Theorem wess

Description: Subset theorem for the well-ordering predicate. Exercise 4 of TakeutiZaring p. 31. (Contributed by NM, 19-Apr-1994)

	Ref	Expression
Assertion	wess	$$⊢A\subseteq B\to \left(R\mathrm{We}B\to R\mathrm{We}A\right)$$

Proof

Step	Hyp	Ref	Expression
1		frss	$$⊢A\subseteq B\to \left(R\mathrm{Fr}B\to R\mathrm{Fr}A\right)$$
2		soss	$$⊢A\subseteq B\to \left(R\mathrm{Or}B\to R\mathrm{Or}A\right)$$
3	1 2	anim12d	$$⊢A\subseteq B\to \left(\left(R\mathrm{Fr}B\wedge R\mathrm{Or}B\right)\to \left(R\mathrm{Fr}A\wedge R\mathrm{Or}A\right)\right)$$
4		df-we	$$⊢R\mathrm{We}B\leftrightarrow \left(R\mathrm{Fr}B\wedge R\mathrm{Or}B\right)$$
5		df-we	$$⊢R\mathrm{We}A\leftrightarrow \left(R\mathrm{Fr}A\wedge R\mathrm{Or}A\right)$$
6	3 4 5	3imtr4g	$$⊢A\subseteq B\to \left(R\mathrm{We}B\to R\mathrm{We}A\right)$$