Theorem dfopg

Description: Value of the ordered pair when the arguments are sets. (Contributed by Mario Carneiro, 26-Apr-2015) (Avoid depending on this detail.)

	Ref	Expression
Assertion	dfopg	$$⊢\left(A\in V\wedge B\in W\right)\to ⟨A,B⟩=\left\{\left\{A\right\},\left\{A,B\right\}\right\}$$

Proof

Step	Hyp	Ref	Expression
1		elex	$$⊢A\in V\to A\in \mathrm{V}$$
2		elex	$$⊢B\in W\to B\in \mathrm{V}$$
3		dfopif	$$⊢⟨A,B⟩=if\left(\left(A\in \mathrm{V}\wedge B\in \mathrm{V}\right),\left\{\left\{A\right\},\left\{A,B\right\}\right\},\varnothing \right)$$
4		iftrue	$$⊢\left(A\in \mathrm{V}\wedge B\in \mathrm{V}\right)\to if\left(\left(A\in \mathrm{V}\wedge B\in \mathrm{V}\right),\left\{\left\{A\right\},\left\{A,B\right\}\right\},\varnothing \right)=\left\{\left\{A\right\},\left\{A,B\right\}\right\}$$
5	3 4	syl5eq	$$⊢\left(A\in \mathrm{V}\wedge B\in \mathrm{V}\right)\to ⟨A,B⟩=\left\{\left\{A\right\},\left\{A,B\right\}\right\}$$
6	1 2 5	syl2an	$$⊢\left(A\in V\wedge B\in W\right)\to ⟨A,B⟩=\left\{\left\{A\right\},\left\{A,B\right\}\right\}$$