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Mirrors > Home > MPE Home > Th. List > numufl | Structured version Visualization version GIF version |
Description: Consequence of filssufilg 23766: a set whose double powerset is well-orderable satisfies the ultrafilter lemma. (Contributed by Mario Carneiro, 26-Aug-2015.) |
Ref | Expression |
---|---|
numufl | ⊢ (𝒫 𝒫 𝑋 ∈ dom card → 𝑋 ∈ UFL) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | filssufilg 23766 | . . . 4 ⊢ ((𝑓 ∈ (Fil‘𝑋) ∧ 𝒫 𝒫 𝑋 ∈ dom card) → ∃𝑔 ∈ (UFil‘𝑋)𝑓 ⊆ 𝑔) | |
2 | 1 | ancoms 458 | . . 3 ⊢ ((𝒫 𝒫 𝑋 ∈ dom card ∧ 𝑓 ∈ (Fil‘𝑋)) → ∃𝑔 ∈ (UFil‘𝑋)𝑓 ⊆ 𝑔) |
3 | 2 | ralrimiva 3140 | . 2 ⊢ (𝒫 𝒫 𝑋 ∈ dom card → ∀𝑓 ∈ (Fil‘𝑋)∃𝑔 ∈ (UFil‘𝑋)𝑓 ⊆ 𝑔) |
4 | pwexr 7748 | . . . 4 ⊢ (𝒫 𝒫 𝑋 ∈ dom card → 𝒫 𝑋 ∈ V) | |
5 | pwexb 7749 | . . . 4 ⊢ (𝑋 ∈ V ↔ 𝒫 𝑋 ∈ V) | |
6 | 4, 5 | sylibr 233 | . . 3 ⊢ (𝒫 𝒫 𝑋 ∈ dom card → 𝑋 ∈ V) |
7 | isufl 23768 | . . 3 ⊢ (𝑋 ∈ V → (𝑋 ∈ UFL ↔ ∀𝑓 ∈ (Fil‘𝑋)∃𝑔 ∈ (UFil‘𝑋)𝑓 ⊆ 𝑔)) | |
8 | 6, 7 | syl 17 | . 2 ⊢ (𝒫 𝒫 𝑋 ∈ dom card → (𝑋 ∈ UFL ↔ ∀𝑓 ∈ (Fil‘𝑋)∃𝑔 ∈ (UFil‘𝑋)𝑓 ⊆ 𝑔)) |
9 | 3, 8 | mpbird 257 | 1 ⊢ (𝒫 𝒫 𝑋 ∈ dom card → 𝑋 ∈ UFL) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∈ wcel 2098 ∀wral 3055 ∃wrex 3064 Vcvv 3468 ⊆ wss 3943 𝒫 cpw 4597 dom cdm 5669 ‘cfv 6536 cardccrd 9929 Filcfil 23700 UFilcufil 23754 UFLcufl 23755 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-rpss 7709 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-oadd 8468 df-er 8702 df-en 8939 df-dom 8940 df-fin 8942 df-fi 9405 df-dju 9895 df-card 9933 df-fbas 21233 df-fg 21234 df-fil 23701 df-ufil 23756 df-ufl 23757 |
This theorem is referenced by: fiufl 23771 acufl 23772 |
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