Theorem hausflimi 24106

Description: One direction of hausflim 24107. 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.

Step	Hyp	Ref	Expression
1		simpl 487	. . . . . . . . 9 ⊢ ((𝐽 ∈ Haus ∧ ((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) ∧ 𝑥 ≠ 𝑦)) → 𝐽 ∈ Haus)
2		simprll 790	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Haus ∧ ((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) ∧ 𝑥 ≠ 𝑦)) → 𝑥 ∈ (𝐽 fLim 𝐹))
3		eqid 2769	. . . . . . . . . . 11 ⊢ ∪ 𝐽 = ∪ 𝐽
4	3	flimelbas 24094	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐽 fLim 𝐹) → 𝑥 ∈ ∪ 𝐽)
5	2, 4	syl 18	. . . . . . . . 9 ⊢ ((𝐽 ∈ Haus ∧ ((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) ∧ 𝑥 ≠ 𝑦)) → 𝑥 ∈ ∪ 𝐽)
6		simprlr 791	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Haus ∧ ((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) ∧ 𝑥 ≠ 𝑦)) → 𝑦 ∈ (𝐽 fLim 𝐹))
7	3	flimelbas 24094	. . . . . . . . . 10 ⊢ (𝑦 ∈ (𝐽 fLim 𝐹) → 𝑦 ∈ ∪ 𝐽)
8	6, 7	syl 18	. . . . . . . . 9 ⊢ ((𝐽 ∈ Haus ∧ ((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) ∧ 𝑥 ≠ 𝑦)) → 𝑦 ∈ ∪ 𝐽)
9		simprr 784	. . . . . . . . 9 ⊢ ((𝐽 ∈ Haus ∧ ((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) ∧ 𝑥 ≠ 𝑦)) → 𝑥 ≠ 𝑦)
10	3	hausnei 23454	. . . . . . . . 9 ⊢ ((𝐽 ∈ Haus ∧ (𝑥 ∈ ∪ 𝐽 ∧ 𝑦 ∈ ∪ 𝐽 ∧ 𝑥 ≠ 𝑦)) → ∃𝑢 ∈ 𝐽 ∃𝑣 ∈ 𝐽 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑣 ∧ (𝑢 ∩ 𝑣) = ∅))
11	1, 5, 8, 9, 10	syl13anc 1397	. . . . . . . 8 ⊢ ((𝐽 ∈ Haus ∧ ((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) ∧ 𝑥 ≠ 𝑦)) → ∃𝑢 ∈ 𝐽 ∃𝑣 ∈ 𝐽 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑣 ∧ (𝑢 ∩ 𝑣) = ∅))
12		df-3an 1103	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑣 ∧ (𝑢 ∩ 𝑣) = ∅) ↔ ((𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑣) ∧ (𝑢 ∩ 𝑣) = ∅))
13		simprl 782	. . . . . . . . . . . . . 14 ⊢ ((𝐽 ∈ Haus ∧ ((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) ∧ 𝑥 ≠ 𝑦)) → (𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)))
14		hausflimlem 24105	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑣 ∈ 𝐽) ∧ (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑣)) → (𝑢 ∩ 𝑣) ≠ ∅)
15	14	3expa 1134	. . . . . . . . . . . . . 14 ⊢ ((((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑣 ∈ 𝐽)) ∧ (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑣)) → (𝑢 ∩ 𝑣) ≠ ∅)
16	13, 15	sylanl1 692	. . . . . . . . . . . . 13 ⊢ ((((𝐽 ∈ Haus ∧ ((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) ∧ 𝑥 ≠ 𝑦)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑣 ∈ 𝐽)) ∧ (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑣)) → (𝑢 ∩ 𝑣) ≠ ∅)
17	16	a1d 26	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ Haus ∧ ((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) ∧ 𝑥 ≠ 𝑦)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑣 ∈ 𝐽)) ∧ (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑣)) → (𝑥 ≠ 𝑦 → (𝑢 ∩ 𝑣) ≠ ∅))
18	17	necon4d 2988	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ Haus ∧ ((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) ∧ 𝑥 ≠ 𝑦)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑣 ∈ 𝐽)) ∧ (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑣)) → ((𝑢 ∩ 𝑣) = ∅ → 𝑥 = 𝑦))
19	18	expimpd 458	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Haus ∧ ((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) ∧ 𝑥 ≠ 𝑦)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑣 ∈ 𝐽)) → (((𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑣) ∧ (𝑢 ∩ 𝑣) = ∅) → 𝑥 = 𝑦))
20	12, 19	biimtrid 245	. . . . . . . . 9 ⊢ (((𝐽 ∈ Haus ∧ ((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) ∧ 𝑥 ≠ 𝑦)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑣 ∈ 𝐽)) → ((𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑣 ∧ (𝑢 ∩ 𝑣) = ∅) → 𝑥 = 𝑦))
21	20	rexlimdvva 3228	. . . . . . . 8 ⊢ ((𝐽 ∈ Haus ∧ ((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) ∧ 𝑥 ≠ 𝑦)) → (∃𝑢 ∈ 𝐽 ∃𝑣 ∈ 𝐽 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑣 ∧ (𝑢 ∩ 𝑣) = ∅) → 𝑥 = 𝑦))
22	11, 21	mpd 16	. . . . . . 7 ⊢ ((𝐽 ∈ Haus ∧ ((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) ∧ 𝑥 ≠ 𝑦)) → 𝑥 = 𝑦)
23	22	expr 461	. . . . . 6 ⊢ ((𝐽 ∈ Haus ∧ (𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹))) → (𝑥 ≠ 𝑦 → 𝑥 = 𝑦))
24	23	necon1bd 2982	. . . . 5 ⊢ ((𝐽 ∈ Haus ∧ (𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹))) → (¬ 𝑥 = 𝑦 → 𝑥 = 𝑦))
25	24	pm2.18d 128	. . . 4 ⊢ ((𝐽 ∈ Haus ∧ (𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹))) → 𝑥 = 𝑦)
26	25	ex 417	. . 3 ⊢ (𝐽 ∈ Haus → ((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) → 𝑥 = 𝑦))
27	26	alrimivv 1955	. 2 ⊢ (𝐽 ∈ Haus → ∀𝑥∀𝑦((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) → 𝑥 = 𝑦))
28		eleq1w 2852	. . 3 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ (𝐽 fLim 𝐹) ↔ 𝑦 ∈ (𝐽 fLim 𝐹)))
29	28	mo4 2600	. 2 ⊢ (∃*𝑥 𝑥 ∈ (𝐽 fLim 𝐹) ↔ ∀𝑥∀𝑦((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) → 𝑥 = 𝑦))
30	27, 29	sylibr 237	1 ⊢ (𝐽 ∈ Haus → ∃*𝑥 𝑥 ∈ (𝐽 fLim 𝐹))

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1101  ∀wal 1565   = wceq 1567   ∈ wcel 2149  ∃*wmo 2571   ≠ wne 2964  ∃wrex 3095   ∩ cin 3910  ∅c0 4292  ∪ cuni 4874  (class class class)co 7411  Hauscha 23434   fLim cflim 24060

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5337  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-reu 3376  df-rab 3423  df-v 3463  df-sbc 3752  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4491  df-pw 4567  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7414  df-oprab 7415  df-mpo 7416  df-fbas 21488  df-top 23020  df-nei 23224  df-haus 23441  df-fil 23972  df-flim 24065

This theorem is referenced by:  hausflim  24107  hausflf  24123  metsscmetcld  25443  minveclem4a  25558