Theorem trwf

Description: The class of well-founded sets is transitive. (Contributed by Eric Schmidt, 9-Sep-2025)

	Ref	Expression
Assertion	trwf	$$⊢\mathrm{Tr}\bigcup R_1\left[\mathrm{On}\right]$$

Proof

Step	Hyp	Ref	Expression
1		r1elssi	$$⊢x\in \bigcup R_1\left[\mathrm{On}\right]\to x\subseteq \bigcup R_1\left[\mathrm{On}\right]$$
2	1	rgen	$$⊢\forall x\in \bigcup R_1\left[\mathrm{On}\right]x\subseteq \bigcup R_1\left[\mathrm{On}\right]$$
3		dftr3	$$⊢\mathrm{Tr}\bigcup R_1\left[\mathrm{On}\right]\leftrightarrow \forall x\in \bigcup R_1\left[\mathrm{On}\right]x\subseteq \bigcup R_1\left[\mathrm{On}\right]$$
4	2 3	mpbir	$$⊢\mathrm{Tr}\bigcup R_1\left[\mathrm{On}\right]$$