Theorem mppsthm

Description: A provable pre-statement is a theorem. (Contributed by Mario Carneiro, 18-Jul-2016)

	Ref	Expression
Hypotheses	mppsthm.j	$$⊢J=\mathrm{mPPSt}\left(T\right)$$
	mppsthm.u	$$⊢U=\mathrm{mThm}\left(T\right)$$
Assertion	mppsthm	$$⊢J\subseteq U$$

Proof

Step	Hyp	Ref	Expression
1		mppsthm.j	$$⊢J=\mathrm{mPPSt}\left(T\right)$$
2		mppsthm.u	$$⊢U=\mathrm{mThm}\left(T\right)$$
3		eqid	$$⊢\mathrm{mStRed}\left(T\right)\left(x\right)=\mathrm{mStRed}\left(T\right)\left(x\right)$$
4		eqid	$$⊢\mathrm{mStRed}\left(T\right)=\mathrm{mStRed}\left(T\right)$$
5	4 1 2	mthmi	$$⊢\left(x\in J\wedge \mathrm{mStRed}\left(T\right)\left(x\right)=\mathrm{mStRed}\left(T\right)\left(x\right)\right)\to x\in U$$
6	3 5	mpan2	$$⊢x\in J\to x\in U$$
7	6	ssriv	$$⊢J\subseteq U$$