Theorem canth3

Description: Cantor's theorem in terms of cardinals. This theorem tells us that no matter how large a cardinal number is, there is a still larger cardinal number. Theorem 18.12 of Monk1 p. 133. (Contributed by NM, 5-Nov-2003)

	Ref	Expression
Assertion	canth3	$$⊢A\in V\to \mathrm{card}\left(A\right)\in \mathrm{card}\left(𝒫A\right)$$

Proof

Step	Hyp	Ref	Expression
1		canth2g	$$⊢A\in V\to A\prec 𝒫A$$
2		pwexg	$$⊢A\in V\to 𝒫A\in \mathrm{V}$$
3		cardsdom	$$⊢\left(A\in V\wedge 𝒫A\in \mathrm{V}\right)\to \left(\mathrm{card}\left(A\right)\in \mathrm{card}\left(𝒫A\right)\leftrightarrow A\prec 𝒫A\right)$$
4	2 3	mpdan	$$⊢A\in V\to \left(\mathrm{card}\left(A\right)\in \mathrm{card}\left(𝒫A\right)\leftrightarrow A\prec 𝒫A\right)$$
5	1 4	mpbird	$$⊢A\in V\to \mathrm{card}\left(A\right)\in \mathrm{card}\left(𝒫A\right)$$