Theorem ist0

Description: The predicate "is a T_0 space". Every pair of distinct points is topologically distinguishable. For the way this definition is usually encountered, see ist0-3 . (Contributed by Jeff Hankins, 1-Feb-2010)

	Ref	Expression
Hypothesis	ist0.1	⊢ 𝑋 = ∪ 𝐽
Assertion	ist0	⊢ ( 𝐽 ∈ Kol2 ↔ ( 𝐽 ∈ Top ∧ ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ( ∀ 𝑜 ∈ 𝐽 ( 𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜 ) → 𝑥 = 𝑦 ) ) )

Proof

Step	Hyp	Ref	Expression
1		ist0.1	⊢ 𝑋 = ∪ 𝐽
2		unieq	⊢ ( 𝑗 = 𝐽 → ∪ 𝑗 = ∪ 𝐽 )
3	2 1	eqtr4di	⊢ ( 𝑗 = 𝐽 → ∪ 𝑗 = 𝑋 )
4		raleq	⊢ ( 𝑗 = 𝐽 → ( ∀ 𝑜 ∈ 𝑗 ( 𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜 ) ↔ ∀ 𝑜 ∈ 𝐽 ( 𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜 ) ) )
5	4	imbi1d	⊢ ( 𝑗 = 𝐽 → ( ( ∀ 𝑜 ∈ 𝑗 ( 𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜 ) → 𝑥 = 𝑦 ) ↔ ( ∀ 𝑜 ∈ 𝐽 ( 𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜 ) → 𝑥 = 𝑦 ) ) )
6	3 5	raleqbidv	⊢ ( 𝑗 = 𝐽 → ( ∀ 𝑦 ∈ ∪ 𝑗 ( ∀ 𝑜 ∈ 𝑗 ( 𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜 ) → 𝑥 = 𝑦 ) ↔ ∀ 𝑦 ∈ 𝑋 ( ∀ 𝑜 ∈ 𝐽 ( 𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜 ) → 𝑥 = 𝑦 ) ) )
7	3 6	raleqbidv	⊢ ( 𝑗 = 𝐽 → ( ∀ 𝑥 ∈ ∪ 𝑗 ∀ 𝑦 ∈ ∪ 𝑗 ( ∀ 𝑜 ∈ 𝑗 ( 𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜 ) → 𝑥 = 𝑦 ) ↔ ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ( ∀ 𝑜 ∈ 𝐽 ( 𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜 ) → 𝑥 = 𝑦 ) ) )
8		df-t0	⊢ Kol2 = { 𝑗 ∈ Top ∣ ∀ 𝑥 ∈ ∪ 𝑗 ∀ 𝑦 ∈ ∪ 𝑗 ( ∀ 𝑜 ∈ 𝑗 ( 𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜 ) → 𝑥 = 𝑦 ) }
9	7 8	elrab2	⊢ ( 𝐽 ∈ Kol2 ↔ ( 𝐽 ∈ Top ∧ ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ( ∀ 𝑜 ∈ 𝐽 ( 𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜 ) → 𝑥 = 𝑦 ) ) )