Theorem snfil 23924

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 4598	. . . 4 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
2		eqimss 3994	. . . . 5 ⊢ (𝑥 = 𝐴 → 𝑥 ⊆ 𝐴)
3	2	pm4.71ri 568	. . . 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		simpl 486	. 2 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ≠ ∅) → 𝐴 ∈ 𝐵)
7		eqid 2762	. . . 4 ⊢ 𝐴 = 𝐴
8		eqsbc1 3790	. . . 4 ⊢ (𝐴 ∈ 𝐵 → ([𝐴 / 𝑥]𝑥 = 𝐴 ↔ 𝐴 = 𝐴))
9	7, 8	mpbiri 260	. . 3 ⊢ (𝐴 ∈ 𝐵 → [𝐴 / 𝑥]𝑥 = 𝐴)
10	9	adantr 484	. 2 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ≠ ∅) → [𝐴 / 𝑥]𝑥 = 𝐴)
11		simpr 488	. . . . 5 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ≠ ∅) → 𝐴 ≠ ∅)
12	11	necomd 3012	. . . 4 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ≠ ∅) → ∅ ≠ 𝐴)
13	12	neneqd 2962	. . 3 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ≠ ∅) → ¬ ∅ = 𝐴)
14		0ex 5257	. . . 4 ⊢ ∅ ∈ V
15		eqsbc1 3790	. . . 4 ⊢ (∅ ∈ V → ([∅ / 𝑥]𝑥 = 𝐴 ↔ ∅ = 𝐴))
16	14, 15	ax-mp 5	. . 3 ⊢ ([∅ / 𝑥]𝑥 = 𝐴 ↔ ∅ = 𝐴)
17	13, 16	sylnibr 331	. 2 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ≠ ∅) → ¬ [∅ / 𝑥]𝑥 = 𝐴)
18		sseq1 3961	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑥 ⊆ 𝑦 ↔ 𝐴 ⊆ 𝑦))
19	18	anbi2d 639	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝑦 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝑦) ↔ (𝑦 ⊆ 𝐴 ∧ 𝐴 ⊆ 𝑦)))
20		eqss 3951	. . . . . . 7 ⊢ (𝑦 = 𝐴 ↔ (𝑦 ⊆ 𝐴 ∧ 𝐴 ⊆ 𝑦))
21	20	biimpri 230	. . . . . 6 ⊢ ((𝑦 ⊆ 𝐴 ∧ 𝐴 ⊆ 𝑦) → 𝑦 = 𝐴)
22	19, 21	biimtrdi 255	. . . . 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 1143	. . 3 ⊢ (((𝐴 ∈ 𝐵 ∧ 𝐴 ≠ ∅) ∧ 𝑦 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝑦) → (𝑥 = 𝐴 → 𝑦 = 𝐴))
25		sbcid 3761	. . 3 ⊢ ([𝑥 / 𝑥]𝑥 = 𝐴 ↔ 𝑥 = 𝐴)
26		eqsbc1 3790	. . . 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 298	. 2 ⊢ (((𝐴 ∈ 𝐵 ∧ 𝐴 ≠ ∅) ∧ 𝑦 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝑦) → ([𝑥 / 𝑥]𝑥 = 𝐴 → [𝑦 / 𝑥]𝑥 = 𝐴))
29		ineq12 4167	. . . . . 6 ⊢ ((𝑦 = 𝐴 ∧ 𝑥 = 𝐴) → (𝑦 ∩ 𝑥) = (𝐴 ∩ 𝐴))
30		inidm 4178	. . . . . 6 ⊢ (𝐴 ∩ 𝐴) = 𝐴
31	29, 30	eqtrdi 2813	. . . . 5 ⊢ ((𝑦 = 𝐴 ∧ 𝑥 = 𝐴) → (𝑦 ∩ 𝑥) = 𝐴)
32	27, 25, 31	syl2anb 607	. . . 4 ⊢ (([𝑦 / 𝑥]𝑥 = 𝐴 ∧ [𝑥 / 𝑥]𝑥 = 𝐴) → (𝑦 ∩ 𝑥) = 𝐴)
33		vex 3458	. . . . . 6 ⊢ 𝑦 ∈ V
34	33	inex1 5273	. . . . 5 ⊢ (𝑦 ∩ 𝑥) ∈ V
35		eqsbc1 3790	. . . . 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 236	. . 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 23918	1 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ≠ ∅) → {𝐴} ∈ (Fil‘𝐴))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∧ w3a 1098   = wceq 1560   ∈ wcel 2142   ≠ wne 2957  Vcvv 3454  [wsbc 3744   ∩ cin 3903   ⊆ wss 3904  ∅c0 4285  {csn 4582  ‘cfv 6521  Filcfil 23905

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5246  ax-nul 5256  ax-pow 5322  ax-pr 5390

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3062  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-iota 6477  df-fun 6523  df-fv 6529  df-fbas 21421  df-fil 23906

This theorem is referenced by:  snfbas  23926