Theorem snfil 22472

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 4583	. . . 4 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
2		eqimss 4023	. . . . 5 ⊢ (𝑥 = 𝐴 → 𝑥 ⊆ 𝐴)
3	2	pm4.71ri 563	. . . 4 ⊢ (𝑥 = 𝐴 ↔ (𝑥 ⊆ 𝐴 ∧ 𝑥 = 𝐴))
4	1, 3	bitri 277	. . 3 ⊢ (𝑥 ∈ {𝐴} ↔ (𝑥 ⊆ 𝐴 ∧ 𝑥 = 𝐴))
5	4	a1i 11	. 2 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ≠ ∅) → (𝑥 ∈ {𝐴} ↔ (𝑥 ⊆ 𝐴 ∧ 𝑥 = 𝐴)))
6		elex 3512	. . 3 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ V)
7	6	adantr 483	. 2 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ≠ ∅) → 𝐴 ∈ V)
8		eqid 2821	. . . 4 ⊢ 𝐴 = 𝐴
9		eqsbc3 3817	. . . 4 ⊢ (𝐴 ∈ 𝐵 → ([𝐴 / 𝑥]𝑥 = 𝐴 ↔ 𝐴 = 𝐴))
10	8, 9	mpbiri 260	. . 3 ⊢ (𝐴 ∈ 𝐵 → [𝐴 / 𝑥]𝑥 = 𝐴)
11	10	adantr 483	. 2 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ≠ ∅) → [𝐴 / 𝑥]𝑥 = 𝐴)
12		simpr 487	. . . . 5 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ≠ ∅) → 𝐴 ≠ ∅)
13	12	necomd 3071	. . . 4 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ≠ ∅) → ∅ ≠ 𝐴)
14	13	neneqd 3021	. . 3 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ≠ ∅) → ¬ ∅ = 𝐴)
15		0ex 5211	. . . 4 ⊢ ∅ ∈ V
16		eqsbc3 3817	. . . 4 ⊢ (∅ ∈ V → ([∅ / 𝑥]𝑥 = 𝐴 ↔ ∅ = 𝐴))
17	15, 16	ax-mp 5	. . 3 ⊢ ([∅ / 𝑥]𝑥 = 𝐴 ↔ ∅ = 𝐴)
18	14, 17	sylnibr 331	. 2 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ≠ ∅) → ¬ [∅ / 𝑥]𝑥 = 𝐴)
19		sseq1 3992	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑥 ⊆ 𝑦 ↔ 𝐴 ⊆ 𝑦))
20	19	anbi2d 630	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝑦 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝑦) ↔ (𝑦 ⊆ 𝐴 ∧ 𝐴 ⊆ 𝑦)))
21		eqss 3982	. . . . . . 7 ⊢ (𝑦 = 𝐴 ↔ (𝑦 ⊆ 𝐴 ∧ 𝐴 ⊆ 𝑦))
22	21	biimpri 230	. . . . . 6 ⊢ ((𝑦 ⊆ 𝐴 ∧ 𝐴 ⊆ 𝑦) → 𝑦 = 𝐴)
23	20, 22	syl6bi 255	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑦 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝑦) → 𝑦 = 𝐴))
24	23	com12 32	. . . 4 ⊢ ((𝑦 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝑦) → (𝑥 = 𝐴 → 𝑦 = 𝐴))
25	24	3adant1 1126	. . 3 ⊢ (((𝐴 ∈ 𝐵 ∧ 𝐴 ≠ ∅) ∧ 𝑦 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝑦) → (𝑥 = 𝐴 → 𝑦 = 𝐴))
26		sbcid 3789	. . 3 ⊢ ([𝑥 / 𝑥]𝑥 = 𝐴 ↔ 𝑥 = 𝐴)
27		eqsbc3 3817	. . . 4 ⊢ (𝑦 ∈ V → ([𝑦 / 𝑥]𝑥 = 𝐴 ↔ 𝑦 = 𝐴))
28	27	elv 3499	. . 3 ⊢ ([𝑦 / 𝑥]𝑥 = 𝐴 ↔ 𝑦 = 𝐴)
29	25, 26, 28	3imtr4g 298	. 2 ⊢ (((𝐴 ∈ 𝐵 ∧ 𝐴 ≠ ∅) ∧ 𝑦 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝑦) → ([𝑥 / 𝑥]𝑥 = 𝐴 → [𝑦 / 𝑥]𝑥 = 𝐴))
30		ineq12 4184	. . . . . 6 ⊢ ((𝑦 = 𝐴 ∧ 𝑥 = 𝐴) → (𝑦 ∩ 𝑥) = (𝐴 ∩ 𝐴))
31		inidm 4195	. . . . . 6 ⊢ (𝐴 ∩ 𝐴) = 𝐴
32	30, 31	syl6eq 2872	. . . . 5 ⊢ ((𝑦 = 𝐴 ∧ 𝑥 = 𝐴) → (𝑦 ∩ 𝑥) = 𝐴)
33	28, 26, 32	syl2anb 599	. . . 4 ⊢ (([𝑦 / 𝑥]𝑥 = 𝐴 ∧ [𝑥 / 𝑥]𝑥 = 𝐴) → (𝑦 ∩ 𝑥) = 𝐴)
34		vex 3497	. . . . . 6 ⊢ 𝑦 ∈ V
35	34	inex1 5221	. . . . 5 ⊢ (𝑦 ∩ 𝑥) ∈ V
36		eqsbc3 3817	. . . . 5 ⊢ ((𝑦 ∩ 𝑥) ∈ V → ([(𝑦 ∩ 𝑥) / 𝑥]𝑥 = 𝐴 ↔ (𝑦 ∩ 𝑥) = 𝐴))
37	35, 36	ax-mp 5	. . . 4 ⊢ ([(𝑦 ∩ 𝑥) / 𝑥]𝑥 = 𝐴 ↔ (𝑦 ∩ 𝑥) = 𝐴)
38	33, 37	sylibr 236	. . 3 ⊢ (([𝑦 / 𝑥]𝑥 = 𝐴 ∧ [𝑥 / 𝑥]𝑥 = 𝐴) → [(𝑦 ∩ 𝑥) / 𝑥]𝑥 = 𝐴)
39	38	a1i 11	. 2 ⊢ (((𝐴 ∈ 𝐵 ∧ 𝐴 ≠ ∅) ∧ 𝑦 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐴) → (([𝑦 / 𝑥]𝑥 = 𝐴 ∧ [𝑥 / 𝑥]𝑥 = 𝐴) → [(𝑦 ∩ 𝑥) / 𝑥]𝑥 = 𝐴))
40	5, 7, 11, 18, 29, 39	isfild 22466	1 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ≠ ∅) → {𝐴} ∈ (Fil‘𝐴))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114   ≠ wne 3016  Vcvv 3494  [wsbc 3772   ∩ cin 3935   ⊆ wss 3936  ∅c0 4291  {csn 4567  ‘cfv 6355  Filcfil 22453

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-br 5067  df-opab 5129  df-mpt 5147  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-iota 6314  df-fun 6357  df-fv 6363  df-fbas 20542  df-fil 22454

This theorem is referenced by:  snfbas  22474