Theorem bj-rabtrALT

Description: Alternate proof of bj-rabtr . (Contributed by BJ, 22-Apr-2019) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	bj-rabtrALT	$$⊢\left\{x\in A|\top \right\}=A$$

Proof

Step	Hyp	Ref	Expression
1		nfrab1	$$⊢\underset{\_}{Ⅎ}x\left\{x\in A|\top \right\}$$
2		nfcv	$$⊢\underset{\_}{Ⅎ}xA$$
3	1 2	cleqf	$$⊢\left\{x\in A|\top \right\}=A\leftrightarrow \forall x\left(x\in \left\{x\in A|\top \right\}\leftrightarrow x\in A\right)$$
4		tru	$$⊢\top$$
5		rabid	$$⊢x\in \left\{x\in A|\top \right\}\leftrightarrow \left(x\in A\wedge \top \right)$$
6	4 5	mpbiran2	$$⊢x\in \left\{x\in A|\top \right\}\leftrightarrow x\in A$$
7	3 6	mpgbir	$$⊢\left\{x\in A|\top \right\}=A$$