Theorem otpsbas

Description: The base set of a topological ordered space. (Contributed by Mario Carneiro, 12-Nov-2015) (Revised by AV, 9-Sep-2021)

	Ref	Expression
Hypothesis	otpsstr.w	⊢ 𝐾 = { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( TopSet ‘ ndx ) , 𝐽 ⟩ , ⟨ ( le ‘ ndx ) , ≤ ⟩ }
Assertion	otpsbas	⊢ ( 𝐵 ∈ 𝑉 → 𝐵 = ( Base ‘ 𝐾 ) )

Proof

Step	Hyp	Ref	Expression
1		otpsstr.w	⊢ 𝐾 = { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( TopSet ‘ ndx ) , 𝐽 ⟩ , ⟨ ( le ‘ ndx ) , ≤ ⟩ }
2	1	otpsstr	⊢ 𝐾 Struct ⟨ 1 , ; 1 0 ⟩
3		baseid	⊢ Base = Slot ( Base ‘ ndx )
4		snsstp1	⊢ { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ } ⊆ { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( TopSet ‘ ndx ) , 𝐽 ⟩ , ⟨ ( le ‘ ndx ) , ≤ ⟩ }
5	4 1	sseqtrri	⊢ { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ } ⊆ 𝐾
6	2 3 5	strfv	⊢ ( 𝐵 ∈ 𝑉 → 𝐵 = ( Base ‘ 𝐾 ) )