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Theorem ist0 21344
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 21369. (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 4580 . . . 4 (𝑗 = 𝐽 𝑗 = 𝐽)
2 ist0.1 . . . 4 𝑋 = 𝐽
31, 2syl6eqr 2822 . . 3 (𝑗 = 𝐽 𝑗 = 𝑋)
4 raleq 3286 . . . . 5 (𝑗 = 𝐽 → (∀𝑜𝑗 (𝑥𝑜𝑦𝑜) ↔ ∀𝑜𝐽 (𝑥𝑜𝑦𝑜)))
54imbi1d 330 . . . 4 (𝑗 = 𝐽 → ((∀𝑜𝑗 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦) ↔ (∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦)))
63, 5raleqbidv 3300 . . 3 (𝑗 = 𝐽 → (∀𝑦 𝑗(∀𝑜𝑗 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦) ↔ ∀𝑦𝑋 (∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦)))
73, 6raleqbidv 3300 . 2 (𝑗 = 𝐽 → (∀𝑥 𝑗𝑦 𝑗(∀𝑜𝑗 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦) ↔ ∀𝑥𝑋𝑦𝑋 (∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦)))
8 df-t0 21337 . 2 Kol2 = {𝑗 ∈ Top ∣ ∀𝑥 𝑗𝑦 𝑗(∀𝑜𝑗 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦)}
97, 8elrab2 3516 1 (𝐽 ∈ Kol2 ↔ (𝐽 ∈ Top ∧ ∀𝑥𝑋𝑦𝑋 (∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1630  wcel 2144  wral 3060   cuni 4572  Topctop 20917  Kol2ct0 21330
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-uni 4573  df-t0 21337
This theorem is referenced by:  t0sep  21348  t0top  21353  ist0-2  21368  cnt0  21370
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