Theorem r1ord3

Description: Ordering relation for the cumulative hierarchy of sets. Part of Theorem 3.3(i) of BellMachover p. 478. (Contributed by NM, 22-Sep-2003)

	Ref	Expression
Assertion	r1ord3	|- ( ( A e. On /\ B e. On ) -> ( A C_ B -> ( R1 ` A ) C_ ( R1 ` B ) ) )

Proof

Step	Hyp	Ref	Expression
1		r1fnon	|- R1 Fn On
2	1	fndmi	|- dom R1 = On
3	2	eleq2i	|- ( A e. dom R1 <-> A e. On )
4	2	eleq2i	|- ( B e. dom R1 <-> B e. On )
5		r1ord3g	|- ( ( A e. dom R1 /\ B e. dom R1 ) -> ( A C_ B -> ( R1 ` A ) C_ ( R1 ` B ) ) )
6	3 4 5	syl2anbr	|- ( ( A e. On /\ B e. On ) -> ( A C_ B -> ( R1 ` A ) C_ ( R1 ` B ) ) )