Theorem brsdom

Description: Strict dominance relation, meaning " B is strictly greater in size than A ". Definition of Mendelson p. 255. (Contributed by NM, 25-Jun-1998)

	Ref	Expression
Assertion	brsdom	$$⊢A\prec B\leftrightarrow \left(A\preccurlyeq B\wedge \neg A\approx B\right)$$

Proof

Step	Hyp	Ref	Expression
1		df-sdom	$$⊢\prec =\preccurlyeq \setminus \approx$$
2	1	eleq2i	$$⊢⟨A,B⟩\in \prec \leftrightarrow ⟨A,B⟩\in \left(\preccurlyeq \setminus \approx \right)$$
3		df-br	$$⊢A\prec B\leftrightarrow ⟨A,B⟩\in \prec$$
4		df-br	$$⊢A\preccurlyeq B\leftrightarrow ⟨A,B⟩\in \preccurlyeq$$
5		df-br	$$⊢A\approx B\leftrightarrow ⟨A,B⟩\in \approx$$
6	5	notbii	$$⊢\neg A\approx B\leftrightarrow \neg ⟨A,B⟩\in \approx$$
7	4 6	anbi12i	$$⊢\left(A\preccurlyeq B\wedge \neg A\approx B\right)\leftrightarrow \left(⟨A,B⟩\in \preccurlyeq \wedge \neg ⟨A,B⟩\in \approx \right)$$
8		eldif	$$⊢⟨A,B⟩\in \left(\preccurlyeq \setminus \approx \right)\leftrightarrow \left(⟨A,B⟩\in \preccurlyeq \wedge \neg ⟨A,B⟩\in \approx \right)$$
9	7 8	bitr4i	$$⊢\left(A\preccurlyeq B\wedge \neg A\approx B\right)\leftrightarrow ⟨A,B⟩\in \left(\preccurlyeq \setminus \approx \right)$$
10	2 3 9	3bitr4i	$$⊢A\prec B\leftrightarrow \left(A\preccurlyeq B\wedge \neg A\approx B\right)$$