Theorem alephsucpw

Description: The power set of an aleph dominates the successor aleph. (The Generalized Continuum Hypothesis says they are equinumerous, see gch3 or gchaleph2 .) (Contributed by NM, 27-Aug-2005)

	Ref	Expression
Assertion	alephsucpw	$$⊢\aleph \left(\mathrm{suc}A\right)\preccurlyeq 𝒫\aleph \left(A\right)$$

Proof

Step	Hyp	Ref	Expression
1		alephsucpw2	$$⊢\neg 𝒫\aleph \left(A\right)\prec \aleph \left(\mathrm{suc}A\right)$$
2		fvex	$$⊢\aleph \left(\mathrm{suc}A\right)\in \mathrm{V}$$
3		fvex	$$⊢\aleph \left(A\right)\in \mathrm{V}$$
4	3	pwex	$$⊢𝒫\aleph \left(A\right)\in \mathrm{V}$$
5		domtri	$$⊢\left(\aleph \left(\mathrm{suc}A\right)\in \mathrm{V}\wedge 𝒫\aleph \left(A\right)\in \mathrm{V}\right)\to \left(\aleph \left(\mathrm{suc}A\right)\preccurlyeq 𝒫\aleph \left(A\right)\leftrightarrow \neg 𝒫\aleph \left(A\right)\prec \aleph \left(\mathrm{suc}A\right)\right)$$
6	2 4 5	mp2an	$$⊢\aleph \left(\mathrm{suc}A\right)\preccurlyeq 𝒫\aleph \left(A\right)\leftrightarrow \neg 𝒫\aleph \left(A\right)\prec \aleph \left(\mathrm{suc}A\right)$$
7	1 6	mpbir	$$⊢\aleph \left(\mathrm{suc}A\right)\preccurlyeq 𝒫\aleph \left(A\right)$$