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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 5069 | . 2 ⊢ (∩ 𝐹 = ∅ → (∩ 𝐹 ≼ 1o ↔ ∅ ≼ 1o)) | |
2 | uffixsn 22533 | . . . . . . . . 9 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ∈ ∩ 𝐹) → {𝑥} ∈ 𝐹) | |
3 | intss1 4891 | . . . . . . . . 9 ⊢ ({𝑥} ∈ 𝐹 → ∩ 𝐹 ⊆ {𝑥}) | |
4 | 2, 3 | syl 17 | . . . . . . . 8 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ∈ ∩ 𝐹) → ∩ 𝐹 ⊆ {𝑥}) |
5 | simpr 487 | . . . . . . . . 9 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ∈ ∩ 𝐹) → 𝑥 ∈ ∩ 𝐹) | |
6 | 5 | snssd 4742 | . . . . . . . 8 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ∈ ∩ 𝐹) → {𝑥} ⊆ ∩ 𝐹) |
7 | 4, 6 | eqssd 3984 | . . . . . . 7 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ∈ ∩ 𝐹) → ∩ 𝐹 = {𝑥}) |
8 | 7 | ex 415 | . . . . . 6 ⊢ (𝐹 ∈ (UFil‘𝑋) → (𝑥 ∈ ∩ 𝐹 → ∩ 𝐹 = {𝑥})) |
9 | 8 | eximdv 1918 | . . . . 5 ⊢ (𝐹 ∈ (UFil‘𝑋) → (∃𝑥 𝑥 ∈ ∩ 𝐹 → ∃𝑥∩ 𝐹 = {𝑥})) |
10 | n0 4310 | . . . . 5 ⊢ (∩ 𝐹 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ ∩ 𝐹) | |
11 | en1 8576 | . . . . 5 ⊢ (∩ 𝐹 ≈ 1o ↔ ∃𝑥∩ 𝐹 = {𝑥}) | |
12 | 9, 10, 11 | 3imtr4g 298 | . . . 4 ⊢ (𝐹 ∈ (UFil‘𝑋) → (∩ 𝐹 ≠ ∅ → ∩ 𝐹 ≈ 1o)) |
13 | 12 | imp 409 | . . 3 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ ∩ 𝐹 ≠ ∅) → ∩ 𝐹 ≈ 1o) |
14 | endom 8536 | . . 3 ⊢ (∩ 𝐹 ≈ 1o → ∩ 𝐹 ≼ 1o) | |
15 | 13, 14 | syl 17 | . 2 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ ∩ 𝐹 ≠ ∅) → ∩ 𝐹 ≼ 1o) |
16 | 1on 8109 | . . 3 ⊢ 1o ∈ On | |
17 | 0domg 8644 | . . 3 ⊢ (1o ∈ On → ∅ ≼ 1o) | |
18 | 16, 17 | mp1i 13 | . 2 ⊢ (𝐹 ∈ (UFil‘𝑋) → ∅ ≼ 1o) |
19 | 1, 15, 18 | pm2.61ne 3102 | 1 ⊢ (𝐹 ∈ (UFil‘𝑋) → ∩ 𝐹 ≼ 1o) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∃wex 1780 ∈ wcel 2114 ≠ wne 3016 ⊆ wss 3936 ∅c0 4291 {csn 4567 ∩ cint 4876 class class class wbr 5066 Oncon0 6191 ‘cfv 6355 1oc1o 8095 ≈ cen 8506 ≼ cdom 8507 UFilcufil 22507 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-ord 6194 df-on 6195 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-1o 8102 df-en 8510 df-dom 8511 df-fbas 20542 df-fg 20543 df-fil 22454 df-ufil 22509 |
This theorem is referenced by: (None) |
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