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Theorem ist0 22025
 Description: The predicate "is a T0 space". Every pair of distinct points is topologically distinguishable. For the way this definition is usually encountered, see ist0-3 22050. (Contributed by Jeff Hankins, 1-Feb-2010.)
Hypothesis
Ref Expression
ist0.1 𝑋 = 𝐽
Assertion
Ref Expression
ist0 (𝐽 ∈ Kol2 ↔ (𝐽 ∈ Top ∧ ∀𝑥𝑋𝑦𝑋 (∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦)))
Distinct variable groups:   𝑥,𝑜,𝑦,𝐽   𝑜,𝑋,𝑥,𝑦

Proof of Theorem ist0
Dummy variable 𝑗 is distinct from all other variables.
StepHypRef Expression
1 unieq 4812 . . . 4 (𝑗 = 𝐽 𝑗 = 𝐽)
2 ist0.1 . . . 4 𝑋 = 𝐽
31, 2eqtr4di 2811 . . 3 (𝑗 = 𝐽 𝑗 = 𝑋)
4 raleq 3323 . . . . 5 (𝑗 = 𝐽 → (∀𝑜𝑗 (𝑥𝑜𝑦𝑜) ↔ ∀𝑜𝐽 (𝑥𝑜𝑦𝑜)))
54imbi1d 345 . . . 4 (𝑗 = 𝐽 → ((∀𝑜𝑗 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦) ↔ (∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦)))
63, 5raleqbidv 3319 . . 3 (𝑗 = 𝐽 → (∀𝑦 𝑗(∀𝑜𝑗 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦) ↔ ∀𝑦𝑋 (∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦)))
73, 6raleqbidv 3319 . 2 (𝑗 = 𝐽 → (∀𝑥 𝑗𝑦 𝑗(∀𝑜𝑗 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦) ↔ ∀𝑥𝑋𝑦𝑋 (∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦)))
8 df-t0 22018 . 2 Kol2 = {𝑗 ∈ Top ∣ ∀𝑥 𝑗𝑦 𝑗(∀𝑜𝑗 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦)}
97, 8elrab2 3607 1 (𝐽 ∈ Kol2 ↔ (𝐽 ∈ Top ∧ ∀𝑥𝑋𝑦𝑋 (∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3070  ∪ cuni 4801  Topctop 21598  Kol2ct0 22011 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ral 3075  df-rab 3079  df-v 3411  df-in 3867  df-ss 3877  df-uni 4802  df-t0 22018 This theorem is referenced by:  t0sep  22029  t0top  22034  ist0-2  22049  cnt0  22051  ist0cld  31308
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