Theorem ssltright

Description: A surreal is less than its right options. Theorem 0(i) of Conway p. 16. (Contributed by Scott Fenton, 7-Aug-2024)

	Ref	Expression
Assertion	ssltright	|- ( A e. No -> { A } <

Proof

Step	Hyp	Ref	Expression
1		snex	|- { A } e. _V
2	1	a1i	|- ( A e. No -> { A } e. _V )
3		fvexd	|- ( A e. No -> ( _Right ` A ) e. _V )
4		snssi	|- ( A e. No -> { A } C_ No )
5		rightf	|- _Right : No --> ~P No
6	5	ffvelcdmi	|- ( A e. No -> ( _Right ` A ) e. ~P No )
7	6	elpwid	|- ( A e. No -> ( _Right ` A ) C_ No )
8		velsn	|- ( x e. { A } <-> x = A )
9		rightval	|- ( _Right ` A ) = { y e. ( _Old ` ( bday ` A ) ) | A
10	9	a1i	|- ( A e. No -> ( _Right ` A ) = { y e. ( _Old ` ( bday ` A ) ) | A
11	10	eleq2d	|- ( A e. No -> ( y e. ( _Right ` A ) <-> y e. { y e. ( _Old ` ( bday ` A ) ) | A
12		rabid	|- ( y e. { y e. ( _Old ` ( bday ` A ) ) | A ( y e. ( _Old ` ( bday ` A ) ) /\ A
13	11 12	bitrdi	|- ( A e. No -> ( y e. ( _Right ` A ) <-> ( y e. ( _Old ` ( bday ` A ) ) /\ A
14	13	simplbda	|- ( ( A e. No /\ y e. ( _Right ` A ) ) -> A
15		breq1	|- ( x = A -> ( x A
16	14 15	imbitrrid	|- ( x = A -> ( ( A e. No /\ y e. ( _Right ` A ) ) -> x
17	16	expd	|- ( x = A -> ( A e. No -> ( y e. ( _Right ` A ) -> x
18	8 17	sylbi	|- ( x e. { A } -> ( A e. No -> ( y e. ( _Right ` A ) -> x
19	18	3imp21	|- ( ( A e. No /\ x e. { A } /\ y e. ( _Right ` A ) ) -> x
20	2 3 4 7 19	ssltd	|- ( A e. No -> { A } <