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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 5150 | . 2 ⊢ (∩ 𝐹 = ∅ → (∩ 𝐹 ≼ 1o ↔ ∅ ≼ 1o)) | |
2 | uffixsn 23420 | . . . . . . . . 9 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ∈ ∩ 𝐹) → {𝑥} ∈ 𝐹) | |
3 | intss1 4966 | . . . . . . . . 9 ⊢ ({𝑥} ∈ 𝐹 → ∩ 𝐹 ⊆ {𝑥}) | |
4 | 2, 3 | syl 17 | . . . . . . . 8 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ∈ ∩ 𝐹) → ∩ 𝐹 ⊆ {𝑥}) |
5 | simpr 485 | . . . . . . . . 9 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ∈ ∩ 𝐹) → 𝑥 ∈ ∩ 𝐹) | |
6 | 5 | snssd 4811 | . . . . . . . 8 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ∈ ∩ 𝐹) → {𝑥} ⊆ ∩ 𝐹) |
7 | 4, 6 | eqssd 3998 | . . . . . . 7 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ∈ ∩ 𝐹) → ∩ 𝐹 = {𝑥}) |
8 | 7 | ex 413 | . . . . . 6 ⊢ (𝐹 ∈ (UFil‘𝑋) → (𝑥 ∈ ∩ 𝐹 → ∩ 𝐹 = {𝑥})) |
9 | 8 | eximdv 1920 | . . . . 5 ⊢ (𝐹 ∈ (UFil‘𝑋) → (∃𝑥 𝑥 ∈ ∩ 𝐹 → ∃𝑥∩ 𝐹 = {𝑥})) |
10 | n0 4345 | . . . . 5 ⊢ (∩ 𝐹 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ ∩ 𝐹) | |
11 | en1 9017 | . . . . 5 ⊢ (∩ 𝐹 ≈ 1o ↔ ∃𝑥∩ 𝐹 = {𝑥}) | |
12 | 9, 10, 11 | 3imtr4g 295 | . . . 4 ⊢ (𝐹 ∈ (UFil‘𝑋) → (∩ 𝐹 ≠ ∅ → ∩ 𝐹 ≈ 1o)) |
13 | 12 | imp 407 | . . 3 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ ∩ 𝐹 ≠ ∅) → ∩ 𝐹 ≈ 1o) |
14 | endom 8971 | . . 3 ⊢ (∩ 𝐹 ≈ 1o → ∩ 𝐹 ≼ 1o) | |
15 | 13, 14 | syl 17 | . 2 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ ∩ 𝐹 ≠ ∅) → ∩ 𝐹 ≼ 1o) |
16 | 1on 8474 | . . 3 ⊢ 1o ∈ On | |
17 | 0domg 9096 | . . 3 ⊢ (1o ∈ On → ∅ ≼ 1o) | |
18 | 16, 17 | mp1i 13 | . 2 ⊢ (𝐹 ∈ (UFil‘𝑋) → ∅ ≼ 1o) |
19 | 1, 15, 18 | pm2.61ne 3027 | 1 ⊢ (𝐹 ∈ (UFil‘𝑋) → ∩ 𝐹 ≼ 1o) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∃wex 1781 ∈ wcel 2106 ≠ wne 2940 ⊆ wss 3947 ∅c0 4321 {csn 4627 ∩ cint 4949 class class class wbr 5147 Oncon0 6361 ‘cfv 6540 1oc1o 8455 ≈ cen 8932 ≼ cdom 8933 UFilcufil 23394 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-ord 6364 df-on 6365 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-ov 7408 df-oprab 7409 df-mpo 7410 df-1o 8462 df-en 8936 df-dom 8937 df-fbas 20933 df-fg 20934 df-fil 23341 df-ufil 23396 |
This theorem is referenced by: (None) |
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