Theorem chrelat4i

Description: A consequence of relative atomicity. Extensionality principle: two lattice elements are equal iff they majorize the same atoms. (Contributed by NM, 30-Jun-2004) (New usage is discouraged.)

	Ref	Expression
Hypotheses	chrelat3.1	$$⊢A\in {\mathbf{C}}_{\mathscr{H}}$$
	chrelat3.2	$$⊢B\in {\mathbf{C}}_{\mathscr{H}}$$
Assertion	chrelat4i	$$⊢A=B\leftrightarrow \forall x\in \mathrm{HAtoms}\left(x\subseteq A\leftrightarrow x\subseteq B\right)$$

Proof

Step	Hyp	Ref	Expression
1		chrelat3.1	$$⊢A\in {\mathbf{C}}_{\mathscr{H}}$$
2		chrelat3.2	$$⊢B\in {\mathbf{C}}_{\mathscr{H}}$$
3	1 2	chrelat3i	$$⊢A\subseteq B\leftrightarrow \forall x\in \mathrm{HAtoms}\left(x\subseteq A\to x\subseteq B\right)$$
4	2 1	chrelat3i	$$⊢B\subseteq A\leftrightarrow \forall x\in \mathrm{HAtoms}\left(x\subseteq B\to x\subseteq A\right)$$
5	3 4	anbi12i	$$⊢\left(A\subseteq B\wedge B\subseteq A\right)\leftrightarrow \left(\forall x\in \mathrm{HAtoms}\left(x\subseteq A\to x\subseteq B\right)\wedge \forall x\in \mathrm{HAtoms}\left(x\subseteq B\to x\subseteq A\right)\right)$$
6		eqss	$$⊢A=B\leftrightarrow \left(A\subseteq B\wedge B\subseteq A\right)$$
7		ralbiim	$$⊢\forall x\in \mathrm{HAtoms}\left(x\subseteq A\leftrightarrow x\subseteq B\right)\leftrightarrow \left(\forall x\in \mathrm{HAtoms}\left(x\subseteq A\to x\subseteq B\right)\wedge \forall x\in \mathrm{HAtoms}\left(x\subseteq B\to x\subseteq A\right)\right)$$
8	5 6 7	3bitr4i	$$⊢A=B\leftrightarrow \forall x\in \mathrm{HAtoms}\left(x\subseteq A\leftrightarrow x\subseteq B\right)$$