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Theorem hausflimi 24004
Description: One direction of hausflim 24005. 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 482 . . . . . . . . 9 ((𝐽 ∈ Haus ∧ ((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) ∧ 𝑥𝑦)) → 𝐽 ∈ Haus)
2 simprll 779 . . . . . . . . . 10 ((𝐽 ∈ Haus ∧ ((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) ∧ 𝑥𝑦)) → 𝑥 ∈ (𝐽 fLim 𝐹))
3 eqid 2735 . . . . . . . . . . 11 𝐽 = 𝐽
43flimelbas 23992 . . . . . . . . . 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 780 . . . . . . . . . 10 ((𝐽 ∈ Haus ∧ ((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) ∧ 𝑥𝑦)) → 𝑦 ∈ (𝐽 fLim 𝐹))
73flimelbas 23992 . . . . . . . . . 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 773 . . . . . . . . 9 ((𝐽 ∈ Haus ∧ ((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) ∧ 𝑥𝑦)) → 𝑥𝑦)
103hausnei 23352 . . . . . . . . 9 ((𝐽 ∈ Haus ∧ (𝑥 𝐽𝑦 𝐽𝑥𝑦)) → ∃𝑢𝐽𝑣𝐽 (𝑥𝑢𝑦𝑣 ∧ (𝑢𝑣) = ∅))
111, 5, 8, 9, 10syl13anc 1371 . . . . . . . 8 ((𝐽 ∈ Haus ∧ ((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) ∧ 𝑥𝑦)) → ∃𝑢𝐽𝑣𝐽 (𝑥𝑢𝑦𝑣 ∧ (𝑢𝑣) = ∅))
12 df-3an 1088 . . . . . . . . . 10 ((𝑥𝑢𝑦𝑣 ∧ (𝑢𝑣) = ∅) ↔ ((𝑥𝑢𝑦𝑣) ∧ (𝑢𝑣) = ∅))
13 simprl 771 . . . . . . . . . . . . . 14 ((𝐽 ∈ Haus ∧ ((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) ∧ 𝑥𝑦)) → (𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)))
14 hausflimlem 24003 . . . . . . . . . . . . . . 15 (((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) ∧ (𝑢𝐽𝑣𝐽) ∧ (𝑥𝑢𝑦𝑣)) → (𝑢𝑣) ≠ ∅)
15143expa 1117 . . . . . . . . . . . . . 14 ((((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) ∧ (𝑢𝐽𝑣𝐽)) ∧ (𝑥𝑢𝑦𝑣)) → (𝑢𝑣) ≠ ∅)
1613, 15sylanl1 680 . . . . . . . . . . . . 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 2962 . . . . . . . . . . 11 ((((𝐽 ∈ Haus ∧ ((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) ∧ 𝑥𝑦)) ∧ (𝑢𝐽𝑣𝐽)) ∧ (𝑥𝑢𝑦𝑣)) → ((𝑢𝑣) = ∅ → 𝑥 = 𝑦))
1918expimpd 453 . . . . . . . . . 10 (((𝐽 ∈ Haus ∧ ((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) ∧ 𝑥𝑦)) ∧ (𝑢𝐽𝑣𝐽)) → (((𝑥𝑢𝑦𝑣) ∧ (𝑢𝑣) = ∅) → 𝑥 = 𝑦))
2012, 19biimtrid 242 . . . . . . . . 9 (((𝐽 ∈ Haus ∧ ((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) ∧ 𝑥𝑦)) ∧ (𝑢𝐽𝑣𝐽)) → ((𝑥𝑢𝑦𝑣 ∧ (𝑢𝑣) = ∅) → 𝑥 = 𝑦))
2120rexlimdvva 3211 . . . . . . . 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 456 . . . . . 6 ((𝐽 ∈ Haus ∧ (𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹))) → (𝑥𝑦𝑥 = 𝑦))
2423necon1bd 2956 . . . . 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 412 . . 3 (𝐽 ∈ Haus → ((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) → 𝑥 = 𝑦))
2726alrimivv 1926 . 2 (𝐽 ∈ Haus → ∀𝑥𝑦((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) → 𝑥 = 𝑦))
28 eleq1w 2822 . . 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 234 1 (𝐽 ∈ Haus → ∃*𝑥 𝑥 ∈ (𝐽 fLim 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086  wal 1535   = wceq 1537  wcel 2106  ∃*wmo 2536  wne 2938  wrex 3068  cin 3962  c0 4339   cuni 4912  (class class class)co 7431  Hauscha 23332   fLim cflim 23958
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-fbas 21379  df-top 22916  df-nei 23122  df-haus 23339  df-fil 23870  df-flim 23963
This theorem is referenced by:  hausflim  24005  hausflf  24021  metsscmetcld  25363  minveclem4a  25478
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