![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > numufl | Structured version Visualization version GIF version |
Description: Consequence of filssufilg 23833: 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 23833 | . . . 4 ⊢ ((𝑓 ∈ (Fil‘𝑋) ∧ 𝒫 𝒫 𝑋 ∈ dom card) → ∃𝑔 ∈ (UFil‘𝑋)𝑓 ⊆ 𝑔) | |
2 | 1 | ancoms 457 | . . 3 ⊢ ((𝒫 𝒫 𝑋 ∈ dom card ∧ 𝑓 ∈ (Fil‘𝑋)) → ∃𝑔 ∈ (UFil‘𝑋)𝑓 ⊆ 𝑔) |
3 | 2 | ralrimiva 3142 | . 2 ⊢ (𝒫 𝒫 𝑋 ∈ dom card → ∀𝑓 ∈ (Fil‘𝑋)∃𝑔 ∈ (UFil‘𝑋)𝑓 ⊆ 𝑔) |
4 | pwexr 7771 | . . . 4 ⊢ (𝒫 𝒫 𝑋 ∈ dom card → 𝒫 𝑋 ∈ V) | |
5 | pwexb 7772 | . . . 4 ⊢ (𝑋 ∈ V ↔ 𝒫 𝑋 ∈ V) | |
6 | 4, 5 | sylibr 233 | . . 3 ⊢ (𝒫 𝒫 𝑋 ∈ dom card → 𝑋 ∈ V) |
7 | isufl 23835 | . . 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 256 | 1 ⊢ (𝒫 𝒫 𝑋 ∈ dom card → 𝑋 ∈ UFL) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∈ wcel 2098 ∀wral 3057 ∃wrex 3066 Vcvv 3471 ⊆ wss 3947 𝒫 cpw 4604 dom cdm 5680 ‘cfv 6551 cardccrd 9964 Filcfil 23767 UFilcufil 23821 UFLcufl 23822 |
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 2166 ax-ext 2698 ax-rep 5287 ax-sep 5301 ax-nul 5308 ax-pow 5367 ax-pr 5431 ax-un 7744 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4325 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4911 df-int 4952 df-iun 5000 df-br 5151 df-opab 5213 df-mpt 5234 df-tr 5268 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5635 df-se 5636 df-we 5637 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-rn 5691 df-res 5692 df-ima 5693 df-pred 6308 df-ord 6375 df-on 6376 df-lim 6377 df-suc 6378 df-iota 6503 df-fun 6553 df-fn 6554 df-f 6555 df-f1 6556 df-fo 6557 df-f1o 6558 df-fv 6559 df-isom 6560 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-rpss 7732 df-om 7875 df-1st 7997 df-2nd 7998 df-frecs 8291 df-wrecs 8322 df-recs 8396 df-rdg 8435 df-1o 8491 df-oadd 8495 df-er 8729 df-en 8969 df-dom 8970 df-fin 8972 df-fi 9440 df-dju 9930 df-card 9968 df-fbas 21281 df-fg 21282 df-fil 23768 df-ufil 23823 df-ufl 23824 |
This theorem is referenced by: fiufl 23838 acufl 23839 |
Copyright terms: Public domain | W3C validator |