Theorem xphe

Description: Any Cartesian product is hereditary in its second class. (Contributed by RP, 27-Mar-2020) (Proof shortened by OpenAI, 3-Jul-2020)

	Ref	Expression
Assertion	xphe	|- ( A X. B ) hereditary B

Proof

Step	Hyp	Ref	Expression
1		imassrn	|- ( ( A X. B ) " B ) C_ ran ( A X. B )
2		rnxpss	|- ran ( A X. B ) C_ B
3	1 2	sstri	|- ( ( A X. B ) " B ) C_ B
4		df-he	|- ( ( A X. B ) hereditary B <-> ( ( A X. B ) " B ) C_ B )
5	3 4	mpbir	|- ( A X. B ) hereditary B