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Mathbox for Zhi Wang |
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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 48172. (Contributed by Zhi Wang, 20-Sep-2024.) |
Ref | Expression |
---|---|
setc2othin | β’ (SetCatβ2o) β ThinCat |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqidd 2726 | . . 3 β’ (β€ β (SetCatβ2o) = (SetCatβ2o)) | |
2 | 2oex 8494 | . . . 4 β’ 2o β V | |
3 | 2 | a1i 11 | . . 3 β’ (β€ β 2o β V) |
4 | elpri 4647 | . . . . . . 7 β’ (π₯ β {β , {β }} β (π₯ = β β¨ π₯ = {β })) | |
5 | 0ex 5302 | . . . . . . . . 9 β’ β β V | |
6 | sneq 4634 | . . . . . . . . . 10 β’ (π¦ = β β {π¦} = {β }) | |
7 | 6 | eqeq2d 2736 | . . . . . . . . 9 β’ (π¦ = β β (π₯ = {π¦} β π₯ = {β })) |
8 | 5, 7 | spcev 3586 | . . . . . . . 8 β’ (π₯ = {β } β βπ¦ π₯ = {π¦}) |
9 | 8 | orim2i 908 | . . . . . . 7 β’ ((π₯ = β β¨ π₯ = {β }) β (π₯ = β β¨ βπ¦ π₯ = {π¦})) |
10 | mo0sn 47997 | . . . . . . . 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 8492 | . . . . . 6 β’ 2o = {β , {β }} | |
14 | 12, 13 | eleq2s 2843 | . . . . 5 β’ (π₯ β 2o β β*π§ π§ β π₯) |
15 | 14 | rgen 3053 | . . . 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 48172 | . 2 β’ (β€ β (SetCatβ2o) β ThinCat) |
18 | 17 | mptru 1540 | 1 β’ (SetCatβ2o) β ThinCat |
Colors of variables: wff setvar class |
Syntax hints: β¨ wo 845 = wceq 1533 β€wtru 1534 βwex 1773 β wcel 2098 β*wmo 2526 βwral 3051 Vcvv 3463 β c0 4318 {csn 4624 {cpr 4626 βcfv 6542 2oc2o 8477 SetCatcsetc 18061 ThinCatcthinc 48136 |
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 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7737 ax-cnex 11192 ax-resscn 11193 ax-1cn 11194 ax-icn 11195 ax-addcl 11196 ax-addrcl 11197 ax-mulcl 11198 ax-mulrcl 11199 ax-mulcom 11200 ax-addass 11201 ax-mulass 11202 ax-distr 11203 ax-i2m1 11204 ax-1ne0 11205 ax-1rid 11206 ax-rnegex 11207 ax-rrecex 11208 ax-cnre 11209 ax-pre-lttri 11210 ax-pre-lttrn 11211 ax-pre-ltadd 11212 ax-pre-mulgt0 11213 |
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 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3960 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 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 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 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-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-om 7868 df-1st 7989 df-2nd 7990 df-frecs 8283 df-wrecs 8314 df-recs 8388 df-rdg 8427 df-1o 8483 df-2o 8484 df-er 8721 df-map 8843 df-en 8961 df-dom 8962 df-sdom 8963 df-fin 8964 df-pnf 11278 df-mnf 11279 df-xr 11280 df-ltxr 11281 df-le 11282 df-sub 11474 df-neg 11475 df-nn 12241 df-2 12303 df-3 12304 df-4 12305 df-5 12306 df-6 12307 df-7 12308 df-8 12309 df-9 12310 df-n0 12501 df-z 12587 df-dec 12706 df-uz 12851 df-fz 13515 df-struct 17113 df-slot 17148 df-ndx 17160 df-base 17178 df-hom 17254 df-cco 17255 df-cat 17645 df-cid 17646 df-setc 18062 df-thinc 48137 |
This theorem is referenced by: (None) |
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