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Theorem hausflimi 21778
Description: One direction of hausflim 21779. 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 473 . . . . . . . . 9 ((𝐽 ∈ Haus ∧ ((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) ∧ 𝑥𝑦)) → 𝐽 ∈ Haus)
2 simprll 802 . . . . . . . . . 10 ((𝐽 ∈ Haus ∧ ((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) ∧ 𝑥𝑦)) → 𝑥 ∈ (𝐽 fLim 𝐹))
3 eqid 2621 . . . . . . . . . . 11 𝐽 = 𝐽
43flimelbas 21766 . . . . . . . . . 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 803 . . . . . . . . . 10 ((𝐽 ∈ Haus ∧ ((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) ∧ 𝑥𝑦)) → 𝑦 ∈ (𝐽 fLim 𝐹))
73flimelbas 21766 . . . . . . . . . 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 796 . . . . . . . . 9 ((𝐽 ∈ Haus ∧ ((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) ∧ 𝑥𝑦)) → 𝑥𝑦)
103hausnei 21126 . . . . . . . . 9 ((𝐽 ∈ Haus ∧ (𝑥 𝐽𝑦 𝐽𝑥𝑦)) → ∃𝑢𝐽𝑣𝐽 (𝑥𝑢𝑦𝑣 ∧ (𝑢𝑣) = ∅))
111, 5, 8, 9, 10syl13anc 1327 . . . . . . . 8 ((𝐽 ∈ Haus ∧ ((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) ∧ 𝑥𝑦)) → ∃𝑢𝐽𝑣𝐽 (𝑥𝑢𝑦𝑣 ∧ (𝑢𝑣) = ∅))
12 df-3an 1039 . . . . . . . . . 10 ((𝑥𝑢𝑦𝑣 ∧ (𝑢𝑣) = ∅) ↔ ((𝑥𝑢𝑦𝑣) ∧ (𝑢𝑣) = ∅))
13 simprl 794 . . . . . . . . . . . . . 14 ((𝐽 ∈ Haus ∧ ((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) ∧ 𝑥𝑦)) → (𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)))
14 hausflimlem 21777 . . . . . . . . . . . . . . 15 (((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) ∧ (𝑢𝐽𝑣𝐽) ∧ (𝑥𝑢𝑦𝑣)) → (𝑢𝑣) ≠ ∅)
15143expa 1264 . . . . . . . . . . . . . 14 ((((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) ∧ (𝑢𝐽𝑣𝐽)) ∧ (𝑥𝑢𝑦𝑣)) → (𝑢𝑣) ≠ ∅)
1613, 15sylanl1 682 . . . . . . . . . . . . 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 2817 . . . . . . . . . . 11 ((((𝐽 ∈ Haus ∧ ((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) ∧ 𝑥𝑦)) ∧ (𝑢𝐽𝑣𝐽)) ∧ (𝑥𝑢𝑦𝑣)) → ((𝑢𝑣) = ∅ → 𝑥 = 𝑦))
1918expimpd 629 . . . . . . . . . 10 (((𝐽 ∈ Haus ∧ ((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) ∧ 𝑥𝑦)) ∧ (𝑢𝐽𝑣𝐽)) → (((𝑥𝑢𝑦𝑣) ∧ (𝑢𝑣) = ∅) → 𝑥 = 𝑦))
2012, 19syl5bi 232 . . . . . . . . 9 (((𝐽 ∈ Haus ∧ ((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) ∧ 𝑥𝑦)) ∧ (𝑢𝐽𝑣𝐽)) → ((𝑥𝑢𝑦𝑣 ∧ (𝑢𝑣) = ∅) → 𝑥 = 𝑦))
2120rexlimdvva 3036 . . . . . . . 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 643 . . . . . 6 ((𝐽 ∈ Haus ∧ (𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹))) → (𝑥𝑦𝑥 = 𝑦))
2423necon1bd 2811 . . . . 5 ((𝐽 ∈ Haus ∧ (𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹))) → (¬ 𝑥 = 𝑦𝑥 = 𝑦))
2524pm2.18d 124 . . . 4 ((𝐽 ∈ Haus ∧ (𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹))) → 𝑥 = 𝑦)
2625ex 450 . . 3 (𝐽 ∈ Haus → ((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) → 𝑥 = 𝑦))
2726alrimivv 1855 . 2 (𝐽 ∈ Haus → ∀𝑥𝑦((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) → 𝑥 = 𝑦))
28 eleq1 2688 . . 3 (𝑥 = 𝑦 → (𝑥 ∈ (𝐽 fLim 𝐹) ↔ 𝑦 ∈ (𝐽 fLim 𝐹)))
2928mo4 2516 . 2 (∃*𝑥 𝑥 ∈ (𝐽 fLim 𝐹) ↔ ∀𝑥𝑦((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) → 𝑥 = 𝑦))
3027, 29sylibr 224 1 (𝐽 ∈ Haus → ∃*𝑥 𝑥 ∈ (𝐽 fLim 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1037  wal 1480   = wceq 1482  wcel 1989  ∃*wmo 2470  wne 2793  wrex 2912  cin 3571  c0 3913   cuni 4434  (class class class)co 6647  Hauscha 21106   fLim cflim 21732
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-rep 4769  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-nel 2897  df-ral 2916  df-rex 2917  df-reu 2918  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-iun 4520  df-br 4652  df-opab 4711  df-mpt 4728  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894  df-ov 6650  df-oprab 6651  df-mpt2 6652  df-fbas 19737  df-top 20693  df-nei 20896  df-haus 21113  df-fil 21644  df-flim 21737
This theorem is referenced by:  hausflim  21779  hausflf  21795  cmetss  23107  minveclem4a  23195
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