Theorem chrelat2

Description: A consequence of relative atomicity. (Contributed by NM, 1-Jul-2004) (New usage is discouraged.)

	Ref	Expression
Assertion	chrelat2	$$⊢\left(A\in {\mathbf{C}}_{\mathscr{H}}\wedge B\in {\mathbf{C}}_{\mathscr{H}}\right)\to \left(\neg A\subseteq B\leftrightarrow \exists x\in \mathrm{HAtoms}\left(x\subseteq A\wedge \neg x\subseteq B\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		sseq1	$$⊢A=if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,\mathscr{H}\right)\to \left(A\subseteq B\leftrightarrow if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,\mathscr{H}\right)\subseteq B\right)$$
2	1	notbid	$$⊢A=if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,\mathscr{H}\right)\to \left(\neg A\subseteq B\leftrightarrow \neg if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,\mathscr{H}\right)\subseteq B\right)$$
3		sseq2	$$⊢A=if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,\mathscr{H}\right)\to \left(x\subseteq A\leftrightarrow x\subseteq if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,\mathscr{H}\right)\right)$$
4	3	anbi1d	$$⊢A=if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,\mathscr{H}\right)\to \left(\left(x\subseteq A\wedge \neg x\subseteq B\right)\leftrightarrow \left(x\subseteq if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,\mathscr{H}\right)\wedge \neg x\subseteq B\right)\right)$$
5	4	rexbidv	$$⊢A=if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,\mathscr{H}\right)\to \left(\exists x\in \mathrm{HAtoms}\left(x\subseteq A\wedge \neg x\subseteq B\right)\leftrightarrow \exists x\in \mathrm{HAtoms}\left(x\subseteq if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,\mathscr{H}\right)\wedge \neg x\subseteq B\right)\right)$$
6	2 5	bibi12d	$$⊢A=if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,\mathscr{H}\right)\to \left(\left(\neg A\subseteq B\leftrightarrow \exists x\in \mathrm{HAtoms}\left(x\subseteq A\wedge \neg x\subseteq B\right)\right)\leftrightarrow \left(\neg if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,\mathscr{H}\right)\subseteq B\leftrightarrow \exists x\in \mathrm{HAtoms}\left(x\subseteq if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,\mathscr{H}\right)\wedge \neg x\subseteq B\right)\right)\right)$$
7		sseq2	$$⊢B=if\left(B\in {\mathbf{C}}_{\mathscr{H}},B,\mathscr{H}\right)\to \left(if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,\mathscr{H}\right)\subseteq B\leftrightarrow if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,\mathscr{H}\right)\subseteq if\left(B\in {\mathbf{C}}_{\mathscr{H}},B,\mathscr{H}\right)\right)$$
8	7	notbid	$$⊢B=if\left(B\in {\mathbf{C}}_{\mathscr{H}},B,\mathscr{H}\right)\to \left(\neg if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,\mathscr{H}\right)\subseteq B\leftrightarrow \neg if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,\mathscr{H}\right)\subseteq if\left(B\in {\mathbf{C}}_{\mathscr{H}},B,\mathscr{H}\right)\right)$$
9		sseq2	$$⊢B=if\left(B\in {\mathbf{C}}_{\mathscr{H}},B,\mathscr{H}\right)\to \left(x\subseteq B\leftrightarrow x\subseteq if\left(B\in {\mathbf{C}}_{\mathscr{H}},B,\mathscr{H}\right)\right)$$
10	9	notbid	$$⊢B=if\left(B\in {\mathbf{C}}_{\mathscr{H}},B,\mathscr{H}\right)\to \left(\neg x\subseteq B\leftrightarrow \neg x\subseteq if\left(B\in {\mathbf{C}}_{\mathscr{H}},B,\mathscr{H}\right)\right)$$
11	10	anbi2d	$$⊢B=if\left(B\in {\mathbf{C}}_{\mathscr{H}},B,\mathscr{H}\right)\to \left(\left(x\subseteq if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,\mathscr{H}\right)\wedge \neg x\subseteq B\right)\leftrightarrow \left(x\subseteq if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,\mathscr{H}\right)\wedge \neg x\subseteq if\left(B\in {\mathbf{C}}_{\mathscr{H}},B,\mathscr{H}\right)\right)\right)$$
12	11	rexbidv	$$⊢B=if\left(B\in {\mathbf{C}}_{\mathscr{H}},B,\mathscr{H}\right)\to \left(\exists x\in \mathrm{HAtoms}\left(x\subseteq if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,\mathscr{H}\right)\wedge \neg x\subseteq B\right)\leftrightarrow \exists x\in \mathrm{HAtoms}\left(x\subseteq if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,\mathscr{H}\right)\wedge \neg x\subseteq if\left(B\in {\mathbf{C}}_{\mathscr{H}},B,\mathscr{H}\right)\right)\right)$$
13	8 12	bibi12d	$$⊢B=if\left(B\in {\mathbf{C}}_{\mathscr{H}},B,\mathscr{H}\right)\to \left(\left(\neg if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,\mathscr{H}\right)\subseteq B\leftrightarrow \exists x\in \mathrm{HAtoms}\left(x\subseteq if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,\mathscr{H}\right)\wedge \neg x\subseteq B\right)\right)\leftrightarrow \left(\neg if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,\mathscr{H}\right)\subseteq if\left(B\in {\mathbf{C}}_{\mathscr{H}},B,\mathscr{H}\right)\leftrightarrow \exists x\in \mathrm{HAtoms}\left(x\subseteq if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,\mathscr{H}\right)\wedge \neg x\subseteq if\left(B\in {\mathbf{C}}_{\mathscr{H}},B,\mathscr{H}\right)\right)\right)\right)$$
14		ifchhv	$$⊢if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,\mathscr{H}\right)\in {\mathbf{C}}_{\mathscr{H}}$$
15		ifchhv	$$⊢if\left(B\in {\mathbf{C}}_{\mathscr{H}},B,\mathscr{H}\right)\in {\mathbf{C}}_{\mathscr{H}}$$
16	14 15	chrelat2i	$$⊢\neg if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,\mathscr{H}\right)\subseteq if\left(B\in {\mathbf{C}}_{\mathscr{H}},B,\mathscr{H}\right)\leftrightarrow \exists x\in \mathrm{HAtoms}\left(x\subseteq if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,\mathscr{H}\right)\wedge \neg x\subseteq if\left(B\in {\mathbf{C}}_{\mathscr{H}},B,\mathscr{H}\right)\right)$$
17	6 13 16	dedth2h	$$⊢\left(A\in {\mathbf{C}}_{\mathscr{H}}\wedge B\in {\mathbf{C}}_{\mathscr{H}}\right)\to \left(\neg A\subseteq B\leftrightarrow \exists x\in \mathrm{HAtoms}\left(x\subseteq A\wedge \neg x\subseteq B\right)\right)$$