Theorem hausflimi 23331

Description: One direction of hausflim 23332. 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 483	. . . . . . . . 9 ⊢ ((𝐽 ∈ Haus ∧ ((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) ∧ 𝑥 ≠ 𝑦)) → 𝐽 ∈ Haus)
2		simprll 777	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Haus ∧ ((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) ∧ 𝑥 ≠ 𝑦)) → 𝑥 ∈ (𝐽 fLim 𝐹))
3		eqid 2736	. . . . . . . . . . 11 ⊢ ∪ 𝐽 = ∪ 𝐽
4	3	flimelbas 23319	. . . . . . . . . 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 778	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Haus ∧ ((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) ∧ 𝑥 ≠ 𝑦)) → 𝑦 ∈ (𝐽 fLim 𝐹))
7	3	flimelbas 23319	. . . . . . . . . 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 771	. . . . . . . . 9 ⊢ ((𝐽 ∈ Haus ∧ ((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) ∧ 𝑥 ≠ 𝑦)) → 𝑥 ≠ 𝑦)
10	3	hausnei 22679	. . . . . . . . 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 1372	. . . . . . . 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 1089	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑣 ∧ (𝑢 ∩ 𝑣) = ∅) ↔ ((𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑣) ∧ (𝑢 ∩ 𝑣) = ∅))
13		simprl 769	. . . . . . . . . . . . . 14 ⊢ ((𝐽 ∈ Haus ∧ ((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) ∧ 𝑥 ≠ 𝑦)) → (𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)))
14		hausflimlem 23330	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑣 ∈ 𝐽) ∧ (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑣)) → (𝑢 ∩ 𝑣) ≠ ∅)
15	14	3expa 1118	. . . . . . . . . . . . . 14 ⊢ ((((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑣 ∈ 𝐽)) ∧ (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑣)) → (𝑢 ∩ 𝑣) ≠ ∅)
16	13, 15	sylanl1 678	. . . . . . . . . . . . 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 2967	. . . . . . . . . . 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 454	. . . . . . . . . 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 241	. . . . . . . . 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 3205	. . . . . . . 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 457	. . . . . 6 ⊢ ((𝐽 ∈ Haus ∧ (𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹))) → (𝑥 ≠ 𝑦 → 𝑥 = 𝑦))
24	23	necon1bd 2961	. . . . 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 413	. . 3 ⊢ (𝐽 ∈ Haus → ((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) → 𝑥 = 𝑦))
27	26	alrimivv 1931	. 2 ⊢ (𝐽 ∈ Haus → ∀𝑥∀𝑦((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) → 𝑥 = 𝑦))
28		eleq1w 2820	. . 3 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ (𝐽 fLim 𝐹) ↔ 𝑦 ∈ (𝐽 fLim 𝐹)))
29	28	mo4 2564	. 2 ⊢ (∃*𝑥 𝑥 ∈ (𝐽 fLim 𝐹) ↔ ∀𝑥∀𝑦((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) → 𝑥 = 𝑦))
30	27, 29	sylibr 233	1 ⊢ (𝐽 ∈ Haus → ∃*𝑥 𝑥 ∈ (𝐽 fLim 𝐹))

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1087  ∀wal 1539   = wceq 1541   ∈ wcel 2106  ∃*wmo 2536   ≠ wne 2943  ∃wrex 3073   ∩ cin 3909  ∅c0 4282  ∪ cuni 4865  (class class class)co 7357  Hauscha 22659   fLim cflim 23285

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-fbas 20793  df-top 22243  df-nei 22449  df-haus 22666  df-fil 23197  df-flim 23290

This theorem is referenced by:  hausflim  23332  hausflf  23348  metsscmetcld  24679  minveclem4a  24794