Theorem oldbday

Description: A surreal is part of the set older than ordinal A iff its birthday is less than A . Remark in Conway p. 29. (Contributed by Scott Fenton, 19-Aug-2024)

	Ref	Expression
Assertion	oldbday	|- ( ( A e. On /\ X e. No ) -> ( X e. ( _Old ` A ) <-> ( bday ` X ) e. A ) )

Proof

Step	Hyp	Ref	Expression
1		oldbdayim	|- ( ( A e. On /\ X e. ( _Old ` A ) ) -> ( bday ` X ) e. A )
2	1	ex	|- ( A e. On -> ( X e. ( _Old ` A ) -> ( bday ` X ) e. A ) )
3	2	adantr	|- ( ( A e. On /\ X e. No ) -> ( X e. ( _Old ` A ) -> ( bday ` X ) e. A ) )
4		simpl	|- ( ( A e. On /\ X e. No ) -> A e. On )
5		onelon	|- ( ( A e. On /\ b e. A ) -> b e. On )
6		madebday	|- ( ( b e. On /\ y e. No ) -> ( y e. ( _M ` b ) <-> ( bday ` y ) C_ b ) )
7	6	biimprd	|- ( ( b e. On /\ y e. No ) -> ( ( bday ` y ) C_ b -> y e. ( _M ` b ) ) )
8	5 7	sylan	|- ( ( ( A e. On /\ b e. A ) /\ y e. No ) -> ( ( bday ` y ) C_ b -> y e. ( _M ` b ) ) )
9	8	anasss	|- ( ( A e. On /\ ( b e. A /\ y e. No ) ) -> ( ( bday ` y ) C_ b -> y e. ( _M ` b ) ) )
10	9	ralrimivva	|- ( A e. On -> A. b e. A A. y e. No ( ( bday ` y ) C_ b -> y e. ( _M ` b ) ) )
11	10	adantr	|- ( ( A e. On /\ X e. No ) -> A. b e. A A. y e. No ( ( bday ` y ) C_ b -> y e. ( _M ` b ) ) )
12		simpr	|- ( ( A e. On /\ X e. No ) -> X e. No )
13		madebdaylemold	|- ( ( A e. On /\ A. b e. A A. y e. No ( ( bday ` y ) C_ b -> y e. ( _M ` b ) ) /\ X e. No ) -> ( ( bday ` X ) e. A -> X e. ( _Old ` A ) ) )
14	4 11 12 13	syl3anc	|- ( ( A e. On /\ X e. No ) -> ( ( bday ` X ) e. A -> X e. ( _Old ` A ) ) )
15	3 14	impbid	|- ( ( A e. On /\ X e. No ) -> ( X e. ( _Old ` A ) <-> ( bday ` X ) e. A ) )