Theorem ist0-3 20959

Description: The predicate "is a T₀ space," expressed in more familiar terms. (Contributed by Jeff Hankins, 1-Feb-2010.)

Assertion

Ref	Expression
ist0-3	⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Kol2 ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥 ≠ 𝑦 → ∃𝑜 ∈ 𝐽 ((𝑥 ∈ 𝑜 ∧ ¬ 𝑦 ∈ 𝑜) ∨ (¬ 𝑥 ∈ 𝑜 ∧ 𝑦 ∈ 𝑜)))))

Distinct variable groups:   𝑥,𝑦,𝑜,𝐽   𝑜,𝑋,𝑥,𝑦

Proof of Theorem ist0-3

Step	Hyp	Ref	Expression
1		ist0-2 20958	. 2 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Kol2 ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜) → 𝑥 = 𝑦)))
2		con34b 305	. . . 4 ⊢ ((∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜) → 𝑥 = 𝑦) ↔ (¬ 𝑥 = 𝑦 → ¬ ∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜)))
3		df-ne 2782	. . . . 5 ⊢ (𝑥 ≠ 𝑦 ↔ ¬ 𝑥 = 𝑦)
4		xor 931	. . . . . . . 8 ⊢ (¬ (𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜) ↔ ((𝑥 ∈ 𝑜 ∧ ¬ 𝑦 ∈ 𝑜) ∨ (𝑦 ∈ 𝑜 ∧ ¬ 𝑥 ∈ 𝑜)))
5		ancom 465	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝑜 ∧ ¬ 𝑥 ∈ 𝑜) ↔ (¬ 𝑥 ∈ 𝑜 ∧ 𝑦 ∈ 𝑜))
6	5	orbi2i 540	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝑜 ∧ ¬ 𝑦 ∈ 𝑜) ∨ (𝑦 ∈ 𝑜 ∧ ¬ 𝑥 ∈ 𝑜)) ↔ ((𝑥 ∈ 𝑜 ∧ ¬ 𝑦 ∈ 𝑜) ∨ (¬ 𝑥 ∈ 𝑜 ∧ 𝑦 ∈ 𝑜)))
7	4, 6	bitri 263	. . . . . . 7 ⊢ (¬ (𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜) ↔ ((𝑥 ∈ 𝑜 ∧ ¬ 𝑦 ∈ 𝑜) ∨ (¬ 𝑥 ∈ 𝑜 ∧ 𝑦 ∈ 𝑜)))
8	7	rexbii 3023	. . . . . 6 ⊢ (∃𝑜 ∈ 𝐽 ¬ (𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜) ↔ ∃𝑜 ∈ 𝐽 ((𝑥 ∈ 𝑜 ∧ ¬ 𝑦 ∈ 𝑜) ∨ (¬ 𝑥 ∈ 𝑜 ∧ 𝑦 ∈ 𝑜)))
9		rexnal 2978	. . . . . 6 ⊢ (∃𝑜 ∈ 𝐽 ¬ (𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜) ↔ ¬ ∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜))
10	8, 9	bitr3i 265	. . . . 5 ⊢ (∃𝑜 ∈ 𝐽 ((𝑥 ∈ 𝑜 ∧ ¬ 𝑦 ∈ 𝑜) ∨ (¬ 𝑥 ∈ 𝑜 ∧ 𝑦 ∈ 𝑜)) ↔ ¬ ∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜))
11	3, 10	imbi12i 339	. . . 4 ⊢ ((𝑥 ≠ 𝑦 → ∃𝑜 ∈ 𝐽 ((𝑥 ∈ 𝑜 ∧ ¬ 𝑦 ∈ 𝑜) ∨ (¬ 𝑥 ∈ 𝑜 ∧ 𝑦 ∈ 𝑜))) ↔ (¬ 𝑥 = 𝑦 → ¬ ∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜)))
12	2, 11	bitr4i 266	. . 3 ⊢ ((∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜) → 𝑥 = 𝑦) ↔ (𝑥 ≠ 𝑦 → ∃𝑜 ∈ 𝐽 ((𝑥 ∈ 𝑜 ∧ ¬ 𝑦 ∈ 𝑜) ∨ (¬ 𝑥 ∈ 𝑜 ∧ 𝑦 ∈ 𝑜))))
13	12	2ralbii 2964	. 2 ⊢ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜) → 𝑥 = 𝑦) ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥 ≠ 𝑦 → ∃𝑜 ∈ 𝐽 ((𝑥 ∈ 𝑜 ∧ ¬ 𝑦 ∈ 𝑜) ∨ (¬ 𝑥 ∈ 𝑜 ∧ 𝑦 ∈ 𝑜))))
14	1, 13	syl6bb 275	1 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Kol2 ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥 ≠ 𝑦 → ∃𝑜 ∈ 𝐽 ((𝑥 ∈ 𝑜 ∧ ¬ 𝑦 ∈ 𝑜) ∨ (¬ 𝑥 ∈ 𝑜 ∧ 𝑦 ∈ 𝑜)))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  ‘cfv 5804  TopOnctopon 20518  Kol2ct0 20920

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-topon 20523  df-t0 20927

This theorem is referenced by: (None)