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Mirrors > Home > MPE Home > Th. List > numufl | Structured version Visualization version GIF version |
Description: Consequence of filssufilg 23802: 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 23802 | . . . 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 3141 | . 2 ⊢ (𝒫 𝒫 𝑋 ∈ dom card → ∀𝑓 ∈ (Fil‘𝑋)∃𝑔 ∈ (UFil‘𝑋)𝑓 ⊆ 𝑔) |
4 | pwexr 7761 | . . . 4 ⊢ (𝒫 𝒫 𝑋 ∈ dom card → 𝒫 𝑋 ∈ V) | |
5 | pwexb 7762 | . . . 4 ⊢ (𝑋 ∈ V ↔ 𝒫 𝑋 ∈ V) | |
6 | 4, 5 | sylibr 233 | . . 3 ⊢ (𝒫 𝒫 𝑋 ∈ dom card → 𝑋 ∈ V) |
7 | isufl 23804 | . . 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 2099 ∀wral 3056 ∃wrex 3065 Vcvv 3469 ⊆ wss 3944 𝒫 cpw 4598 dom cdm 5672 ‘cfv 6542 cardccrd 9950 Filcfil 23736 UFilcufil 23790 UFLcufl 23791 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-rpss 7722 df-om 7865 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-oadd 8484 df-er 8718 df-en 8956 df-dom 8957 df-fin 8959 df-fi 9426 df-dju 9916 df-card 9954 df-fbas 21263 df-fg 21264 df-fil 23737 df-ufil 23792 df-ufl 23793 |
This theorem is referenced by: fiufl 23807 acufl 23808 |
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