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	$$⊢\left(A\in \mathrm{On}\wedge X\in \mathrm{No}\right)\to \left(X\in \mathrm{Old}\left(A\right)\leftrightarrow \mathrm{bday}\left(X\right)\in A\right)$$

Proof

Step	Hyp	Ref	Expression
1		oldbdayim	$$⊢X\in \mathrm{Old}\left(A\right)\to \mathrm{bday}\left(X\right)\in A$$
2		simpl	$$⊢\left(A\in \mathrm{On}\wedge X\in \mathrm{No}\right)\to A\in \mathrm{On}$$
3		onelon	$$⊢\left(A\in \mathrm{On}\wedge b\in A\right)\to b\in \mathrm{On}$$
4		madebday	Could not format ( ( b e. On /\ y e. No ) -> ( y e. ( _Made ` b ) <-> ( bday ` y ) C_ b ) ) : No typesetting found for |- ( ( b e. On /\ y e. No ) -> ( y e. ( _Made ` b ) <-> ( bday ` y ) C_ b ) ) with typecode |-
5	4	biimprd	Could not format ( ( b e. On /\ y e. No ) -> ( ( bday ` y ) C_ b -> y e. ( _Made ` b ) ) ) : No typesetting found for |- ( ( b e. On /\ y e. No ) -> ( ( bday ` y ) C_ b -> y e. ( _Made ` b ) ) ) with typecode |-
6	3 5	sylan	Could not format ( ( ( A e. On /\ b e. A ) /\ y e. No ) -> ( ( bday ` y ) C_ b -> y e. ( _Made ` b ) ) ) : No typesetting found for |- ( ( ( A e. On /\ b e. A ) /\ y e. No ) -> ( ( bday ` y ) C_ b -> y e. ( _Made ` b ) ) ) with typecode |-
7	6	anasss	Could not format ( ( A e. On /\ ( b e. A /\ y e. No ) ) -> ( ( bday ` y ) C_ b -> y e. ( _Made ` b ) ) ) : No typesetting found for |- ( ( A e. On /\ ( b e. A /\ y e. No ) ) -> ( ( bday ` y ) C_ b -> y e. ( _Made ` b ) ) ) with typecode |-
8	7	ralrimivva	Could not format ( A e. On -> A. b e. A A. y e. No ( ( bday ` y ) C_ b -> y e. ( _Made ` b ) ) ) : No typesetting found for |- ( A e. On -> A. b e. A A. y e. No ( ( bday ` y ) C_ b -> y e. ( _Made ` b ) ) ) with typecode |-
9	8	adantr	Could not format ( ( A e. On /\ X e. No ) -> A. b e. A A. y e. No ( ( bday ` y ) C_ b -> y e. ( _Made ` b ) ) ) : No typesetting found for |- ( ( A e. On /\ X e. No ) -> A. b e. A A. y e. No ( ( bday ` y ) C_ b -> y e. ( _Made ` b ) ) ) with typecode |-
10		simpr	$$⊢\left(A\in \mathrm{On}\wedge X\in \mathrm{No}\right)\to X\in \mathrm{No}$$
11		madebdaylemold	Could not format ( ( A e. On /\ A. b e. A A. y e. No ( ( bday ` y ) C_ b -> y e. ( _Made ` b ) ) /\ X e. No ) -> ( ( bday ` X ) e. A -> X e. ( _Old ` A ) ) ) : No typesetting found for |- ( ( A e. On /\ A. b e. A A. y e. No ( ( bday ` y ) C_ b -> y e. ( _Made ` b ) ) /\ X e. No ) -> ( ( bday ` X ) e. A -> X e. ( _Old ` A ) ) ) with typecode |-
12	2 9 10 11	syl3anc	$$⊢\left(A\in \mathrm{On}\wedge X\in \mathrm{No}\right)\to \left(\mathrm{bday}\left(X\right)\in A\to X\in \mathrm{Old}\left(A\right)\right)$$
13	1 12	impbid2	$$⊢\left(A\in \mathrm{On}\wedge X\in \mathrm{No}\right)\to \left(X\in \mathrm{Old}\left(A\right)\leftrightarrow \mathrm{bday}\left(X\right)\in A\right)$$