Theorem idssen

Description: Equality implies equinumerosity. (Contributed by NM, 30-Apr-1998) (Revised by Mario Carneiro, 15-Nov-2014)

	Ref	Expression
Assertion	idssen	$$⊢\mathrm{I}\subseteq \approx$$

Proof

Step	Hyp	Ref	Expression
1		reli	$$⊢\mathrm{Rel}\mathrm{I}$$
2		vex	$$⊢y\in \mathrm{V}$$
3	2	ideq	$$⊢x\mathrm{I}y\leftrightarrow x=y$$
4		eqeng	$$⊢x\in \mathrm{V}\to \left(x=y\to x\approx y\right)$$
5	4	elv	$$⊢x=y\to x\approx y$$
6	3 5	sylbi	$$⊢x\mathrm{I}y\to x\approx y$$
7		df-br	$$⊢x\mathrm{I}y\leftrightarrow ⟨x,y⟩\in \mathrm{I}$$
8		df-br	$$⊢x\approx y\leftrightarrow ⟨x,y⟩\in \approx$$
9	6 7 8	3imtr3i	$$⊢⟨x,y⟩\in \mathrm{I}\to ⟨x,y⟩\in \approx$$
10	1 9	relssi	$$⊢\mathrm{I}\subseteq \approx$$