Theorem lrcut

Description: A surreal is equal to the cut of its left and right sets. (Contributed by Scott Fenton, 19-Aug-2024)

	Ref	Expression
Assertion	lrcut	Could not format assertion : No typesetting found for |- ( X e. No -> ( ( _Left ` X ) |s ( _Right ` X ) ) = X ) with typecode |-

Proof

Step	Hyp	Ref	Expression
1		bdayelon	$$⊢\mathrm{bday}\left(X\right)\in \mathrm{On}$$
2	1	oneli	$$⊢b\in \mathrm{bday}\left(X\right)\to b\in \mathrm{On}$$
3		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 |-
4	3	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 |-
5	2 4	sylan	Could not format ( ( b e. ( bday ` X ) /\ y e. No ) -> ( ( bday ` y ) C_ b -> y e. ( _Made ` b ) ) ) : No typesetting found for |- ( ( b e. ( bday ` X ) /\ y e. No ) -> ( ( bday ` y ) C_ b -> y e. ( _Made ` b ) ) ) with typecode |-
6	5	rgen2	Could not format A. b e. ( bday ` X ) A. y e. No ( ( bday ` y ) C_ b -> y e. ( _Made ` b ) ) : No typesetting found for |- A. b e. ( bday ` X ) A. y e. No ( ( bday ` y ) C_ b -> y e. ( _Made ` b ) ) with typecode |-
7		madebdaylemlrcut	Could not format ( ( A. b e. ( bday ` X ) A. y e. No ( ( bday ` y ) C_ b -> y e. ( _Made ` b ) ) /\ X e. No ) -> ( ( _Left ` X ) |s ( _Right ` X ) ) = X ) : No typesetting found for |- ( ( A. b e. ( bday ` X ) A. y e. No ( ( bday ` y ) C_ b -> y e. ( _Made ` b ) ) /\ X e. No ) -> ( ( _Left ` X ) |s ( _Right ` X ) ) = X ) with typecode |-
8	6 7	mpan	Could not format ( X e. No -> ( ( _Left ` X ) |s ( _Right ` X ) ) = X ) : No typesetting found for |- ( X e. No -> ( ( _Left ` X ) |s ( _Right ` X ) ) = X ) with typecode |-