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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 48856. (Contributed by Zhi Wang, 20-Sep-2024.) |
Ref | Expression |
---|---|
setc2othin | ⊢ (SetCat‘2o) ∈ ThinCat |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqidd 2736 | . . 3 ⊢ (⊤ → (SetCat‘2o) = (SetCat‘2o)) | |
2 | 2oex 8516 | . . . 4 ⊢ 2o ∈ V | |
3 | 2 | a1i 11 | . . 3 ⊢ (⊤ → 2o ∈ V) |
4 | elpri 4654 | . . . . . . 7 ⊢ (𝑥 ∈ {∅, {∅}} → (𝑥 = ∅ ∨ 𝑥 = {∅})) | |
5 | 0ex 5313 | . . . . . . . . 9 ⊢ ∅ ∈ V | |
6 | sneq 4641 | . . . . . . . . . 10 ⊢ (𝑦 = ∅ → {𝑦} = {∅}) | |
7 | 6 | eqeq2d 2746 | . . . . . . . . 9 ⊢ (𝑦 = ∅ → (𝑥 = {𝑦} ↔ 𝑥 = {∅})) |
8 | 5, 7 | spcev 3606 | . . . . . . . 8 ⊢ (𝑥 = {∅} → ∃𝑦 𝑥 = {𝑦}) |
9 | 8 | orim2i 910 | . . . . . . 7 ⊢ ((𝑥 = ∅ ∨ 𝑥 = {∅}) → (𝑥 = ∅ ∨ ∃𝑦 𝑥 = {𝑦})) |
10 | mo0sn 48664 | . . . . . . . 8 ⊢ (∃*𝑧 𝑧 ∈ 𝑥 ↔ (𝑥 = ∅ ∨ ∃𝑦 𝑥 = {𝑦})) | |
11 | 10 | biimpri 228 | . . . . . . 7 ⊢ ((𝑥 = ∅ ∨ ∃𝑦 𝑥 = {𝑦}) → ∃*𝑧 𝑧 ∈ 𝑥) |
12 | 4, 9, 11 | 3syl 18 | . . . . . 6 ⊢ (𝑥 ∈ {∅, {∅}} → ∃*𝑧 𝑧 ∈ 𝑥) |
13 | df2o2 8514 | . . . . . 6 ⊢ 2o = {∅, {∅}} | |
14 | 12, 13 | eleq2s 2857 | . . . . 5 ⊢ (𝑥 ∈ 2o → ∃*𝑧 𝑧 ∈ 𝑥) |
15 | 14 | rgen 3061 | . . . 4 ⊢ ∀𝑥 ∈ 2o ∃*𝑧 𝑧 ∈ 𝑥 |
16 | 15 | a1i 11 | . . 3 ⊢ (⊤ → ∀𝑥 ∈ 2o ∃*𝑧 𝑧 ∈ 𝑥) |
17 | 1, 3, 16 | setcthin 48856 | . 2 ⊢ (⊤ → (SetCat‘2o) ∈ ThinCat) |
18 | 17 | mptru 1544 | 1 ⊢ (SetCat‘2o) ∈ ThinCat |
Colors of variables: wff setvar class |
Syntax hints: ∨ wo 847 = wceq 1537 ⊤wtru 1538 ∃wex 1776 ∈ wcel 2106 ∃*wmo 2536 ∀wral 3059 Vcvv 3478 ∅c0 4339 {csn 4631 {cpr 4633 ‘cfv 6563 2oc2o 8499 SetCatcsetc 18129 ThinCatcthinc 48819 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-fz 13545 df-struct 17181 df-slot 17216 df-ndx 17228 df-base 17246 df-hom 17322 df-cco 17323 df-cat 17713 df-cid 17714 df-setc 18130 df-thinc 48820 |
This theorem is referenced by: (None) |
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