Theorem ufildom1 23955

Description: An ultrafilter is generated by at most one element (because free ultrafilters have no generators and fixed ultrafilters have exactly one). (Contributed by Mario Carneiro, 24-May-2015.) (Revised by Stefan O'Rear, 2-Aug-2015.)

Assertion

Ref	Expression
ufildom1	⊢ (𝐹 ∈ (UFil‘𝑋) → ∩ 𝐹 ≼ 1o)

Proof of Theorem ufildom1

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq1 5169	. 2 ⊢ (∩ 𝐹 = ∅ → (∩ 𝐹 ≼ 1o ↔ ∅ ≼ 1o))
2		uffixsn 23954	. . . . . . . . 9 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ∈ ∩ 𝐹) → {𝑥} ∈ 𝐹)
3		intss1 4987	. . . . . . . . 9 ⊢ ({𝑥} ∈ 𝐹 → ∩ 𝐹 ⊆ {𝑥})
4	2, 3	syl 17	. . . . . . . 8 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ∈ ∩ 𝐹) → ∩ 𝐹 ⊆ {𝑥})
5		simpr 484	. . . . . . . . 9 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ∈ ∩ 𝐹) → 𝑥 ∈ ∩ 𝐹)
6	5	snssd 4834	. . . . . . . 8 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ∈ ∩ 𝐹) → {𝑥} ⊆ ∩ 𝐹)
7	4, 6	eqssd 4026	. . . . . . 7 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ∈ ∩ 𝐹) → ∩ 𝐹 = {𝑥})
8	7	ex 412	. . . . . 6 ⊢ (𝐹 ∈ (UFil‘𝑋) → (𝑥 ∈ ∩ 𝐹 → ∩ 𝐹 = {𝑥}))
9	8	eximdv 1916	. . . . 5 ⊢ (𝐹 ∈ (UFil‘𝑋) → (∃𝑥 𝑥 ∈ ∩ 𝐹 → ∃𝑥∩ 𝐹 = {𝑥}))
10		n0 4376	. . . . 5 ⊢ (∩ 𝐹 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ ∩ 𝐹)
11		en1 9086	. . . . 5 ⊢ (∩ 𝐹 ≈ 1o ↔ ∃𝑥∩ 𝐹 = {𝑥})
12	9, 10, 11	3imtr4g 296	. . . 4 ⊢ (𝐹 ∈ (UFil‘𝑋) → (∩ 𝐹 ≠ ∅ → ∩ 𝐹 ≈ 1o))
13	12	imp 406	. . 3 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ ∩ 𝐹 ≠ ∅) → ∩ 𝐹 ≈ 1o)
14		endom 9039	. . 3 ⊢ (∩ 𝐹 ≈ 1o → ∩ 𝐹 ≼ 1o)
15	13, 14	syl 17	. 2 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ ∩ 𝐹 ≠ ∅) → ∩ 𝐹 ≼ 1o)
16		1on 8534	. . 3 ⊢ 1o ∈ On
17		0domg 9166	. . 3 ⊢ (1o ∈ On → ∅ ≼ 1o)
18	16, 17	mp1i 13	. 2 ⊢ (𝐹 ∈ (UFil‘𝑋) → ∅ ≼ 1o)
19	1, 15, 18	pm2.61ne 3033	1 ⊢ (𝐹 ∈ (UFil‘𝑋) → ∩ 𝐹 ≼ 1o)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537  ∃wex 1777   ∈ wcel 2108   ≠ wne 2946   ⊆ wss 3976  ∅c0 4352  {csn 4648  ∩ cint 4970   class class class wbr 5166  Oncon0 6395  ‘cfv 6573  1_oc1o 8515   ≈ cen 9000   ≼ cdom 9001  UFilcufil 23928

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-ord 6398  df-on 6399  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-1o 8522  df-en 9004  df-dom 9005  df-fbas 21384  df-fg 21385  df-fil 23875  df-ufil 23930

This theorem is referenced by: (None)