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Theorem hausflimi 22003
Description: One direction of hausflim 22004. 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 468 . . . . . . . . 9 ((𝐽 ∈ Haus ∧ ((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) ∧ 𝑥𝑦)) → 𝐽 ∈ Haus)
2 simprll 756 . . . . . . . . . 10 ((𝐽 ∈ Haus ∧ ((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) ∧ 𝑥𝑦)) → 𝑥 ∈ (𝐽 fLim 𝐹))
3 eqid 2770 . . . . . . . . . . 11 𝐽 = 𝐽
43flimelbas 21991 . . . . . . . . . 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 757 . . . . . . . . . 10 ((𝐽 ∈ Haus ∧ ((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) ∧ 𝑥𝑦)) → 𝑦 ∈ (𝐽 fLim 𝐹))
73flimelbas 21991 . . . . . . . . . 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 748 . . . . . . . . 9 ((𝐽 ∈ Haus ∧ ((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) ∧ 𝑥𝑦)) → 𝑥𝑦)
103hausnei 21352 . . . . . . . . 9 ((𝐽 ∈ Haus ∧ (𝑥 𝐽𝑦 𝐽𝑥𝑦)) → ∃𝑢𝐽𝑣𝐽 (𝑥𝑢𝑦𝑣 ∧ (𝑢𝑣) = ∅))
111, 5, 8, 9, 10syl13anc 1477 . . . . . . . 8 ((𝐽 ∈ Haus ∧ ((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) ∧ 𝑥𝑦)) → ∃𝑢𝐽𝑣𝐽 (𝑥𝑢𝑦𝑣 ∧ (𝑢𝑣) = ∅))
12 df-3an 1072 . . . . . . . . . 10 ((𝑥𝑢𝑦𝑣 ∧ (𝑢𝑣) = ∅) ↔ ((𝑥𝑢𝑦𝑣) ∧ (𝑢𝑣) = ∅))
13 simprl 746 . . . . . . . . . . . . . 14 ((𝐽 ∈ Haus ∧ ((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) ∧ 𝑥𝑦)) → (𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)))
14 hausflimlem 22002 . . . . . . . . . . . . . . 15 (((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) ∧ (𝑢𝐽𝑣𝐽) ∧ (𝑥𝑢𝑦𝑣)) → (𝑢𝑣) ≠ ∅)
15143expa 1110 . . . . . . . . . . . . . 14 ((((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) ∧ (𝑢𝐽𝑣𝐽)) ∧ (𝑥𝑢𝑦𝑣)) → (𝑢𝑣) ≠ ∅)
1613, 15sylanl1 651 . . . . . . . . . . . . 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 2966 . . . . . . . . . . 11 ((((𝐽 ∈ Haus ∧ ((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) ∧ 𝑥𝑦)) ∧ (𝑢𝐽𝑣𝐽)) ∧ (𝑥𝑢𝑦𝑣)) → ((𝑢𝑣) = ∅ → 𝑥 = 𝑦))
1918expimpd 441 . . . . . . . . . 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 3185 . . . . . . . 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 444 . . . . . 6 ((𝐽 ∈ Haus ∧ (𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹))) → (𝑥𝑦𝑥 = 𝑦))
2423necon1bd 2960 . . . . 5 ((𝐽 ∈ Haus ∧ (𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹))) → (¬ 𝑥 = 𝑦𝑥 = 𝑦))
2524pm2.18d 125 . . . 4 ((𝐽 ∈ Haus ∧ (𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹))) → 𝑥 = 𝑦)
2625ex 397 . . 3 (𝐽 ∈ Haus → ((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) → 𝑥 = 𝑦))
2726alrimivv 2007 . 2 (𝐽 ∈ Haus → ∀𝑥𝑦((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) → 𝑥 = 𝑦))
28 eleq1w 2832 . . 3 (𝑥 = 𝑦 → (𝑥 ∈ (𝐽 fLim 𝐹) ↔ 𝑦 ∈ (𝐽 fLim 𝐹)))
2928mo4 2665 . 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 382  w3a 1070  wal 1628   = wceq 1630  wcel 2144  ∃*wmo 2618  wne 2942  wrex 3061  cin 3720  c0 4061   cuni 4572  (class class class)co 6792  Hauscha 21332   fLim cflim 21957
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-nel 3046  df-ral 3065  df-rex 3066  df-reu 3067  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-fbas 19957  df-top 20918  df-nei 21122  df-haus 21339  df-fil 21869  df-flim 21962
This theorem is referenced by:  hausflim  22004  hausflf  22020  cmetss  23331  minveclem4a  23419
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