![]() |
Mathbox for Zhi Wang |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > setc2othin | Structured version Visualization version GIF version |
Description: The category (SetCatβ2o) is thin. A special case of setcthin 47161. (Contributed by Zhi Wang, 20-Sep-2024.) |
Ref | Expression |
---|---|
setc2othin | β’ (SetCatβ2o) β ThinCat |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqidd 2734 | . . 3 β’ (β€ β (SetCatβ2o) = (SetCatβ2o)) | |
2 | 2oex 8424 | . . . 4 β’ 2o β V | |
3 | 2 | a1i 11 | . . 3 β’ (β€ β 2o β V) |
4 | elpri 4609 | . . . . . . 7 β’ (π₯ β {β , {β }} β (π₯ = β β¨ π₯ = {β })) | |
5 | 0ex 5265 | . . . . . . . . 9 β’ β β V | |
6 | sneq 4597 | . . . . . . . . . 10 β’ (π¦ = β β {π¦} = {β }) | |
7 | 6 | eqeq2d 2744 | . . . . . . . . 9 β’ (π¦ = β β (π₯ = {π¦} β π₯ = {β })) |
8 | 5, 7 | spcev 3564 | . . . . . . . 8 β’ (π₯ = {β } β βπ¦ π₯ = {π¦}) |
9 | 8 | orim2i 910 | . . . . . . 7 β’ ((π₯ = β β¨ π₯ = {β }) β (π₯ = β β¨ βπ¦ π₯ = {π¦})) |
10 | mo0sn 46986 | . . . . . . . 8 β’ (β*π§ π§ β π₯ β (π₯ = β β¨ βπ¦ π₯ = {π¦})) | |
11 | 10 | biimpri 227 | . . . . . . 7 β’ ((π₯ = β β¨ βπ¦ π₯ = {π¦}) β β*π§ π§ β π₯) |
12 | 4, 9, 11 | 3syl 18 | . . . . . 6 β’ (π₯ β {β , {β }} β β*π§ π§ β π₯) |
13 | df2o2 8422 | . . . . . 6 β’ 2o = {β , {β }} | |
14 | 12, 13 | eleq2s 2852 | . . . . 5 β’ (π₯ β 2o β β*π§ π§ β π₯) |
15 | 14 | rgen 3063 | . . . 4 β’ βπ₯ β 2o β*π§ π§ β π₯ |
16 | 15 | a1i 11 | . . 3 β’ (β€ β βπ₯ β 2o β*π§ π§ β π₯) |
17 | 1, 3, 16 | setcthin 47161 | . 2 β’ (β€ β (SetCatβ2o) β ThinCat) |
18 | 17 | mptru 1549 | 1 β’ (SetCatβ2o) β ThinCat |
Colors of variables: wff setvar class |
Syntax hints: β¨ wo 846 = wceq 1542 β€wtru 1543 βwex 1782 β wcel 2107 β*wmo 2533 βwral 3061 Vcvv 3444 β c0 4283 {csn 4587 {cpr 4589 βcfv 6497 2oc2o 8407 SetCatcsetc 17966 ThinCatcthinc 47125 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-2o 8414 df-er 8651 df-map 8770 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-nn 12159 df-2 12221 df-3 12222 df-4 12223 df-5 12224 df-6 12225 df-7 12226 df-8 12227 df-9 12228 df-n0 12419 df-z 12505 df-dec 12624 df-uz 12769 df-fz 13431 df-struct 17024 df-slot 17059 df-ndx 17071 df-base 17089 df-hom 17162 df-cco 17163 df-cat 17553 df-cid 17554 df-setc 17967 df-thinc 47126 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |