Theorem t1sep2 23256

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 23217	. . . . . 6 ⊢ (𝐽 ∈ Fre → 𝐽 ∈ Top)
2		t1sep.1	. . . . . . 7 ⊢ 𝑋 = ∪ 𝐽
3	2	toptopon 22804	. . . . . 6 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
4	1, 3	sylib 218	. . . . 5 ⊢ (𝐽 ∈ Fre → 𝐽 ∈ (TopOn‘𝑋))
5		ist1-2 23234	. . . . 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 267	. . 3 ⊢ (𝐽 ∈ Fre → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 → 𝑦 ∈ 𝑜) → 𝑥 = 𝑦))
8		eleq1 2816	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝑜 ↔ 𝐴 ∈ 𝑜))
9	8	imbi1d 341	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ 𝑜 → 𝑦 ∈ 𝑜) ↔ (𝐴 ∈ 𝑜 → 𝑦 ∈ 𝑜)))
10	9	ralbidv 3156	. . . . 5 ⊢ (𝑥 = 𝐴 → (∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 → 𝑦 ∈ 𝑜) ↔ ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → 𝑦 ∈ 𝑜)))
11		eqeq1 2733	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 = 𝑦 ↔ 𝐴 = 𝑦))
12	10, 11	imbi12d 344	. . . 4 ⊢ (𝑥 = 𝐴 → ((∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 → 𝑦 ∈ 𝑜) → 𝑥 = 𝑦) ↔ (∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → 𝑦 ∈ 𝑜) → 𝐴 = 𝑦)))
13		eleq1 2816	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (𝑦 ∈ 𝑜 ↔ 𝐵 ∈ 𝑜))
14	13	imbi2d 340	. . . . . 6 ⊢ (𝑦 = 𝐵 → ((𝐴 ∈ 𝑜 → 𝑦 ∈ 𝑜) ↔ (𝐴 ∈ 𝑜 → 𝐵 ∈ 𝑜)))
15	14	ralbidv 3156	. . . . 5 ⊢ (𝑦 = 𝐵 → (∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → 𝑦 ∈ 𝑜) ↔ ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → 𝐵 ∈ 𝑜)))
16		eqeq2 2741	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝐴 = 𝑦 ↔ 𝐴 = 𝐵))
17	15, 16	imbi12d 344	. . . 4 ⊢ (𝑦 = 𝐵 → ((∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → 𝑦 ∈ 𝑜) → 𝐴 = 𝑦) ↔ (∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → 𝐵 ∈ 𝑜) → 𝐴 = 𝐵)))
18	12, 17	rspc2v 3599	. . 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 506	. 2 ⊢ ((𝐽 ∈ Fre ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → 𝐵 ∈ 𝑜) → 𝐴 = 𝐵))
20	19	3impb 1114	1 ⊢ ((𝐽 ∈ Fre ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → 𝐵 ∈ 𝑜) → 𝐴 = 𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3044  ∪ cuni 4871  ‘cfv 6511  Topctop 22780  TopOnctopon 22797  Frect1 23194

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-iota 6464  df-fun 6513  df-fv 6519  df-topgen 17406  df-top 22781  df-topon 22798  df-cld 22906  df-t1 23201

This theorem is referenced by:  t1sep  23257  isr0  23624