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Theorem t1sep2 23224
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 = βˆͺ 𝐽
Assertion
Ref Expression
t1sep2 ((𝐽 ∈ Fre ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (βˆ€π‘œ ∈ 𝐽 (𝐴 ∈ π‘œ β†’ 𝐡 ∈ π‘œ) β†’ 𝐴 = 𝐡))
Distinct variable groups:   𝐴,π‘œ   𝐡,π‘œ   π‘œ,𝐽   π‘œ,𝑋
Proof of Theorem t1sep2
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 t1top 23185 . . . . . 6 (𝐽 ∈ Fre β†’ 𝐽 ∈ Top)
2 t1sep.1 . . . . . . 7 𝑋 = βˆͺ 𝐽
32toptopon 22770 . . . . . 6 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOnβ€˜π‘‹))
41, 3sylib 217 . . . . 5 (𝐽 ∈ Fre β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
5 ist1-2 23202 . . . . 5 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ (𝐽 ∈ Fre ↔ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (βˆ€π‘œ ∈ 𝐽 (π‘₯ ∈ π‘œ β†’ 𝑦 ∈ π‘œ) β†’ π‘₯ = 𝑦)))
64, 5syl 17 . . . 4 (𝐽 ∈ Fre β†’ (𝐽 ∈ Fre ↔ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (βˆ€π‘œ ∈ 𝐽 (π‘₯ ∈ π‘œ β†’ 𝑦 ∈ π‘œ) β†’ π‘₯ = 𝑦)))
76ibi 267 . . 3 (𝐽 ∈ Fre β†’ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (βˆ€π‘œ ∈ 𝐽 (π‘₯ ∈ π‘œ β†’ 𝑦 ∈ π‘œ) β†’ π‘₯ = 𝑦))
8 eleq1 2815 . . . . . . 7 (π‘₯ = 𝐴 β†’ (π‘₯ ∈ π‘œ ↔ 𝐴 ∈ π‘œ))
98imbi1d 341 . . . . . 6 (π‘₯ = 𝐴 β†’ ((π‘₯ ∈ π‘œ β†’ 𝑦 ∈ π‘œ) ↔ (𝐴 ∈ π‘œ β†’ 𝑦 ∈ π‘œ)))
109ralbidv 3171 . . . . 5 (π‘₯ = 𝐴 β†’ (βˆ€π‘œ ∈ 𝐽 (π‘₯ ∈ π‘œ β†’ 𝑦 ∈ π‘œ) ↔ βˆ€π‘œ ∈ 𝐽 (𝐴 ∈ π‘œ β†’ 𝑦 ∈ π‘œ)))
11 eqeq1 2730 . . . . 5 (π‘₯ = 𝐴 β†’ (π‘₯ = 𝑦 ↔ 𝐴 = 𝑦))
1210, 11imbi12d 344 . . . 4 (π‘₯ = 𝐴 β†’ ((βˆ€π‘œ ∈ 𝐽 (π‘₯ ∈ π‘œ β†’ 𝑦 ∈ π‘œ) β†’ π‘₯ = 𝑦) ↔ (βˆ€π‘œ ∈ 𝐽 (𝐴 ∈ π‘œ β†’ 𝑦 ∈ π‘œ) β†’ 𝐴 = 𝑦)))
13 eleq1 2815 . . . . . . 7 (𝑦 = 𝐡 β†’ (𝑦 ∈ π‘œ ↔ 𝐡 ∈ π‘œ))
1413imbi2d 340 . . . . . 6 (𝑦 = 𝐡 β†’ ((𝐴 ∈ π‘œ β†’ 𝑦 ∈ π‘œ) ↔ (𝐴 ∈ π‘œ β†’ 𝐡 ∈ π‘œ)))
1514ralbidv 3171 . . . . 5 (𝑦 = 𝐡 β†’ (βˆ€π‘œ ∈ 𝐽 (𝐴 ∈ π‘œ β†’ 𝑦 ∈ π‘œ) ↔ βˆ€π‘œ ∈ 𝐽 (𝐴 ∈ π‘œ β†’ 𝐡 ∈ π‘œ)))
16 eqeq2 2738 . . . . 5 (𝑦 = 𝐡 β†’ (𝐴 = 𝑦 ↔ 𝐴 = 𝐡))
1715, 16imbi12d 344 . . . 4 (𝑦 = 𝐡 β†’ ((βˆ€π‘œ ∈ 𝐽 (𝐴 ∈ π‘œ β†’ 𝑦 ∈ π‘œ) β†’ 𝐴 = 𝑦) ↔ (βˆ€π‘œ ∈ 𝐽 (𝐴 ∈ π‘œ β†’ 𝐡 ∈ π‘œ) β†’ 𝐴 = 𝐡)))
1812, 17rspc2v 3617 . . 3 ((𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (βˆ€π‘œ ∈ 𝐽 (π‘₯ ∈ π‘œ β†’ 𝑦 ∈ π‘œ) β†’ π‘₯ = 𝑦) β†’ (βˆ€π‘œ ∈ 𝐽 (𝐴 ∈ π‘œ β†’ 𝐡 ∈ π‘œ) β†’ 𝐴 = 𝐡)))
197, 18mpan9 506 . 2 ((𝐽 ∈ Fre ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋)) β†’ (βˆ€π‘œ ∈ 𝐽 (𝐴 ∈ π‘œ β†’ 𝐡 ∈ π‘œ) β†’ 𝐴 = 𝐡))
20193impb 1112 1 ((𝐽 ∈ Fre ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (βˆ€π‘œ ∈ 𝐽 (𝐴 ∈ π‘œ β†’ 𝐡 ∈ π‘œ) β†’ 𝐴 = 𝐡))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  βˆ€wral 3055  βˆͺ cuni 4902  β€˜cfv 6536  Topctop 22746  TopOnctopon 22763  Frect1 23162
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-iota 6488  df-fun 6538  df-fv 6544  df-topgen 17396  df-top 22747  df-topon 22764  df-cld 22874  df-t1 23169
This theorem is referenced by:  t1sep  23225  isr0  23592
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