Theorem cofcutr1d

Description: If X is the cut of A and B , then A is cofinal with ( _LeftX ) . First half of theorem 2.9 of Gonshor p. 12. (Contributed by Scott Fenton, 23-Jan-2025)

	Ref	Expression
Hypotheses	cofcutrd.1	|- ( ph -> A <
	cofcutrd.2	|- ( ph -> X = ( A |s B ) )
Assertion	cofcutr1d	|- ( ph -> A. x e. ( _Left ` X ) E. y e. A x <_s y )

Proof

Step	Hyp	Ref	Expression
1		cofcutrd.1	|- ( ph -> A <
2		cofcutrd.2	|- ( ph -> X = ( A |s B ) )
3		cofcutr	|- ( ( A < ( A. x e. ( _Left ` X ) E. y e. A x <_s y /\ A. z e. ( _Right ` X ) E. w e. B w <_s z ) )
4	1 2 3	syl2anc	|- ( ph -> ( A. x e. ( _Left ` X ) E. y e. A x <_s y /\ A. z e. ( _Right ` X ) E. w e. B w <_s z ) )
5	4	simpld	|- ( ph -> A. x e. ( _Left ` X ) E. y e. A x <_s y )