Theorem snfil 23847

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 4571	. . . 4 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
2		eqimss 3973	. . . . 5 ⊢ (𝑥 = 𝐴 → 𝑥 ⊆ 𝐴)
3	2	pm4.71ri 565	. . . 4 ⊢ (𝑥 = 𝐴 ↔ (𝑥 ⊆ 𝐴 ∧ 𝑥 = 𝐴))
4	1, 3	bitri 276	. . 3 ⊢ (𝑥 ∈ {𝐴} ↔ (𝑥 ⊆ 𝐴 ∧ 𝑥 = 𝐴))
5	4	a1i 11	. 2 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ≠ ∅) → (𝑥 ∈ {𝐴} ↔ (𝑥 ⊆ 𝐴 ∧ 𝑥 = 𝐴)))
6		simpl 483	. 2 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ≠ ∅) → 𝐴 ∈ 𝐵)
7		eqid 2739	. . . 4 ⊢ 𝐴 = 𝐴
8		eqsbc1 3769	. . . 4 ⊢ (𝐴 ∈ 𝐵 → ([𝐴 / 𝑥]𝑥 = 𝐴 ↔ 𝐴 = 𝐴))
9	7, 8	mpbiri 259	. . 3 ⊢ (𝐴 ∈ 𝐵 → [𝐴 / 𝑥]𝑥 = 𝐴)
10	9	adantr 481	. 2 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ≠ ∅) → [𝐴 / 𝑥]𝑥 = 𝐴)
11		simpr 485	. . . . 5 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ≠ ∅) → 𝐴 ≠ ∅)
12	11	necomd 2989	. . . 4 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ≠ ∅) → ∅ ≠ 𝐴)
13	12	neneqd 2939	. . 3 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ≠ ∅) → ¬ ∅ = 𝐴)
14		0ex 5229	. . . 4 ⊢ ∅ ∈ V
15		eqsbc1 3769	. . . 4 ⊢ (∅ ∈ V → ([∅ / 𝑥]𝑥 = 𝐴 ↔ ∅ = 𝐴))
16	14, 15	ax-mp 5	. . 3 ⊢ ([∅ / 𝑥]𝑥 = 𝐴 ↔ ∅ = 𝐴)
17	13, 16	sylnibr 330	. 2 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ≠ ∅) → ¬ [∅ / 𝑥]𝑥 = 𝐴)
18		sseq1 3940	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑥 ⊆ 𝑦 ↔ 𝐴 ⊆ 𝑦))
19	18	anbi2d 636	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝑦 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝑦) ↔ (𝑦 ⊆ 𝐴 ∧ 𝐴 ⊆ 𝑦)))
20		eqss 3930	. . . . . . 7 ⊢ (𝑦 = 𝐴 ↔ (𝑦 ⊆ 𝐴 ∧ 𝐴 ⊆ 𝑦))
21	20	biimpri 229	. . . . . 6 ⊢ ((𝑦 ⊆ 𝐴 ∧ 𝐴 ⊆ 𝑦) → 𝑦 = 𝐴)
22	19, 21	biimtrdi 254	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑦 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝑦) → 𝑦 = 𝐴))
23	22	com12 32	. . . 4 ⊢ ((𝑦 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝑦) → (𝑥 = 𝐴 → 𝑦 = 𝐴))
24	23	3adant1 1136	. . 3 ⊢ (((𝐴 ∈ 𝐵 ∧ 𝐴 ≠ ∅) ∧ 𝑦 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝑦) → (𝑥 = 𝐴 → 𝑦 = 𝐴))
25		sbcid 3740	. . 3 ⊢ ([𝑥 / 𝑥]𝑥 = 𝐴 ↔ 𝑥 = 𝐴)
26		eqsbc1 3769	. . . 4 ⊢ (𝑦 ∈ V → ([𝑦 / 𝑥]𝑥 = 𝐴 ↔ 𝑦 = 𝐴))
27	26	elv 3436	. . 3 ⊢ ([𝑦 / 𝑥]𝑥 = 𝐴 ↔ 𝑦 = 𝐴)
28	24, 25, 27	3imtr4g 297	. 2 ⊢ (((𝐴 ∈ 𝐵 ∧ 𝐴 ≠ ∅) ∧ 𝑦 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝑦) → ([𝑥 / 𝑥]𝑥 = 𝐴 → [𝑦 / 𝑥]𝑥 = 𝐴))
29		ineq12 4144	. . . . . 6 ⊢ ((𝑦 = 𝐴 ∧ 𝑥 = 𝐴) → (𝑦 ∩ 𝑥) = (𝐴 ∩ 𝐴))
30		inidm 4155	. . . . . 6 ⊢ (𝐴 ∩ 𝐴) = 𝐴
31	29, 30	eqtrdi 2790	. . . . 5 ⊢ ((𝑦 = 𝐴 ∧ 𝑥 = 𝐴) → (𝑦 ∩ 𝑥) = 𝐴)
32	27, 25, 31	syl2anb 604	. . . 4 ⊢ (([𝑦 / 𝑥]𝑥 = 𝐴 ∧ [𝑥 / 𝑥]𝑥 = 𝐴) → (𝑦 ∩ 𝑥) = 𝐴)
33		vex 3435	. . . . . 6 ⊢ 𝑦 ∈ V
34	33	inex1 5245	. . . . 5 ⊢ (𝑦 ∩ 𝑥) ∈ V
35		eqsbc1 3769	. . . . 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 235	. . 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 23841	1 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ≠ ∅) → {𝐴} ∈ (Fil‘𝐴))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119   ≠ wne 2934  Vcvv 3431  [wsbc 3723   ∩ cin 3882   ⊆ wss 3883  ∅c0 4261  {csn 4555  ‘cfv 6485  Filcfil 23828

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-nel 3039  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fun 6487  df-fv 6493  df-fbas 21344  df-fil 23829

This theorem is referenced by:  snfbas  23849