Theorem tz7.2

Description: Similar to Theorem 7.2 of TakeutiZaring p. 35, except that the Axiom of Regularity is not required due to the antecedent _E Fr A . (Contributed by NM, 4-May-1994)

	Ref	Expression
Assertion	tz7.2	$$⊢\left(\mathrm{Tr}A\wedge \mathrm{E}\mathrm{Fr}A\wedge B\in A\right)\to \left(B\subseteq A\wedge B\ne A\right)$$

Proof

Step	Hyp	Ref	Expression
1		trss	$$⊢\mathrm{Tr}A\to \left(B\in A\to B\subseteq A\right)$$
2		efrirr	$$⊢\mathrm{E}\mathrm{Fr}A\to \neg A\in A$$
3		eleq1	$$⊢B=A\to \left(B\in A\leftrightarrow A\in A\right)$$
4	3	notbid	$$⊢B=A\to \left(\neg B\in A\leftrightarrow \neg A\in A\right)$$
5	2 4	syl5ibrcom	$$⊢\mathrm{E}\mathrm{Fr}A\to \left(B=A\to \neg B\in A\right)$$
6	5	necon2ad	$$⊢\mathrm{E}\mathrm{Fr}A\to \left(B\in A\to B\ne A\right)$$
7	1 6	anim12ii	$$⊢\left(\mathrm{Tr}A\wedge \mathrm{E}\mathrm{Fr}A\right)\to \left(B\in A\to \left(B\subseteq A\wedge B\ne A\right)\right)$$
8	7	3impia	$$⊢\left(\mathrm{Tr}A\wedge \mathrm{E}\mathrm{Fr}A\wedge B\in A\right)\to \left(B\subseteq A\wedge B\ne A\right)$$