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Theorem hausflimi 23277
Description: One direction of hausflim 23278. A filter in a Hausdorff space has at most one limit. (Contributed by FL, 14-Nov-2010.) (Revised by Mario Carneiro, 21-Sep-2015.)
Assertion
Ref Expression
hausflimi (𝐽 ∈ Haus → ∃*𝑥 𝑥 ∈ (𝐽 fLim 𝐹))
Distinct variable groups:   𝑥,𝐹   𝑥,𝐽

Proof of Theorem hausflimi
Dummy variables 𝑣 𝑢 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 483 . . . . . . . . 9 ((𝐽 ∈ Haus ∧ ((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) ∧ 𝑥𝑦)) → 𝐽 ∈ Haus)
2 simprll 777 . . . . . . . . . 10 ((𝐽 ∈ Haus ∧ ((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) ∧ 𝑥𝑦)) → 𝑥 ∈ (𝐽 fLim 𝐹))
3 eqid 2736 . . . . . . . . . . 11 𝐽 = 𝐽
43flimelbas 23265 . . . . . . . . . 10 (𝑥 ∈ (𝐽 fLim 𝐹) → 𝑥 𝐽)
52, 4syl 17 . . . . . . . . 9 ((𝐽 ∈ Haus ∧ ((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) ∧ 𝑥𝑦)) → 𝑥 𝐽)
6 simprlr 778 . . . . . . . . . 10 ((𝐽 ∈ Haus ∧ ((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) ∧ 𝑥𝑦)) → 𝑦 ∈ (𝐽 fLim 𝐹))
73flimelbas 23265 . . . . . . . . . 10 (𝑦 ∈ (𝐽 fLim 𝐹) → 𝑦 𝐽)
86, 7syl 17 . . . . . . . . 9 ((𝐽 ∈ Haus ∧ ((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) ∧ 𝑥𝑦)) → 𝑦 𝐽)
9 simprr 771 . . . . . . . . 9 ((𝐽 ∈ Haus ∧ ((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) ∧ 𝑥𝑦)) → 𝑥𝑦)
103hausnei 22625 . . . . . . . . 9 ((𝐽 ∈ Haus ∧ (𝑥 𝐽𝑦 𝐽𝑥𝑦)) → ∃𝑢𝐽𝑣𝐽 (𝑥𝑢𝑦𝑣 ∧ (𝑢𝑣) = ∅))
111, 5, 8, 9, 10syl13anc 1372 . . . . . . . 8 ((𝐽 ∈ Haus ∧ ((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) ∧ 𝑥𝑦)) → ∃𝑢𝐽𝑣𝐽 (𝑥𝑢𝑦𝑣 ∧ (𝑢𝑣) = ∅))
12 df-3an 1089 . . . . . . . . . 10 ((𝑥𝑢𝑦𝑣 ∧ (𝑢𝑣) = ∅) ↔ ((𝑥𝑢𝑦𝑣) ∧ (𝑢𝑣) = ∅))
13 simprl 769 . . . . . . . . . . . . . 14 ((𝐽 ∈ Haus ∧ ((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) ∧ 𝑥𝑦)) → (𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)))
14 hausflimlem 23276 . . . . . . . . . . . . . . 15 (((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) ∧ (𝑢𝐽𝑣𝐽) ∧ (𝑥𝑢𝑦𝑣)) → (𝑢𝑣) ≠ ∅)
15143expa 1118 . . . . . . . . . . . . . 14 ((((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) ∧ (𝑢𝐽𝑣𝐽)) ∧ (𝑥𝑢𝑦𝑣)) → (𝑢𝑣) ≠ ∅)
1613, 15sylanl1 678 . . . . . . . . . . . . 13 ((((𝐽 ∈ Haus ∧ ((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) ∧ 𝑥𝑦)) ∧ (𝑢𝐽𝑣𝐽)) ∧ (𝑥𝑢𝑦𝑣)) → (𝑢𝑣) ≠ ∅)
1716a1d 25 . . . . . . . . . . . 12 ((((𝐽 ∈ Haus ∧ ((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) ∧ 𝑥𝑦)) ∧ (𝑢𝐽𝑣𝐽)) ∧ (𝑥𝑢𝑦𝑣)) → (𝑥𝑦 → (𝑢𝑣) ≠ ∅))
1817necon4d 2965 . . . . . . . . . . 11 ((((𝐽 ∈ Haus ∧ ((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) ∧ 𝑥𝑦)) ∧ (𝑢𝐽𝑣𝐽)) ∧ (𝑥𝑢𝑦𝑣)) → ((𝑢𝑣) = ∅ → 𝑥 = 𝑦))
1918expimpd 454 . . . . . . . . . 10 (((𝐽 ∈ Haus ∧ ((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) ∧ 𝑥𝑦)) ∧ (𝑢𝐽𝑣𝐽)) → (((𝑥𝑢𝑦𝑣) ∧ (𝑢𝑣) = ∅) → 𝑥 = 𝑦))
2012, 19biimtrid 241 . . . . . . . . 9 (((𝐽 ∈ Haus ∧ ((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) ∧ 𝑥𝑦)) ∧ (𝑢𝐽𝑣𝐽)) → ((𝑥𝑢𝑦𝑣 ∧ (𝑢𝑣) = ∅) → 𝑥 = 𝑦))
2120rexlimdvva 3203 . . . . . . . 8 ((𝐽 ∈ Haus ∧ ((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) ∧ 𝑥𝑦)) → (∃𝑢𝐽𝑣𝐽 (𝑥𝑢𝑦𝑣 ∧ (𝑢𝑣) = ∅) → 𝑥 = 𝑦))
2211, 21mpd 15 . . . . . . 7 ((𝐽 ∈ Haus ∧ ((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) ∧ 𝑥𝑦)) → 𝑥 = 𝑦)
2322expr 457 . . . . . 6 ((𝐽 ∈ Haus ∧ (𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹))) → (𝑥𝑦𝑥 = 𝑦))
2423necon1bd 2959 . . . . 5 ((𝐽 ∈ Haus ∧ (𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹))) → (¬ 𝑥 = 𝑦𝑥 = 𝑦))
2524pm2.18d 127 . . . 4 ((𝐽 ∈ Haus ∧ (𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹))) → 𝑥 = 𝑦)
2625ex 413 . . 3 (𝐽 ∈ Haus → ((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) → 𝑥 = 𝑦))
2726alrimivv 1931 . 2 (𝐽 ∈ Haus → ∀𝑥𝑦((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) → 𝑥 = 𝑦))
28 eleq1w 2820 . . 3 (𝑥 = 𝑦 → (𝑥 ∈ (𝐽 fLim 𝐹) ↔ 𝑦 ∈ (𝐽 fLim 𝐹)))
2928mo4 2564 . 2 (∃*𝑥 𝑥 ∈ (𝐽 fLim 𝐹) ↔ ∀𝑥𝑦((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) → 𝑥 = 𝑦))
3027, 29sylibr 233 1 (𝐽 ∈ Haus → ∃*𝑥 𝑥 ∈ (𝐽 fLim 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1087  wal 1539   = wceq 1541  wcel 2106  ∃*wmo 2536  wne 2941  wrex 3071  cin 3907  c0 4280   cuni 4863  (class class class)co 7351  Hauscha 22605   fLim cflim 23231
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 2707  ax-rep 5240  ax-sep 5254  ax-nul 5261  ax-pow 5318  ax-pr 5382
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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3352  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-iun 4954  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-ov 7354  df-oprab 7355  df-mpo 7356  df-fbas 20740  df-top 22189  df-nei 22395  df-haus 22612  df-fil 23143  df-flim 23236
This theorem is referenced by:  hausflim  23278  hausflf  23294  metsscmetcld  24625  minveclem4a  24740
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