Theorem ufildom1 23913

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 5078	. 2 ⊢ (∩ 𝐹 = ∅ → (∩ 𝐹 ≼ 1o ↔ ∅ ≼ 1o))
2		uffixsn 23912	. . . . . . . . 9 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ∈ ∩ 𝐹) → {𝑥} ∈ 𝐹)
3		intss1 4896	. . . . . . . . 9 ⊢ ({𝑥} ∈ 𝐹 → ∩ 𝐹 ⊆ {𝑥})
4	2, 3	syl 17	. . . . . . . 8 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ∈ ∩ 𝐹) → ∩ 𝐹 ⊆ {𝑥})
5		simpr 486	. . . . . . . . 9 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ∈ ∩ 𝐹) → 𝑥 ∈ ∩ 𝐹)
6	5	snssd 4721	. . . . . . . 8 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ∈ ∩ 𝐹) → {𝑥} ⊆ ∩ 𝐹)
7	4, 6	eqssd 3934	. . . . . . 7 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ∈ ∩ 𝐹) → ∩ 𝐹 = {𝑥})
8	7	ex 414	. . . . . 6 ⊢ (𝐹 ∈ (UFil‘𝑋) → (𝑥 ∈ ∩ 𝐹 → ∩ 𝐹 = {𝑥}))
9	8	eximdv 1925	. . . . 5 ⊢ (𝐹 ∈ (UFil‘𝑋) → (∃𝑥 𝑥 ∈ ∩ 𝐹 → ∃𝑥∩ 𝐹 = {𝑥}))
10		n0 4284	. . . . 5 ⊢ (∩ 𝐹 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ ∩ 𝐹)
11		en1 8965	. . . . 5 ⊢ (∩ 𝐹 ≈ 1o ↔ ∃𝑥∩ 𝐹 = {𝑥})
12	9, 10, 11	3imtr4g 298	. . . 4 ⊢ (𝐹 ∈ (UFil‘𝑋) → (∩ 𝐹 ≠ ∅ → ∩ 𝐹 ≈ 1o))
13	12	imp 408	. . 3 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ ∩ 𝐹 ≠ ∅) → ∩ 𝐹 ≈ 1o)
14		endom 8920	. . 3 ⊢ (∩ 𝐹 ≈ 1o → ∩ 𝐹 ≼ 1o)
15	13, 14	syl 17	. 2 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ ∩ 𝐹 ≠ ∅) → ∩ 𝐹 ≼ 1o)
16		1on 8411	. . 3 ⊢ 1o ∈ On
17		0domg 9036	. . 3 ⊢ (1o ∈ On → ∅ ≼ 1o)
18	16, 17	mp1i 13	. 2 ⊢ (𝐹 ∈ (UFil‘𝑋) → ∅ ≼ 1o)
19	1, 15, 18	pm2.61ne 3021	1 ⊢ (𝐹 ∈ (UFil‘𝑋) → ∩ 𝐹 ≼ 1o)

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1548  ∃wex 1787   ∈ wcel 2121   ≠ wne 2936   ⊆ wss 3885  ∅c0 4264  {csn 4558  ∩ cint 4880   class class class wbr 5075  Oncon0 6314  ‘cfv 6489  1_oc1o 8392   ≈ cen 8884   ≼ cdom 8885  UFilcufil 23886

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-sep 5221  ax-nul 5231  ax-pow 5297  ax-pr 5365

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3or 1094  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-nel 3041  df-ral 3056  df-rex 3066  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-pss 3905  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-int 4881  df-br 5076  df-opab 5138  df-mpt 5157  df-tr 5183  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-ord 6317  df-on 6318  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-ov 7363  df-oprab 7364  df-mpo 7365  df-1o 8399  df-en 8888  df-dom 8889  df-fbas 21348  df-fg 21349  df-fil 23833  df-ufil 23888

This theorem is referenced by: (None)