Theorem oaltublim

Description: Given C is a limit ordinal, the sum of any ordinal with an ordinal less than C is less than the sum of the first ordinal with C . Lemma 3.5 of Schloeder p. 7. (Contributed by RP, 29-Jan-2025)

	Ref	Expression
Assertion	oaltublim	|- ( ( A e. On /\ B e. C /\ ( Lim C /\ C e. V ) ) -> ( A +o B ) e. ( A +o C ) )

Proof

Step	Hyp	Ref	Expression
1		limord	|- ( Lim C -> Ord C )
2		elex	|- ( C e. V -> C e. _V )
3	1 2	anim12i	|- ( ( Lim C /\ C e. V ) -> ( Ord C /\ C e. _V ) )
4		elon2	|- ( C e. On <-> ( Ord C /\ C e. _V ) )
5	3 4	sylibr	|- ( ( Lim C /\ C e. V ) -> C e. On )
6	5	3ad2ant3	|- ( ( A e. On /\ B e. C /\ ( Lim C /\ C e. V ) ) -> C e. On )
7		simp1	|- ( ( A e. On /\ B e. C /\ ( Lim C /\ C e. V ) ) -> A e. On )
8	6 7	jca	|- ( ( A e. On /\ B e. C /\ ( Lim C /\ C e. V ) ) -> ( C e. On /\ A e. On ) )
9		simp2	|- ( ( A e. On /\ B e. C /\ ( Lim C /\ C e. V ) ) -> B e. C )
10		oaordi	|- ( ( C e. On /\ A e. On ) -> ( B e. C -> ( A +o B ) e. ( A +o C ) ) )
11	8 9 10	sylc	|- ( ( A e. On /\ B e. C /\ ( Lim C /\ C e. V ) ) -> ( A +o B ) e. ( A +o C ) )