Theorem hausflimi 23967

Description: One direction of hausflim 23968. 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 484	. . . . . . . . 9 ⊢ ((𝐽 ∈ Haus ∧ ((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) ∧ 𝑥 ≠ 𝑦)) → 𝐽 ∈ Haus)
2		simprll 785	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Haus ∧ ((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) ∧ 𝑥 ≠ 𝑦)) → 𝑥 ∈ (𝐽 fLim 𝐹))
3		eqid 2741	. . . . . . . . . . 11 ⊢ ∪ 𝐽 = ∪ 𝐽
4	3	flimelbas 23955	. . . . . . . . . 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 786	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Haus ∧ ((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) ∧ 𝑥 ≠ 𝑦)) → 𝑦 ∈ (𝐽 fLim 𝐹))
7	3	flimelbas 23955	. . . . . . . . . 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 779	. . . . . . . . 9 ⊢ ((𝐽 ∈ Haus ∧ ((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) ∧ 𝑥 ≠ 𝑦)) → 𝑥 ≠ 𝑦)
10	3	hausnei 23315	. . . . . . . . 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 1381	. . . . . . . 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 1095	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑣 ∧ (𝑢 ∩ 𝑣) = ∅) ↔ ((𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑣) ∧ (𝑢 ∩ 𝑣) = ∅))
13		simprl 777	. . . . . . . . . . . . . 14 ⊢ ((𝐽 ∈ Haus ∧ ((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) ∧ 𝑥 ≠ 𝑦)) → (𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)))
14		hausflimlem 23966	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑣 ∈ 𝐽) ∧ (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑣)) → (𝑢 ∩ 𝑣) ≠ ∅)
15	14	3expa 1125	. . . . . . . . . . . . . 14 ⊢ ((((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑣 ∈ 𝐽)) ∧ (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑣)) → (𝑢 ∩ 𝑣) ≠ ∅)
16	13, 15	sylanl1 687	. . . . . . . . . . . . 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 2960	. . . . . . . . . . 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 455	. . . . . . . . . 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 244	. . . . . . . . 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 3198	. . . . . . . 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 458	. . . . . 6 ⊢ ((𝐽 ∈ Haus ∧ (𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹))) → (𝑥 ≠ 𝑦 → 𝑥 = 𝑦))
24	23	necon1bd 2954	. . . . 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 414	. . 3 ⊢ (𝐽 ∈ Haus → ((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) → 𝑥 = 𝑦))
27	26	alrimivv 1936	. 2 ⊢ (𝐽 ∈ Haus → ∀𝑥∀𝑦((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) → 𝑥 = 𝑦))
28		eleq1w 2824	. . 3 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ (𝐽 fLim 𝐹) ↔ 𝑦 ∈ (𝐽 fLim 𝐹)))
29	28	mo4 2572	. 2 ⊢ (∃*𝑥 𝑥 ∈ (𝐽 fLim 𝐹) ↔ ∀𝑥∀𝑦((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ (𝐽 fLim 𝐹)) → 𝑥 = 𝑦))
30	27, 29	sylibr 236	1 ⊢ (𝐽 ∈ Haus → ∃*𝑥 𝑥 ∈ (𝐽 fLim 𝐹))

Syntax hints:   → wi 4   ∧ wa 397   ∧ w3a 1093  ∀wal 1546   = wceq 1548   ∈ wcel 2121  ∃*wmo 2543   ≠ wne 2936  ∃wrex 3065   ∩ cin 3884  ∅c0 4264  ∪ cuni 4841  (class class class)co 7360  Hauscha 23295   fLim cflim 23921

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-rep 5202  ax-sep 5221  ax-nul 5231  ax-pow 5297  ax-pr 5365

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-nel 3041  df-ral 3056  df-rex 3066  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-iun 4926  df-br 5076  df-opab 5138  df-mpt 5157  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-ov 7363  df-oprab 7364  df-mpo 7365  df-fbas 21348  df-top 22881  df-nei 23085  df-haus 23302  df-fil 23833  df-flim 23926

This theorem is referenced by:  hausflim  23968  hausflf  23984  metsscmetcld  25304  minveclem4a  25419