Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > numufl | Structured version Visualization version GIF version |
Description: Consequence of filssufilg 21715: 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 21715 | . . . 4 ⊢ ((𝑓 ∈ (Fil‘𝑋) ∧ 𝒫 𝒫 𝑋 ∈ dom card) → ∃𝑔 ∈ (UFil‘𝑋)𝑓 ⊆ 𝑔) | |
2 | 1 | ancoms 469 | . . 3 ⊢ ((𝒫 𝒫 𝑋 ∈ dom card ∧ 𝑓 ∈ (Fil‘𝑋)) → ∃𝑔 ∈ (UFil‘𝑋)𝑓 ⊆ 𝑔) |
3 | 2 | ralrimiva 2966 | . 2 ⊢ (𝒫 𝒫 𝑋 ∈ dom card → ∀𝑓 ∈ (Fil‘𝑋)∃𝑔 ∈ (UFil‘𝑋)𝑓 ⊆ 𝑔) |
4 | pwexr 6974 | . . . 4 ⊢ (𝒫 𝒫 𝑋 ∈ dom card → 𝒫 𝑋 ∈ V) | |
5 | pwexb 6975 | . . . 4 ⊢ (𝑋 ∈ V ↔ 𝒫 𝑋 ∈ V) | |
6 | 4, 5 | sylibr 224 | . . 3 ⊢ (𝒫 𝒫 𝑋 ∈ dom card → 𝑋 ∈ V) |
7 | isufl 21717 | . . 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 247 | 1 ⊢ (𝒫 𝒫 𝑋 ∈ dom card → 𝑋 ∈ UFL) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∈ wcel 1990 ∀wral 2912 ∃wrex 2913 Vcvv 3200 ⊆ wss 3574 𝒫 cpw 4158 dom cdm 5114 ‘cfv 5888 cardccrd 8761 Filcfil 21649 UFilcufil 21703 UFLcufl 21704 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-rpss 6937 df-om 7066 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-en 7956 df-dom 7957 df-fin 7959 df-fi 8317 df-card 8765 df-cda 8990 df-fbas 19743 df-fg 19744 df-fil 21650 df-ufil 21705 df-ufl 21706 |
This theorem is referenced by: fiufl 21720 acufl 21721 |
Copyright terms: Public domain | W3C validator |