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	$$⊢\phi \to A{\ll }_{s}B$$
	cofcutrd.2	$$⊢\phi \to X=A{|}_{s}B$$
Assertion	cofcutr1d	Could not format assertion : No typesetting found for |- ( ph -> A. x e. ( _Left ` X ) E. y e. A x <_s y ) with typecode |-

Proof

Step	Hyp	Ref	Expression
1		cofcutrd.1	$$⊢\phi \to A{\ll }_{s}B$$
2		cofcutrd.2	$$⊢\phi \to X=A{|}_{s}B$$
3		cofcutr	Could not format ( ( 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 ) ) : No typesetting found for |- ( ( 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 ) ) with typecode |-
4	1 2 3	syl2anc	Could not format ( 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 ) ) : No typesetting found for |- ( 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 ) ) with typecode |-
5	4	simpld	Could not format ( ph -> A. x e. ( _Left ` X ) E. y e. A x <_s y ) : No typesetting found for |- ( ph -> A. x e. ( _Left ` X ) E. y e. A x <_s y ) with typecode |-