Theorem snfil 24238

Description: A singleton is a filter. Example 1 of [BourbakiTop1] p. I.36. (Contributed by FL, 16-Sep-2007.) (Revised by Stefan O'Rear, 2-Aug-2015.)

Assertion

Ref	Expression
snfil	⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ≠ ∅) → {𝐴} ∈ (Fil‘𝐴))

Proof of Theorem snfil

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		velsn 4600	. . . 4 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
2		eqimss 3971	. . . . 5 ⊢ (𝑥 = 𝐴 → 𝑥 ⊆ 𝐴)
3	2	pm4.71ri 570	. . . 4 ⊢ (𝑥 = 𝐴 ↔ (𝑥 ⊆ 𝐴 ∧ 𝑥 = 𝐴))
4	1, 3	bitri 278	. . 3 ⊢ (𝑥 ∈ {𝐴} ↔ (𝑥 ⊆ 𝐴 ∧ 𝑥 = 𝐴))
5	4	a1i 11	. 2 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ≠ ∅) → (𝑥 ∈ {𝐴} ↔ (𝑥 ⊆ 𝐴 ∧ 𝑥 = 𝐴)))
6		simpl 488	. 2 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ≠ ∅) → 𝐴 ∈ 𝐵)
7		eqid 2763	. . . 4 ⊢ 𝐴 = 𝐴
8		eqsbc1 3788	. . . 4 ⊢ (𝐴 ∈ 𝐵 → ([𝐴 / 𝑥]𝑥 = 𝐴 ↔ 𝐴 = 𝐴))
9	7, 8	mpbiri 261	. . 3 ⊢ (𝐴 ∈ 𝐵 → [𝐴 / 𝑥]𝑥 = 𝐴)
10	9	adantr 486	. 2 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ≠ ∅) → [𝐴 / 𝑥]𝑥 = 𝐴)
11		simpr 490	. . . . 5 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ≠ ∅) → 𝐴 ≠ ∅)
12	11	necomd 3014	. . . 4 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ≠ ∅) → ∅ ≠ 𝐴)
13	12	neneqd 2964	. . 3 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ≠ ∅) → ¬ ∅ = 𝐴)
14		0ex 5267	. . . 4 ⊢ ∅ ∈ V
15		eqsbc1 3788	. . . 4 ⊢ (∅ ∈ V → ([∅ / 𝑥]𝑥 = 𝐴 ↔ ∅ = 𝐴))
16	14, 15	ax-mp 5	. . 3 ⊢ ([∅ / 𝑥]𝑥 = 𝐴 ↔ ∅ = 𝐴)
17	13, 16	sylnibr 332	. 2 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ≠ ∅) → ¬ [∅ / 𝑥]𝑥 = 𝐴)
18		sseq1 3938	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑥 ⊆ 𝑦 ↔ 𝐴 ⊆ 𝑦))
19	18	anbi2d 642	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝑦 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝑦) ↔ (𝑦 ⊆ 𝐴 ∧ 𝐴 ⊆ 𝑦)))
20		eqss 3928	. . . . . . 7 ⊢ (𝑦 = 𝐴 ↔ (𝑦 ⊆ 𝐴 ∧ 𝐴 ⊆ 𝑦))
21	20	biimpri 231	. . . . . 6 ⊢ ((𝑦 ⊆ 𝐴 ∧ 𝐴 ⊆ 𝑦) → 𝑦 = 𝐴)
22	19, 21	biimtrdi 256	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑦 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝑦) → 𝑦 = 𝐴))
23	22	com12 33	. . . 4 ⊢ ((𝑦 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝑦) → (𝑥 = 𝐴 → 𝑦 = 𝐴))
24	23	3adant1 1148	. . 3 ⊢ (((𝐴 ∈ 𝐵 ∧ 𝐴 ≠ ∅) ∧ 𝑦 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝑦) → (𝑥 = 𝐴 → 𝑦 = 𝐴))
25		sbcid 3759	. . 3 ⊢ ([𝑥 / 𝑥]𝑥 = 𝐴 ↔ 𝑥 = 𝐴)
26		eqsbc1 3788	. . . 4 ⊢ (𝑦 ∈ V → ([𝑦 / 𝑥]𝑥 = 𝐴 ↔ 𝑦 = 𝐴))
27	26	elv 3459	. . 3 ⊢ ([𝑦 / 𝑥]𝑥 = 𝐴 ↔ 𝑦 = 𝐴)
28	24, 25, 27	3imtr4g 299	. 2 ⊢ (((𝐴 ∈ 𝐵 ∧ 𝐴 ≠ ∅) ∧ 𝑦 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝑦) → ([𝑥 / 𝑥]𝑥 = 𝐴 → [𝑦 / 𝑥]𝑥 = 𝐴))
29		ineq12 4115	. . . . . 6 ⊢ ((𝑦 = 𝐴 ∧ 𝑥 = 𝐴) → (𝑦 ∩ 𝑥) = (𝐴 ∩ 𝐴))
30		inidm 4126	. . . . . 6 ⊢ (𝐴 ∩ 𝐴) = 𝐴
31	29, 30	eqtrdi 2814	. . . . 5 ⊢ ((𝑦 = 𝐴 ∧ 𝑥 = 𝐴) → (𝑦 ∩ 𝑥) = 𝐴)
32	27, 25, 31	syl2anb 610	. . . 4 ⊢ (([𝑦 / 𝑥]𝑥 = 𝐴 ∧ [𝑥 / 𝑥]𝑥 = 𝐴) → (𝑦 ∩ 𝑥) = 𝐴)
33		vex 3458	. . . . . 6 ⊢ 𝑦 ∈ V
34	33	inex1 5284	. . . . 5 ⊢ (𝑦 ∩ 𝑥) ∈ V
35		eqsbc1 3788	. . . . 5 ⊢ ((𝑦 ∩ 𝑥) ∈ V → ([(𝑦 ∩ 𝑥) / 𝑥]𝑥 = 𝐴 ↔ (𝑦 ∩ 𝑥) = 𝐴))
36	34, 35	ax-mp 5	. . . 4 ⊢ ([(𝑦 ∩ 𝑥) / 𝑥]𝑥 = 𝐴 ↔ (𝑦 ∩ 𝑥) = 𝐴)
37	32, 36	sylibr 237	. . 3 ⊢ (([𝑦 / 𝑥]𝑥 = 𝐴 ∧ [𝑥 / 𝑥]𝑥 = 𝐴) → [(𝑦 ∩ 𝑥) / 𝑥]𝑥 = 𝐴)
38	37	a1i 11	. 2 ⊢ (((𝐴 ∈ 𝐵 ∧ 𝐴 ≠ ∅) ∧ 𝑦 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐴) → (([𝑦 / 𝑥]𝑥 = 𝐴 ∧ [𝑥 / 𝑥]𝑥 = 𝐴) → [(𝑦 ∩ 𝑥) / 𝑥]𝑥 = 𝐴))
39	5, 6, 10, 17, 28, 38	isfild 24232	1 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ≠ ∅) → {𝐴} ∈ (Fil‘𝐴))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147   ≠ wne 2959  Vcvv 3454  [wsbc 3742   ⊆ wss 3900   ∩ cin 4092  ∅c0 4281  {csn 4588  ‘cfv 6566  Filcfil 24219

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-iota 6520  df-fun 6568  df-fv 6574  df-fbas 21726  df-fil 24220

This theorem is used by:  snfbas  24240