Theorem enssdom

Description: Equinumerosity implies dominance. (Contributed by NM, 31-Mar-1998)

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

Proof

Step	Hyp	Ref	Expression
1		relen	$$⊢\mathrm{Rel}\approx$$
2		f1of1	$$⊢f:x\underset{1-1\; onto}{⟶}y\to f:x\underset{1-1}{⟶}y$$
3	2	eximi	$$⊢\exists ff:x\underset{1-1\; onto}{⟶}y\to \exists ff:x\underset{1-1}{⟶}y$$
4		opabidw	$$⊢⟨x,y⟩\in \left\{⟨x,y⟩|\exists ff:x\underset{1-1\; onto}{⟶}y\right\}\leftrightarrow \exists ff:x\underset{1-1\; onto}{⟶}y$$
5		opabidw	$$⊢⟨x,y⟩\in \left\{⟨x,y⟩|\exists ff:x\underset{1-1}{⟶}y\right\}\leftrightarrow \exists ff:x\underset{1-1}{⟶}y$$
6	3 4 5	3imtr4i	$$⊢⟨x,y⟩\in \left\{⟨x,y⟩|\exists ff:x\underset{1-1\; onto}{⟶}y\right\}\to ⟨x,y⟩\in \left\{⟨x,y⟩|\exists ff:x\underset{1-1}{⟶}y\right\}$$
7		df-en	$$⊢\approx =\left\{⟨x,y⟩|\exists ff:x\underset{1-1\; onto}{⟶}y\right\}$$
8	7	eleq2i	$$⊢⟨x,y⟩\in \approx \leftrightarrow ⟨x,y⟩\in \left\{⟨x,y⟩|\exists ff:x\underset{1-1\; onto}{⟶}y\right\}$$
9		df-dom	$$⊢\preccurlyeq =\left\{⟨x,y⟩|\exists ff:x\underset{1-1}{⟶}y\right\}$$
10	9	eleq2i	$$⊢⟨x,y⟩\in \preccurlyeq \leftrightarrow ⟨x,y⟩\in \left\{⟨x,y⟩|\exists ff:x\underset{1-1}{⟶}y\right\}$$
11	6 8 10	3imtr4i	$$⊢⟨x,y⟩\in \approx \to ⟨x,y⟩\in \preccurlyeq$$
12	1 11	relssi	$$⊢\approx \subseteq \preccurlyeq$$