right)\right)\right)$$
7	6	rspcv	$$⊢A\in {\mathbf{C}}_{\mathscr{H}}\to \left(\forall x\in {\mathbf{C}}_{\mathscr{H}}\left(x\subseteq B\to \left(x=B\vee x=0_{\mathscr{H}}\right)\right)\to \left(A\subseteq B\to \left(A=B\vee A=0_{\mathscr{H}}\right)\right)\right)$$
8	7	adantld	$$⊢A\in {\mathbf{C}}_{\mathscr{H}}\to \left(\left(B\ne 0_{\mathscr{H}}\wedge \forall x\in {\mathbf{C}}_{\mathscr{H}}\left(x\subseteq B\to \left(x=B\vee x=0_{\mathscr{H}}\right)\right)\right)\to \left(A\subseteq B\to \left(A=B\vee A=0_{\mathscr{H}}\right)\right)\right)$$
9	8	adantld	$$⊢A\in {\mathbf{C}}_{\mathscr{H}}\to \left(\left(B\in {\mathbf{C}}_{\mathscr{H}}\wedge \left(B\ne 0_{\mathscr{H}}\wedge \forall x\in {\mathbf{C}}_{\mathscr{H}}\left(x\subseteq B\to \left(x=B\vee x=0_{\mathscr{H}}\right)\right)\right)\right)\to \left(A\subseteq B\to \left(A=B\vee A=0_{\mathscr{H}}\right)\right)\right)$$
10	9	imp	$$⊢\left(A\in {\mathbf{C}}_{\mathscr{H}}\wedge \left(B\in {\mathbf{C}}_{\mathscr{H}}\wedge \left(B\ne 0_{\mathscr{H}}\wedge \forall x\in {\mathbf{C}}_{\mathscr{H}}\left(x\subseteq B\to \left(x=B\vee x=0_{\mathscr{H}}\right)\right)\right)\right)\right)\to \left(A\subseteq B\to \left(A=B\vee A=0_{\mathscr{H}}\right)\right)$$
11	1 10	sylan2b	$$⊢\left(A\in {\mathbf{C}}_{\mathscr{H}}\wedge B\in \mathrm{HAtoms}\right)\to \left(A\subseteq B\to \left(A=B\vee A=0_{\mathscr{H}}\right)\right)$$