Theorem hausflimi 22161

Description: One direction of hausflim 22162. 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 476	. . . . . . . . 9 ⊢ ((𝐽 ∈ Haus ∧ ((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) ∧ 𝑥 ≠ 𝑦)) → 𝐽 ∈ Haus)
2		simprll 797	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Haus ∧ ((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) ∧ 𝑥 ≠ 𝑦)) → 𝑥 ∈ (𝐽 fLim 𝐹))
3		eqid 2825	. . . . . . . . . . 11 ⊢ ∪ 𝐽 = ∪ 𝐽
4	3	flimelbas 22149	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐽 fLim 𝐹) → 𝑥 ∈ ∪ 𝐽)
5	2, 4	syl 17	. . . . . . . . 9 ⊢ ((𝐽 ∈ Haus ∧ ((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) ∧ 𝑥 ≠ 𝑦)) → 𝑥 ∈ ∪ 𝐽)
6		simprlr 798	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Haus ∧ ((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) ∧ 𝑥 ≠ 𝑦)) → 𝑦 ∈ (𝐽 fLim 𝐹))
7	3	flimelbas 22149	. . . . . . . . . 10 ⊢ (𝑦 ∈ (𝐽 fLim 𝐹) → 𝑦 ∈ ∪ 𝐽)
8	6, 7	syl 17	. . . . . . . . 9 ⊢ ((𝐽 ∈ Haus ∧ ((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) ∧ 𝑥 ≠ 𝑦)) → 𝑦 ∈ ∪ 𝐽)
9		simprr 789	. . . . . . . . 9 ⊢ ((𝐽 ∈ Haus ∧ ((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) ∧ 𝑥 ≠ 𝑦)) → 𝑥 ≠ 𝑦)
10	3	hausnei 21510	. . . . . . . . 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 1495	. . . . . . . 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 1113	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑣 ∧ (𝑢 ∩ 𝑣) = ∅) ↔ ((𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑣) ∧ (𝑢 ∩ 𝑣) = ∅))
13		simprl 787	. . . . . . . . . . . . . 14 ⊢ ((𝐽 ∈ Haus ∧ ((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) ∧ 𝑥 ≠ 𝑦)) → (𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)))
14		hausflimlem 22160	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑣 ∈ 𝐽) ∧ (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑣)) → (𝑢 ∩ 𝑣) ≠ ∅)
15	14	3expa 1151	. . . . . . . . . . . . . 14 ⊢ ((((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑣 ∈ 𝐽)) ∧ (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑣)) → (𝑢 ∩ 𝑣) ≠ ∅)
16	13, 15	sylanl1 670	. . . . . . . . . . . . 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 25	. . . . . . . . . . . 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 3023	. . . . . . . . . . 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 447	. . . . . . . . . 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	syl5bi 234	. . . . . . . . 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 3248	. . . . . . . 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 15	. . . . . . 7 ⊢ ((𝐽 ∈ Haus ∧ ((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) ∧ 𝑥 ≠ 𝑦)) → 𝑥 = 𝑦)
23	22	expr 450	. . . . . 6 ⊢ ((𝐽 ∈ Haus ∧ (𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹))) → (𝑥 ≠ 𝑦 → 𝑥 = 𝑦))
24	23	necon1bd 3017	. . . . 5 ⊢ ((𝐽 ∈ Haus ∧ (𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹))) → (¬ 𝑥 = 𝑦 → 𝑥 = 𝑦))
25	24	pm2.18d 127	. . . 4 ⊢ ((𝐽 ∈ Haus ∧ (𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹))) → 𝑥 = 𝑦)
26	25	ex 403	. . 3 ⊢ (𝐽 ∈ Haus → ((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) → 𝑥 = 𝑦))
27	26	alrimivv 2027	. 2 ⊢ (𝐽 ∈ Haus → ∀𝑥∀𝑦((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) → 𝑥 = 𝑦))
28		eleq1w 2889	. . 3 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ (𝐽 fLim 𝐹) ↔ 𝑦 ∈ (𝐽 fLim 𝐹)))
29	28	mo4 2638	. 2 ⊢ (∃*𝑥 𝑥 ∈ (𝐽 fLim 𝐹) ↔ ∀𝑥∀𝑦((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) → 𝑥 = 𝑦))
30	27, 29	sylibr 226	1 ⊢ (𝐽 ∈ Haus → ∃*𝑥 𝑥 ∈ (𝐽 fLim 𝐹))

Syntax hints:   → wi 4   ∧ wa 386   ∧ w3a 1111  ∀wal 1654   = wceq 1656   ∈ wcel 2164  ∃*wmo 2603   ≠ wne 2999  ∃wrex 3118   ∩ cin 3797  ∅c0 4146  ∪ cuni 4660  (class class class)co 6910  Hauscha 21490   fLim cflim 22115

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4996  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-nel 3103  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-iun 4744  df-br 4876  df-opab 4938  df-mpt 4955  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-ov 6913  df-oprab 6914  df-mpt2 6915  df-fbas 20110  df-top 21076  df-nei 21280  df-haus 21497  df-fil 22027  df-flim 22120

This theorem is referenced by:  hausflim  22162  hausflf  22178  metsscmetcld  23490  minveclem4a  23605