Theorem isowe

Description: An isomorphism preserves the property of being a well-ordering. Proposition 6.32(3) of TakeutiZaring p. 33. (Contributed by NM, 30-Apr-2004) (Revised by Mario Carneiro, 18-Nov-2014)

	Ref	Expression
Assertion	isowe	$$⊢H{Isom}_{R,S}\left(A,B\right)\to \left(R\mathrm{We}A\leftrightarrow S\mathrm{We}B\right)$$

Proof

Step	Hyp	Ref	Expression
1		isofr	$$⊢H{Isom}_{R,S}\left(A,B\right)\to \left(R\mathrm{Fr}A\leftrightarrow S\mathrm{Fr}B\right)$$
2		isoso	$$⊢H{Isom}_{R,S}\left(A,B\right)\to \left(R\mathrm{Or}A\leftrightarrow S\mathrm{Or}B\right)$$
3	1 2	anbi12d	$$⊢H{Isom}_{R,S}\left(A,B\right)\to \left(\left(R\mathrm{Fr}A\wedge R\mathrm{Or}A\right)\leftrightarrow \left(S\mathrm{Fr}B\wedge S\mathrm{Or}B\right)\right)$$
4		df-we	$$⊢R\mathrm{We}A\leftrightarrow \left(R\mathrm{Fr}A\wedge R\mathrm{Or}A\right)$$
5		df-we	$$⊢S\mathrm{We}B\leftrightarrow \left(S\mathrm{Fr}B\wedge S\mathrm{Or}B\right)$$
6	3 4 5	3bitr4g	$$⊢H{Isom}_{R,S}\left(A,B\right)\to \left(R\mathrm{We}A\leftrightarrow S\mathrm{We}B\right)$$