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Theorem hausflimi 22585
Description: One direction of hausflim 22586. 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 486 . . . . . . . . 9 ((𝐽 ∈ Haus ∧ ((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) ∧ 𝑥𝑦)) → 𝐽 ∈ Haus)
2 simprll 778 . . . . . . . . . 10 ((𝐽 ∈ Haus ∧ ((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) ∧ 𝑥𝑦)) → 𝑥 ∈ (𝐽 fLim 𝐹))
3 eqid 2798 . . . . . . . . . . 11 𝐽 = 𝐽
43flimelbas 22573 . . . . . . . . . 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 779 . . . . . . . . . 10 ((𝐽 ∈ Haus ∧ ((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) ∧ 𝑥𝑦)) → 𝑦 ∈ (𝐽 fLim 𝐹))
73flimelbas 22573 . . . . . . . . . 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 772 . . . . . . . . 9 ((𝐽 ∈ Haus ∧ ((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) ∧ 𝑥𝑦)) → 𝑥𝑦)
103hausnei 21933 . . . . . . . . 9 ((𝐽 ∈ Haus ∧ (𝑥 𝐽𝑦 𝐽𝑥𝑦)) → ∃𝑢𝐽𝑣𝐽 (𝑥𝑢𝑦𝑣 ∧ (𝑢𝑣) = ∅))
111, 5, 8, 9, 10syl13anc 1369 . . . . . . . 8 ((𝐽 ∈ Haus ∧ ((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) ∧ 𝑥𝑦)) → ∃𝑢𝐽𝑣𝐽 (𝑥𝑢𝑦𝑣 ∧ (𝑢𝑣) = ∅))
12 df-3an 1086 . . . . . . . . . 10 ((𝑥𝑢𝑦𝑣 ∧ (𝑢𝑣) = ∅) ↔ ((𝑥𝑢𝑦𝑣) ∧ (𝑢𝑣) = ∅))
13 simprl 770 . . . . . . . . . . . . . 14 ((𝐽 ∈ Haus ∧ ((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) ∧ 𝑥𝑦)) → (𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)))
14 hausflimlem 22584 . . . . . . . . . . . . . . 15 (((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) ∧ (𝑢𝐽𝑣𝐽) ∧ (𝑥𝑢𝑦𝑣)) → (𝑢𝑣) ≠ ∅)
15143expa 1115 . . . . . . . . . . . . . 14 ((((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) ∧ (𝑢𝐽𝑣𝐽)) ∧ (𝑥𝑢𝑦𝑣)) → (𝑢𝑣) ≠ ∅)
1613, 15sylanl1 679 . . . . . . . . . . . . 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 3011 . . . . . . . . . . 11 ((((𝐽 ∈ Haus ∧ ((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) ∧ 𝑥𝑦)) ∧ (𝑢𝐽𝑣𝐽)) ∧ (𝑥𝑢𝑦𝑣)) → ((𝑢𝑣) = ∅ → 𝑥 = 𝑦))
1918expimpd 457 . . . . . . . . . 10 (((𝐽 ∈ Haus ∧ ((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) ∧ 𝑥𝑦)) ∧ (𝑢𝐽𝑣𝐽)) → (((𝑥𝑢𝑦𝑣) ∧ (𝑢𝑣) = ∅) → 𝑥 = 𝑦))
2012, 19syl5bi 245 . . . . . . . . 9 (((𝐽 ∈ Haus ∧ ((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) ∧ 𝑥𝑦)) ∧ (𝑢𝐽𝑣𝐽)) → ((𝑥𝑢𝑦𝑣 ∧ (𝑢𝑣) = ∅) → 𝑥 = 𝑦))
2120rexlimdvva 3253 . . . . . . . 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 460 . . . . . 6 ((𝐽 ∈ Haus ∧ (𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹))) → (𝑥𝑦𝑥 = 𝑦))
2423necon1bd 3005 . . . . 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 416 . . 3 (𝐽 ∈ Haus → ((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) → 𝑥 = 𝑦))
2726alrimivv 1929 . 2 (𝐽 ∈ Haus → ∀𝑥𝑦((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) → 𝑥 = 𝑦))
28 eleq1w 2872 . . 3 (𝑥 = 𝑦 → (𝑥 ∈ (𝐽 fLim 𝐹) ↔ 𝑦 ∈ (𝐽 fLim 𝐹)))
2928mo4 2625 . 2 (∃*𝑥 𝑥 ∈ (𝐽 fLim 𝐹) ↔ ∀𝑥𝑦((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) → 𝑥 = 𝑦))
3027, 29sylibr 237 1 (𝐽 ∈ Haus → ∃*𝑥 𝑥 ∈ (𝐽 fLim 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1084  wal 1536   = wceq 1538  wcel 2111  ∃*wmo 2596  wne 2987  wrex 3107  cin 3880  c0 4243   cuni 4800  (class class class)co 7135  Hauscha 21913   fLim cflim 22539
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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-fbas 20088  df-top 21499  df-nei 21703  df-haus 21920  df-fil 22451  df-flim 22544
This theorem is referenced by:  hausflim  22586  hausflf  22602  metsscmetcld  23919  minveclem4a  24034
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