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Metamath Proof Explorer

Theorem plendxnscandx

Description: The slot for the "less than or equal to" ordering is not the slot for the scalar in an extensible structure. Formerly part of proof for opsrsca . (Contributed by AV, 1-Nov-2024)

		Ref	Expression
	Assertion	plendxnscandx	⊢ ( le ‘ ndx ) ≠ ( Scalar ‘ ndx )

Proof

Step	Hyp	Ref	Expression
1		5re	⊢ 5 ∈ ℝ
2		5lt10	⊢ 5 < ; 1 0
3	1 2	gtneii	⊢ ; 1 0 ≠ 5
4		plendx	⊢ ( le ‘ ndx ) = ; 1 0
5		scandx	⊢ ( Scalar ‘ ndx ) = 5
6	4 5	neeq12i	⊢ ( ( le ‘ ndx ) ≠ ( Scalar ‘ ndx ) ↔ ; 1 0 ≠ 5 )
7	3 6	mpbir	⊢ ( le ‘ ndx ) ≠ ( Scalar ‘ ndx )