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Metamath Proof Explorer

Theorem 1pr

Description: The positive real number 'one'. (Contributed by NM, 13-Mar-1996) (Revised by Mario Carneiro, 12-Jun-2013) (New usage is discouraged.)

		Ref	Expression
	Assertion	1pr	$⊢ 1_{𝑷} \in 𝑷$

Proof

Step	Hyp	Ref	Expression
1		df-1p	$⊢ 1_{𝑷} = \{x \| x <_{𝑸} 1_{𝑸}\}$
2		1nq	$⊢ 1_{𝑸} \in 𝑸$
3		nqpr	$⊢ 1_{𝑸} \in 𝑸 \to \{x \| x <_{𝑸} 1_{𝑸}\} \in 𝑷$
4	2 3	ax-mp	$⊢ \{x \| x <_{𝑸} 1_{𝑸}\} \in 𝑷$
5	1 4	eqeltri	$⊢ 1_{𝑷} \in 𝑷$