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Mathbox for Stefan O'Rear |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > isnumbasgrp | Structured version Visualization version GIF version |
Description: A set is numerable iff it and its Hartogs number can be jointly given the structure of a group. (Contributed by Stefan O'Rear, 9-Jul-2015.) |
Ref | Expression |
---|---|
isnumbasgrp | ⊢ (𝑆 ∈ dom card ↔ (𝑆 ∪ (har‘𝑆)) ∈ (Base “ Grp)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ablgrp 19829 | . . . . 5 ⊢ (𝑥 ∈ Abel → 𝑥 ∈ Grp) | |
2 | 1 | ssriv 4012 | . . . 4 ⊢ Abel ⊆ Grp |
3 | imass2 6134 | . . . 4 ⊢ (Abel ⊆ Grp → (Base “ Abel) ⊆ (Base “ Grp)) | |
4 | 2, 3 | ax-mp 5 | . . 3 ⊢ (Base “ Abel) ⊆ (Base “ Grp) |
5 | isnumbasabl 43065 | . . . 4 ⊢ (𝑆 ∈ dom card ↔ (𝑆 ∪ (har‘𝑆)) ∈ (Base “ Abel)) | |
6 | 5 | biimpi 216 | . . 3 ⊢ (𝑆 ∈ dom card → (𝑆 ∪ (har‘𝑆)) ∈ (Base “ Abel)) |
7 | 4, 6 | sselid 4006 | . 2 ⊢ (𝑆 ∈ dom card → (𝑆 ∪ (har‘𝑆)) ∈ (Base “ Grp)) |
8 | isnumbasgrplem2 43063 | . 2 ⊢ ((𝑆 ∪ (har‘𝑆)) ∈ (Base “ Grp) → 𝑆 ∈ dom card) | |
9 | 7, 8 | impbii 209 | 1 ⊢ (𝑆 ∈ dom card ↔ (𝑆 ∪ (har‘𝑆)) ∈ (Base “ Grp)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∈ wcel 2108 ∪ cun 3974 ⊆ wss 3976 dom cdm 5700 “ cima 5703 ‘cfv 6575 harchar 9627 cardccrd 10006 Basecbs 17260 Grpcgrp 18975 Abelcabl 19825 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7772 ax-inf2 9712 ax-cnex 11242 ax-resscn 11243 ax-1cn 11244 ax-icn 11245 ax-addcl 11246 ax-addrcl 11247 ax-mulcl 11248 ax-mulrcl 11249 ax-mulcom 11250 ax-addass 11251 ax-mulass 11252 ax-distr 11253 ax-i2m1 11254 ax-1ne0 11255 ax-1rid 11256 ax-rnegex 11257 ax-rrecex 11258 ax-cnre 11259 ax-pre-lttri 11260 ax-pre-lttrn 11261 ax-pre-ltadd 11262 ax-pre-mulgt0 11263 ax-pre-sup 11264 ax-addf 11265 ax-mulf 11266 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6334 df-ord 6400 df-on 6401 df-lim 6402 df-suc 6403 df-iota 6527 df-fun 6577 df-fn 6578 df-f 6579 df-f1 6580 df-fo 6581 df-f1o 6582 df-fv 6583 df-isom 6584 df-riota 7406 df-ov 7453 df-oprab 7454 df-mpo 7455 df-om 7906 df-1st 8032 df-2nd 8033 df-supp 8204 df-tpos 8269 df-frecs 8324 df-wrecs 8355 df-recs 8429 df-rdg 8468 df-seqom 8506 df-1o 8524 df-2o 8525 df-oadd 8528 df-omul 8529 df-er 8765 df-ec 8767 df-qs 8771 df-map 8888 df-ixp 8958 df-en 9006 df-dom 9007 df-sdom 9008 df-fin 9009 df-fsupp 9434 df-sup 9513 df-inf 9514 df-oi 9581 df-har 9628 df-wdom 9636 df-dju 9972 df-card 10010 df-acn 10013 df-pnf 11328 df-mnf 11329 df-xr 11330 df-ltxr 11331 df-le 11332 df-sub 11524 df-neg 11525 df-div 11950 df-nn 12296 df-2 12358 df-3 12359 df-4 12360 df-5 12361 df-6 12362 df-7 12363 df-8 12364 df-9 12365 df-n0 12556 df-z 12642 df-dec 12761 df-uz 12906 df-rp 13060 df-fz 13570 df-fzo 13714 df-fl 13845 df-mod 13923 df-seq 14055 df-hash 14382 df-dvds 16305 df-struct 17196 df-sets 17213 df-slot 17231 df-ndx 17243 df-base 17261 df-ress 17290 df-plusg 17326 df-mulr 17327 df-starv 17328 df-sca 17329 df-vsca 17330 df-ip 17331 df-tset 17332 df-ple 17333 df-ds 17335 df-unif 17336 df-hom 17337 df-cco 17338 df-0g 17503 df-prds 17509 df-pws 17511 df-imas 17570 df-qus 17571 df-mgm 18680 df-sgrp 18759 df-mnd 18775 df-mhm 18820 df-grp 18978 df-minusg 18979 df-sbg 18980 df-mulg 19110 df-subg 19165 df-nsg 19166 df-eqg 19167 df-ghm 19255 df-gim 19301 df-gic 19302 df-cmn 19826 df-abl 19827 df-mgp 20164 df-rng 20182 df-ur 20211 df-ring 20264 df-cring 20265 df-oppr 20362 df-dvdsr 20385 df-rhm 20500 df-subrng 20574 df-subrg 20599 df-lmod 20884 df-lss 20955 df-lsp 20995 df-sra 21197 df-rgmod 21198 df-lidl 21243 df-rsp 21244 df-2idl 21285 df-cnfld 21390 df-zring 21483 df-zrh 21539 df-zn 21542 df-dsmm 21777 df-frlm 21792 |
This theorem is referenced by: dfacbasgrp 43067 |
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