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Theorem t1sep2 22864
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 22825 . . . . . 6 (𝐽 ∈ Fre β†’ 𝐽 ∈ Top)
2 t1sep.1 . . . . . . 7 𝑋 = βˆͺ 𝐽
32toptopon 22410 . . . . . 6 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOnβ€˜π‘‹))
41, 3sylib 217 . . . . 5 (𝐽 ∈ Fre β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
5 ist1-2 22842 . . . . 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 266 . . 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 2821 . . . . . . 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 3177 . . . . 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 2736 . . . . 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 2821 . . . . . . 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 3177 . . . . 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 2744 . . . . 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 3621 . . 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 507 . 2 ((𝐽 ∈ Fre ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋)) β†’ (βˆ€π‘œ ∈ 𝐽 (𝐴 ∈ π‘œ β†’ 𝐡 ∈ π‘œ) β†’ 𝐴 = 𝐡))
20193impb 1115 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 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061  βˆͺ cuni 4907  β€˜cfv 6540  Topctop 22386  TopOnctopon 22403  Frect1 22802
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-iota 6492  df-fun 6542  df-fv 6548  df-topgen 17385  df-top 22387  df-topon 22404  df-cld 22514  df-t1 22809
This theorem is referenced by:  t1sep  22865  isr0  23232
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