Theorem naddss2

Description: Ordinal less-than-or-equal is not affected by natural addition. (Contributed by Scott Fenton, 9-Sep-2024)

	Ref	Expression
Assertion	naddss2	|- ( ( A e. On /\ B e. On /\ C e. On ) -> ( A C_ B <-> ( C +no A ) C_ ( C +no B ) ) )

Proof

Step	Hyp	Ref	Expression
1		naddss1	|- ( ( A e. On /\ B e. On /\ C e. On ) -> ( A C_ B <-> ( A +no C ) C_ ( B +no C ) ) )
2		naddcom	|- ( ( A e. On /\ C e. On ) -> ( A +no C ) = ( C +no A ) )
3	2	3adant2	|- ( ( A e. On /\ B e. On /\ C e. On ) -> ( A +no C ) = ( C +no A ) )
4		naddcom	|- ( ( B e. On /\ C e. On ) -> ( B +no C ) = ( C +no B ) )
5	4	3adant1	|- ( ( A e. On /\ B e. On /\ C e. On ) -> ( B +no C ) = ( C +no B ) )
6	3 5	sseq12d	|- ( ( A e. On /\ B e. On /\ C e. On ) -> ( ( A +no C ) C_ ( B +no C ) <-> ( C +no A ) C_ ( C +no B ) ) )
7	1 6	bitrd	|- ( ( A e. On /\ B e. On /\ C e. On ) -> ( A C_ B <-> ( C +no A ) C_ ( C +no B ) ) )