Theorem t1sep2 20983

Description: Any two points in a T₁ space which have no separation are equal. (Contributed by Jeff Hankins, 1-Feb-2010.)

Hypothesis

Ref	Expression
t1sep.1	⊢ 𝑋 = ∪ 𝐽

Assertion

Ref	Expression
t1sep2	⊢ ((𝐽 ∈ Fre ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → 𝐵 ∈ 𝑜) → 𝐴 = 𝐵))

Distinct variable groups:   𝐴,𝑜   𝐵,𝑜   𝑜,𝐽   𝑜,𝑋

Proof of Theorem t1sep2

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		t1top 20944	. . . . . 6 ⊢ (𝐽 ∈ Fre → 𝐽 ∈ Top)
2		t1sep.1	. . . . . . 7 ⊢ 𝑋 = ∪ 𝐽
3	2	toptopon 20548	. . . . . 6 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
4	1, 3	sylib 207	. . . . 5 ⊢ (𝐽 ∈ Fre → 𝐽 ∈ (TopOn‘𝑋))
5		ist1-2 20961	. . . . 5 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Fre ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 → 𝑦 ∈ 𝑜) → 𝑥 = 𝑦)))
6	4, 5	syl 17	. . . 4 ⊢ (𝐽 ∈ Fre → (𝐽 ∈ Fre ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 → 𝑦 ∈ 𝑜) → 𝑥 = 𝑦)))
7	6	ibi 255	. . 3 ⊢ (𝐽 ∈ Fre → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 → 𝑦 ∈ 𝑜) → 𝑥 = 𝑦))
8		eleq1 2676	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝑜 ↔ 𝐴 ∈ 𝑜))
9	8	imbi1d 330	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ 𝑜 → 𝑦 ∈ 𝑜) ↔ (𝐴 ∈ 𝑜 → 𝑦 ∈ 𝑜)))
10	9	ralbidv 2969	. . . . 5 ⊢ (𝑥 = 𝐴 → (∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 → 𝑦 ∈ 𝑜) ↔ ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → 𝑦 ∈ 𝑜)))
11		eqeq1 2614	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 = 𝑦 ↔ 𝐴 = 𝑦))
12	10, 11	imbi12d 333	. . . 4 ⊢ (𝑥 = 𝐴 → ((∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 → 𝑦 ∈ 𝑜) → 𝑥 = 𝑦) ↔ (∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → 𝑦 ∈ 𝑜) → 𝐴 = 𝑦)))
13		eleq1 2676	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (𝑦 ∈ 𝑜 ↔ 𝐵 ∈ 𝑜))
14	13	imbi2d 329	. . . . . 6 ⊢ (𝑦 = 𝐵 → ((𝐴 ∈ 𝑜 → 𝑦 ∈ 𝑜) ↔ (𝐴 ∈ 𝑜 → 𝐵 ∈ 𝑜)))
15	14	ralbidv 2969	. . . . 5 ⊢ (𝑦 = 𝐵 → (∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → 𝑦 ∈ 𝑜) ↔ ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → 𝐵 ∈ 𝑜)))
16		eqeq2 2621	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝐴 = 𝑦 ↔ 𝐴 = 𝐵))
17	15, 16	imbi12d 333	. . . 4 ⊢ (𝑦 = 𝐵 → ((∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → 𝑦 ∈ 𝑜) → 𝐴 = 𝑦) ↔ (∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → 𝐵 ∈ 𝑜) → 𝐴 = 𝐵)))
18	12, 17	rspc2v 3293	. . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 → 𝑦 ∈ 𝑜) → 𝑥 = 𝑦) → (∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → 𝐵 ∈ 𝑜) → 𝐴 = 𝐵)))
19	7, 18	mpan9 485	. 2 ⊢ ((𝐽 ∈ Fre ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → 𝐵 ∈ 𝑜) → 𝐴 = 𝐵))
20	19	3impb 1252	1 ⊢ ((𝐽 ∈ Fre ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → 𝐵 ∈ 𝑜) → 𝐴 = 𝐵))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∪ cuni 4372  ‘cfv 5804  Topctop 20517  TopOnctopon 20518  Frect1 20921

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-topgen 15927  df-top 20521  df-topon 20523  df-cld 20633  df-t1 20928

This theorem is referenced by:  t1sep  20984  isr0  21350