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Theorem ist0 21034
 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 21059. (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 4410 . . . 4 (𝑗 = 𝐽 𝑗 = 𝐽)
2 ist0.1 . . . 4 𝑋 = 𝐽
31, 2syl6eqr 2673 . . 3 (𝑗 = 𝐽 𝑗 = 𝑋)
4 raleq 3127 . . . . 5 (𝑗 = 𝐽 → (∀𝑜𝑗 (𝑥𝑜𝑦𝑜) ↔ ∀𝑜𝐽 (𝑥𝑜𝑦𝑜)))
54imbi1d 331 . . . 4 (𝑗 = 𝐽 → ((∀𝑜𝑗 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦) ↔ (∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦)))
63, 5raleqbidv 3141 . . 3 (𝑗 = 𝐽 → (∀𝑦 𝑗(∀𝑜𝑗 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦) ↔ ∀𝑦𝑋 (∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦)))
73, 6raleqbidv 3141 . 2 (𝑗 = 𝐽 → (∀𝑥 𝑗𝑦 𝑗(∀𝑜𝑗 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦) ↔ ∀𝑥𝑋𝑦𝑋 (∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦)))
8 df-t0 21027 . 2 Kol2 = {𝑗 ∈ Top ∣ ∀𝑥 𝑗𝑦 𝑗(∀𝑜𝑗 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦)}
97, 8elrab2 3348 1 (𝐽 ∈ Kol2 ↔ (𝐽 ∈ Top ∧ ∀𝑥𝑋𝑦𝑋 (∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1480   ∈ wcel 1987  ∀wral 2907  ∪ cuni 4402  Topctop 20617  Kol2ct0 21020 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-uni 4403  df-t0 21027 This theorem is referenced by:  t0sep  21038  t0top  21043  ist0-2  21058  cnt0  21060
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