Theorem ordelssne

Description: For ordinal classes, membership is equivalent to strict inclusion. Corollary 7.8 of TakeutiZaring p. 37. (Contributed by NM, 25-Nov-1995)

	Ref	Expression
Assertion	ordelssne	$$⊢\left(\mathrm{Ord}A\wedge \mathrm{Ord}B\right)\to \left(A\in B\leftrightarrow \left(A\subseteq B\wedge A\ne B\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		ordtr	$$⊢\mathrm{Ord}A\to \mathrm{Tr}A$$
2		tz7.7	$$⊢\left(\mathrm{Ord}B\wedge \mathrm{Tr}A\right)\to \left(A\in B\leftrightarrow \left(A\subseteq B\wedge A\ne B\right)\right)$$
3	1 2	sylan2	$$⊢\left(\mathrm{Ord}B\wedge \mathrm{Ord}A\right)\to \left(A\in B\leftrightarrow \left(A\subseteq B\wedge A\ne B\right)\right)$$
4	3	ancoms	$$⊢\left(\mathrm{Ord}A\wedge \mathrm{Ord}B\right)\to \left(A\in B\leftrightarrow \left(A\subseteq B\wedge A\ne B\right)\right)$$