Theorem enssdom

Description: Equinumerosity implies dominance. (Contributed by NM, 31-Mar-1998) (Proof shortened by TM, 10-Feb-2026)

	Ref	Expression
Assertion	enssdom	$$⊢\approx \subseteq \preccurlyeq$$

Proof

Step	Hyp	Ref	Expression
1		f1of1	$$⊢f:x\underset{1-1\; onto}{⟶}y\to f:x\underset{1-1}{⟶}y$$
2	1	eximi	$$⊢\exists ff:x\underset{1-1\; onto}{⟶}y\to \exists ff:x\underset{1-1}{⟶}y$$
3	2	ssopab2i	$$⊢\left\{⟨x,y⟩|\exists ff:x\underset{1-1\; onto}{⟶}y\right\}\subseteq \left\{⟨x,y⟩|\exists ff:x\underset{1-1}{⟶}y\right\}$$
4		df-en	$$⊢\approx =\left\{⟨x,y⟩|\exists ff:x\underset{1-1\; onto}{⟶}y\right\}$$
5		df-dom	$$⊢\preccurlyeq =\left\{⟨x,y⟩|\exists ff:x\underset{1-1}{⟶}y\right\}$$
6	3 4 5	3sstr4i	$$⊢\approx \subseteq \preccurlyeq$$