Theorem t1sep2 23634

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 23595	. . . . . 6 ⊢ (𝐽 ∈ Fre → 𝐽 ∈ Top)
2		t1sep.1	. . . . . . 7 ⊢ 𝑋 = ∪ 𝐽
3	2	toptopon 23182	. . . . . 6 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
4	1, 3	sylib 221	. . . . 5 ⊢ (𝐽 ∈ Fre → 𝐽 ∈ (TopOn‘𝑋))
5		ist1-2 23612	. . . . 5 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Fre ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 → 𝑦 ∈ 𝑜) → 𝑥 = 𝑦)))
6	4, 5	syl 18	. . . 4 ⊢ (𝐽 ∈ Fre → (𝐽 ∈ Fre ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 → 𝑦 ∈ 𝑜) → 𝑥 = 𝑦)))
7	6	ibi 270	. . 3 ⊢ (𝐽 ∈ Fre → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 → 𝑦 ∈ 𝑜) → 𝑥 = 𝑦))
8		eleq1 2848	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝑜 ↔ 𝐴 ∈ 𝑜))
9	8	imbi1d 344	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ 𝑜 → 𝑦 ∈ 𝑜) ↔ (𝐴 ∈ 𝑜 → 𝑦 ∈ 𝑜)))
10	9	ralbidv 3185	. . . . 5 ⊢ (𝑥 = 𝐴 → (∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 → 𝑦 ∈ 𝑜) ↔ ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → 𝑦 ∈ 𝑜)))
11		eqeq1 2764	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 = 𝑦 ↔ 𝐴 = 𝑦))
12	10, 11	imbi12d 347	. . . 4 ⊢ (𝑥 = 𝐴 → ((∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 → 𝑦 ∈ 𝑜) → 𝑥 = 𝑦) ↔ (∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → 𝑦 ∈ 𝑜) → 𝐴 = 𝑦)))
13		eleq1 2848	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (𝑦 ∈ 𝑜 ↔ 𝐵 ∈ 𝑜))
14	13	imbi2d 343	. . . . . 6 ⊢ (𝑦 = 𝐵 → ((𝐴 ∈ 𝑜 → 𝑦 ∈ 𝑜) ↔ (𝐴 ∈ 𝑜 → 𝐵 ∈ 𝑜)))
15	14	ralbidv 3185	. . . . 5 ⊢ (𝑦 = 𝐵 → (∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → 𝑦 ∈ 𝑜) ↔ ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → 𝐵 ∈ 𝑜)))
16		eqeq2 2772	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝐴 = 𝑦 ↔ 𝐴 = 𝐵))
17	15, 16	imbi12d 347	. . . 4 ⊢ (𝑦 = 𝐵 → ((∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → 𝑦 ∈ 𝑜) → 𝐴 = 𝑦) ↔ (∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → 𝐵 ∈ 𝑜) → 𝐴 = 𝐵)))
18	12, 17	rspc2v 3587	. . 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 516	. 2 ⊢ ((𝐽 ∈ Fre ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → 𝐵 ∈ 𝑜) → 𝐴 = 𝐵))
20	19	3impb 1132	1 ⊢ ((𝐽 ∈ Fre ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → 𝐵 ∈ 𝑜) → 𝐴 = 𝐵))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∀wral 3076  ∪ cuni 4867  ‘cfv 6528  Topctop 23158  TopOnctopon 23175  Frect1 23572

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5543  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-iota 6484  df-fun 6530  df-fv 6536  df-topgen 17561  df-top 23159  df-topon 23176  df-cld 23284  df-t1 23579

This theorem is used by:  t1sep  23635  isr0  24003