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Mirrors > Home > MPE Home > Th. List > ufildom1 | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
ufildom1 | ⊢ (𝐹 ∈ (UFil‘𝑋) → ∩ 𝐹 ≼ 1𝑜) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breq1 4586 | . 2 ⊢ (∩ 𝐹 = ∅ → (∩ 𝐹 ≼ 1𝑜 ↔ ∅ ≼ 1𝑜)) | |
2 | uffixsn 21539 | . . . . . . . . 9 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ∈ ∩ 𝐹) → {𝑥} ∈ 𝐹) | |
3 | intss1 4427 | . . . . . . . . 9 ⊢ ({𝑥} ∈ 𝐹 → ∩ 𝐹 ⊆ {𝑥}) | |
4 | 2, 3 | syl 17 | . . . . . . . 8 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ∈ ∩ 𝐹) → ∩ 𝐹 ⊆ {𝑥}) |
5 | simpr 476 | . . . . . . . . 9 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ∈ ∩ 𝐹) → 𝑥 ∈ ∩ 𝐹) | |
6 | 5 | snssd 4281 | . . . . . . . 8 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ∈ ∩ 𝐹) → {𝑥} ⊆ ∩ 𝐹) |
7 | 4, 6 | eqssd 3585 | . . . . . . 7 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ∈ ∩ 𝐹) → ∩ 𝐹 = {𝑥}) |
8 | 7 | ex 449 | . . . . . 6 ⊢ (𝐹 ∈ (UFil‘𝑋) → (𝑥 ∈ ∩ 𝐹 → ∩ 𝐹 = {𝑥})) |
9 | 8 | eximdv 1833 | . . . . 5 ⊢ (𝐹 ∈ (UFil‘𝑋) → (∃𝑥 𝑥 ∈ ∩ 𝐹 → ∃𝑥∩ 𝐹 = {𝑥})) |
10 | n0 3890 | . . . . 5 ⊢ (∩ 𝐹 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ ∩ 𝐹) | |
11 | en1 7909 | . . . . 5 ⊢ (∩ 𝐹 ≈ 1𝑜 ↔ ∃𝑥∩ 𝐹 = {𝑥}) | |
12 | 9, 10, 11 | 3imtr4g 284 | . . . 4 ⊢ (𝐹 ∈ (UFil‘𝑋) → (∩ 𝐹 ≠ ∅ → ∩ 𝐹 ≈ 1𝑜)) |
13 | 12 | imp 444 | . . 3 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ ∩ 𝐹 ≠ ∅) → ∩ 𝐹 ≈ 1𝑜) |
14 | endom 7868 | . . 3 ⊢ (∩ 𝐹 ≈ 1𝑜 → ∩ 𝐹 ≼ 1𝑜) | |
15 | 13, 14 | syl 17 | . 2 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ ∩ 𝐹 ≠ ∅) → ∩ 𝐹 ≼ 1𝑜) |
16 | 1on 7454 | . . 3 ⊢ 1𝑜 ∈ On | |
17 | 0domg 7972 | . . 3 ⊢ (1𝑜 ∈ On → ∅ ≼ 1𝑜) | |
18 | 16, 17 | mp1i 13 | . 2 ⊢ (𝐹 ∈ (UFil‘𝑋) → ∅ ≼ 1𝑜) |
19 | 1, 15, 18 | pm2.61ne 2867 | 1 ⊢ (𝐹 ∈ (UFil‘𝑋) → ∩ 𝐹 ≼ 1𝑜) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∃wex 1695 ∈ wcel 1977 ≠ wne 2780 ⊆ wss 3540 ∅c0 3874 {csn 4125 ∩ cint 4410 class class class wbr 4583 Oncon0 5640 ‘cfv 5804 1𝑜c1o 7440 ≈ cen 7838 ≼ cdom 7839 UFilcufil 21513 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-ord 5643 df-on 5644 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1o 7447 df-en 7842 df-dom 7843 df-fbas 19564 df-fg 19565 df-fil 21460 df-ufil 21515 |
This theorem is referenced by: (None) |
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