Theorem bj-elabtru

Description: This is as close as we can get to proving extensionality for "the" "universal" class without ax-ext . (Contributed by BJ, 24-Apr-2024) (Proof modification is discouraged.)

	Ref	Expression
Assertion	bj-elabtru	$$⊢A\in \left\{x|\top \right\}\leftrightarrow A\in \left\{y|\top \right\}$$

Proof

Step	Hyp	Ref	Expression
1		bj-denoteslem	$$⊢\exists zz=A\leftrightarrow A\in \left\{x|\top \right\}$$
2		bj-denoteslem	$$⊢\exists zz=A\leftrightarrow A\in \left\{y|\top \right\}$$
3	1 2	bitr3i	$$⊢A\in \left\{x|\top \right\}\leftrightarrow A\in \left\{y|\top \right\}$$