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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > stgr0 | Structured version Visualization version GIF version |
Description: The star graph S0 consists of a single vertex without edges. (Contributed by AV, 11-Sep-2025.) |
Ref | Expression |
---|---|
stgr0 | ⊢ (StarGr‘0) = {〈(Base‘ndx), {0}〉, 〈(.ef‘ndx), ∅〉} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0nn0 12538 | . . 3 ⊢ 0 ∈ ℕ0 | |
2 | stgrfv 47855 | . . 3 ⊢ (0 ∈ ℕ0 → (StarGr‘0) = {〈(Base‘ndx), (0...0)〉, 〈(.ef‘ndx), ( I ↾ {𝑒 ∈ 𝒫 (0...0) ∣ ∃𝑥 ∈ (1...0)𝑒 = {0, 𝑥}})〉}) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ (StarGr‘0) = {〈(Base‘ndx), (0...0)〉, 〈(.ef‘ndx), ( I ↾ {𝑒 ∈ 𝒫 (0...0) ∣ ∃𝑥 ∈ (1...0)𝑒 = {0, 𝑥}})〉} |
4 | fz0sn 13663 | . . . 4 ⊢ (0...0) = {0} | |
5 | 4 | opeq2i 4881 | . . 3 ⊢ 〈(Base‘ndx), (0...0)〉 = 〈(Base‘ndx), {0}〉 |
6 | rabeq0 4393 | . . . . . . 7 ⊢ ({𝑒 ∈ 𝒫 (0...0) ∣ ∃𝑥 ∈ (1...0)𝑒 = {0, 𝑥}} = ∅ ↔ ∀𝑒 ∈ 𝒫 (0...0) ¬ ∃𝑥 ∈ (1...0)𝑒 = {0, 𝑥}) | |
7 | noel 4343 | . . . . . . . . . . . 12 ⊢ ¬ 𝑥 ∈ ∅ | |
8 | 7 | pm2.21i 119 | . . . . . . . . . . 11 ⊢ (𝑥 ∈ ∅ → ¬ 𝑒 = {0, 𝑥}) |
9 | fz10 13581 | . . . . . . . . . . 11 ⊢ (1...0) = ∅ | |
10 | 8, 9 | eleq2s 2856 | . . . . . . . . . 10 ⊢ (𝑥 ∈ (1...0) → ¬ 𝑒 = {0, 𝑥}) |
11 | 10 | a1i 11 | . . . . . . . . 9 ⊢ (𝑒 ∈ 𝒫 (0...0) → (𝑥 ∈ (1...0) → ¬ 𝑒 = {0, 𝑥})) |
12 | 11 | ralrimiv 3142 | . . . . . . . 8 ⊢ (𝑒 ∈ 𝒫 (0...0) → ∀𝑥 ∈ (1...0) ¬ 𝑒 = {0, 𝑥}) |
13 | ralnex 3069 | . . . . . . . 8 ⊢ (∀𝑥 ∈ (1...0) ¬ 𝑒 = {0, 𝑥} ↔ ¬ ∃𝑥 ∈ (1...0)𝑒 = {0, 𝑥}) | |
14 | 12, 13 | sylib 218 | . . . . . . 7 ⊢ (𝑒 ∈ 𝒫 (0...0) → ¬ ∃𝑥 ∈ (1...0)𝑒 = {0, 𝑥}) |
15 | 6, 14 | mprgbir 3065 | . . . . . 6 ⊢ {𝑒 ∈ 𝒫 (0...0) ∣ ∃𝑥 ∈ (1...0)𝑒 = {0, 𝑥}} = ∅ |
16 | 15 | reseq2i 5996 | . . . . 5 ⊢ ( I ↾ {𝑒 ∈ 𝒫 (0...0) ∣ ∃𝑥 ∈ (1...0)𝑒 = {0, 𝑥}}) = ( I ↾ ∅) |
17 | res0 6003 | . . . . 5 ⊢ ( I ↾ ∅) = ∅ | |
18 | 16, 17 | eqtri 2762 | . . . 4 ⊢ ( I ↾ {𝑒 ∈ 𝒫 (0...0) ∣ ∃𝑥 ∈ (1...0)𝑒 = {0, 𝑥}}) = ∅ |
19 | 18 | opeq2i 4881 | . . 3 ⊢ 〈(.ef‘ndx), ( I ↾ {𝑒 ∈ 𝒫 (0...0) ∣ ∃𝑥 ∈ (1...0)𝑒 = {0, 𝑥}})〉 = 〈(.ef‘ndx), ∅〉 |
20 | 5, 19 | preq12i 4742 | . 2 ⊢ {〈(Base‘ndx), (0...0)〉, 〈(.ef‘ndx), ( I ↾ {𝑒 ∈ 𝒫 (0...0) ∣ ∃𝑥 ∈ (1...0)𝑒 = {0, 𝑥}})〉} = {〈(Base‘ndx), {0}〉, 〈(.ef‘ndx), ∅〉} |
21 | 3, 20 | eqtri 2762 | 1 ⊢ (StarGr‘0) = {〈(Base‘ndx), {0}〉, 〈(.ef‘ndx), ∅〉} |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1536 ∈ wcel 2105 ∀wral 3058 ∃wrex 3067 {crab 3432 ∅c0 4338 𝒫 cpw 4604 {csn 4630 {cpr 4632 〈cop 4636 I cid 5581 ↾ cres 5690 ‘cfv 6562 (class class class)co 7430 0cc0 11152 1c1 11153 ℕ0cn0 12523 ...cfz 13543 ndxcnx 17226 Basecbs 17244 .efcedgf 29017 StarGrcstgr 47853 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-nn 12264 df-n0 12524 df-z 12611 df-uz 12876 df-fz 13544 df-stgr 47854 |
This theorem is referenced by: (None) |
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