Theorem sstr2

Description: Transitivity of subclass relationship. Exercise 5 of TakeutiZaring p. 17. (Contributed by NM, 24-Jun-1993) (Proof shortened by Andrew Salmon, 14-Jun-2011)

	Ref	Expression
Assertion	sstr2	$$⊢A\subseteq B\to \left(B\subseteq C\to A\subseteq C\right)$$

Proof

Step	Hyp	Ref	Expression
1		ssel	$$⊢A\subseteq B\to \left(x\in A\to x\in B\right)$$
2	1	imim1d	$$⊢A\subseteq B\to \left(\left(x\in B\to x\in C\right)\to \left(x\in A\to x\in C\right)\right)$$
3	2	alimdv	$$⊢A\subseteq B\to \left(\forall x\left(x\in B\to x\in C\right)\to \forall x\left(x\in A\to x\in C\right)\right)$$
4		dfss2	$$⊢B\subseteq C\leftrightarrow \forall x\left(x\in B\to x\in C\right)$$
5		dfss2	$$⊢A\subseteq C\leftrightarrow \forall x\left(x\in A\to x\in C\right)$$
6	3 4 5	3imtr4g	$$⊢A\subseteq B\to \left(B\subseteq C\to A\subseteq C\right)$$