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	$$⊢\phi \to \forall x\in L\left(X\right)\exists y\in Ax{\le }_{s}y$$

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	$$⊢\left(A{\ll }_{s}B\wedge X=A{|}_{s}B\right)\to \left(\forall x\in L\left(X\right)\exists y\in Ax{\le }_{s}y\wedge \forall z\in R\left(X\right)\exists w\in Bw{\le }_{s}z\right)$$
4	1 2 3	syl2anc	$$⊢\phi \to \left(\forall x\in L\left(X\right)\exists y\in Ax{\le }_{s}y\wedge \forall z\in R\left(X\right)\exists w\in Bw{\le }_{s}z\right)$$
5	4	simpld	$$⊢\phi \to \forall x\in L\left(X\right)\exists y\in Ax{\le }_{s}y$$