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Theorem t1sep2 23286
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 23247 . . . . . 6 (𝐽 ∈ Fre β†’ 𝐽 ∈ Top)
2 t1sep.1 . . . . . . 7 𝑋 = βˆͺ 𝐽
32toptopon 22832 . . . . . 6 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOnβ€˜π‘‹))
41, 3sylib 217 . . . . 5 (𝐽 ∈ Fre β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
5 ist1-2 23264 . . . . 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 2817 . . . . . . 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 3174 . . . . 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 2732 . . . . 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 2817 . . . . . . 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 3174 . . . . 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 2740 . . . . 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 3620 . . 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 1113 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 1085   = wceq 1534   ∈ wcel 2099  βˆ€wral 3058  βˆͺ cuni 4908  β€˜cfv 6548  Topctop 22808  TopOnctopon 22825  Frect1 23224
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-iota 6500  df-fun 6550  df-fv 6556  df-topgen 17425  df-top 22809  df-topon 22826  df-cld 22936  df-t1 23231
This theorem is referenced by:  t1sep  23287  isr0  23654
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