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Mirrors > Home > MPE Home > Th. List > Mathboxes > setc2othin | Structured version Visualization version GIF version |
Description: The category (SetCat‘2o) is thin. A special case of setcthin 48112. (Contributed by Zhi Wang, 20-Sep-2024.) |
Ref | Expression |
---|---|
setc2othin | ⊢ (SetCat‘2o) ∈ ThinCat |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqidd 2728 | . . 3 ⊢ (⊤ → (SetCat‘2o) = (SetCat‘2o)) | |
2 | 2oex 8502 | . . . 4 ⊢ 2o ∈ V | |
3 | 2 | a1i 11 | . . 3 ⊢ (⊤ → 2o ∈ V) |
4 | elpri 4653 | . . . . . . 7 ⊢ (𝑥 ∈ {∅, {∅}} → (𝑥 = ∅ ∨ 𝑥 = {∅})) | |
5 | 0ex 5309 | . . . . . . . . 9 ⊢ ∅ ∈ V | |
6 | sneq 4640 | . . . . . . . . . 10 ⊢ (𝑦 = ∅ → {𝑦} = {∅}) | |
7 | 6 | eqeq2d 2738 | . . . . . . . . 9 ⊢ (𝑦 = ∅ → (𝑥 = {𝑦} ↔ 𝑥 = {∅})) |
8 | 5, 7 | spcev 3593 | . . . . . . . 8 ⊢ (𝑥 = {∅} → ∃𝑦 𝑥 = {𝑦}) |
9 | 8 | orim2i 908 | . . . . . . 7 ⊢ ((𝑥 = ∅ ∨ 𝑥 = {∅}) → (𝑥 = ∅ ∨ ∃𝑦 𝑥 = {𝑦})) |
10 | mo0sn 47937 | . . . . . . . 8 ⊢ (∃*𝑧 𝑧 ∈ 𝑥 ↔ (𝑥 = ∅ ∨ ∃𝑦 𝑥 = {𝑦})) | |
11 | 10 | biimpri 227 | . . . . . . 7 ⊢ ((𝑥 = ∅ ∨ ∃𝑦 𝑥 = {𝑦}) → ∃*𝑧 𝑧 ∈ 𝑥) |
12 | 4, 9, 11 | 3syl 18 | . . . . . 6 ⊢ (𝑥 ∈ {∅, {∅}} → ∃*𝑧 𝑧 ∈ 𝑥) |
13 | df2o2 8500 | . . . . . 6 ⊢ 2o = {∅, {∅}} | |
14 | 12, 13 | eleq2s 2846 | . . . . 5 ⊢ (𝑥 ∈ 2o → ∃*𝑧 𝑧 ∈ 𝑥) |
15 | 14 | rgen 3059 | . . . 4 ⊢ ∀𝑥 ∈ 2o ∃*𝑧 𝑧 ∈ 𝑥 |
16 | 15 | a1i 11 | . . 3 ⊢ (⊤ → ∀𝑥 ∈ 2o ∃*𝑧 𝑧 ∈ 𝑥) |
17 | 1, 3, 16 | setcthin 48112 | . 2 ⊢ (⊤ → (SetCat‘2o) ∈ ThinCat) |
18 | 17 | mptru 1540 | 1 ⊢ (SetCat‘2o) ∈ ThinCat |
Colors of variables: wff setvar class |
Syntax hints: ∨ wo 845 = wceq 1533 ⊤wtru 1534 ∃wex 1773 ∈ wcel 2098 ∃*wmo 2527 ∀wral 3057 Vcvv 3471 ∅c0 4324 {csn 4630 {cpr 4632 ‘cfv 6551 2oc2o 8485 SetCatcsetc 18069 ThinCatcthinc 48076 |
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 ax-cnex 11200 ax-resscn 11201 ax-1cn 11202 ax-icn 11203 ax-addcl 11204 ax-addrcl 11205 ax-mulcl 11206 ax-mulrcl 11207 ax-mulcom 11208 ax-addass 11209 ax-mulass 11210 ax-distr 11211 ax-i2m1 11212 ax-1ne0 11213 ax-1rid 11214 ax-rnegex 11215 ax-rrecex 11216 ax-cnre 11217 ax-pre-lttri 11218 ax-pre-lttrn 11219 ax-pre-ltadd 11220 ax-pre-mulgt0 11221 |
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-tp 4635 df-op 4637 df-uni 4911 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-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-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 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-2o 8492 df-er 8729 df-map 8851 df-en 8969 df-dom 8970 df-sdom 8971 df-fin 8972 df-pnf 11286 df-mnf 11287 df-xr 11288 df-ltxr 11289 df-le 11290 df-sub 11482 df-neg 11483 df-nn 12249 df-2 12311 df-3 12312 df-4 12313 df-5 12314 df-6 12315 df-7 12316 df-8 12317 df-9 12318 df-n0 12509 df-z 12595 df-dec 12714 df-uz 12859 df-fz 13523 df-struct 17121 df-slot 17156 df-ndx 17168 df-base 17186 df-hom 17262 df-cco 17263 df-cat 17653 df-cid 17654 df-setc 18070 df-thinc 48077 |
This theorem is referenced by: (None) |
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