| 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 12541 | . . 3 ⊢ 0 ∈ ℕ0 | |
| 2 | stgrfv 47920 | . . 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 13667 | . . . 4 ⊢ (0...0) = {0} | |
| 5 | 4 | opeq2i 4877 | . . 3 ⊢ 〈(Base‘ndx), (0...0)〉 = 〈(Base‘ndx), {0}〉 |
| 6 | rabeq0 4388 | . . . . . . 7 ⊢ ({𝑒 ∈ 𝒫 (0...0) ∣ ∃𝑥 ∈ (1...0)𝑒 = {0, 𝑥}} = ∅ ↔ ∀𝑒 ∈ 𝒫 (0...0) ¬ ∃𝑥 ∈ (1...0)𝑒 = {0, 𝑥}) | |
| 7 | noel 4338 | . . . . . . . . . . . 12 ⊢ ¬ 𝑥 ∈ ∅ | |
| 8 | 7 | pm2.21i 119 | . . . . . . . . . . 11 ⊢ (𝑥 ∈ ∅ → ¬ 𝑒 = {0, 𝑥}) |
| 9 | fz10 13585 | . . . . . . . . . . 11 ⊢ (1...0) = ∅ | |
| 10 | 8, 9 | eleq2s 2859 | . . . . . . . . . 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 3145 | . . . . . . . 8 ⊢ (𝑒 ∈ 𝒫 (0...0) → ∀𝑥 ∈ (1...0) ¬ 𝑒 = {0, 𝑥}) |
| 13 | ralnex 3072 | . . . . . . . 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 3068 | . . . . . 6 ⊢ {𝑒 ∈ 𝒫 (0...0) ∣ ∃𝑥 ∈ (1...0)𝑒 = {0, 𝑥}} = ∅ |
| 16 | 15 | reseq2i 5994 | . . . . 5 ⊢ ( I ↾ {𝑒 ∈ 𝒫 (0...0) ∣ ∃𝑥 ∈ (1...0)𝑒 = {0, 𝑥}}) = ( I ↾ ∅) |
| 17 | res0 6001 | . . . . 5 ⊢ ( I ↾ ∅) = ∅ | |
| 18 | 16, 17 | eqtri 2765 | . . . 4 ⊢ ( I ↾ {𝑒 ∈ 𝒫 (0...0) ∣ ∃𝑥 ∈ (1...0)𝑒 = {0, 𝑥}}) = ∅ |
| 19 | 18 | opeq2i 4877 | . . 3 ⊢ 〈(.ef‘ndx), ( I ↾ {𝑒 ∈ 𝒫 (0...0) ∣ ∃𝑥 ∈ (1...0)𝑒 = {0, 𝑥}})〉 = 〈(.ef‘ndx), ∅〉 |
| 20 | 5, 19 | preq12i 4738 | . 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 2765 | 1 ⊢ (StarGr‘0) = {〈(Base‘ndx), {0}〉, 〈(.ef‘ndx), ∅〉} |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ∃wrex 3070 {crab 3436 ∅c0 4333 𝒫 cpw 4600 {csn 4626 {cpr 4628 〈cop 4632 I cid 5577 ↾ cres 5687 ‘cfv 6561 (class class class)co 7431 0cc0 11155 1c1 11156 ℕ0cn0 12526 ...cfz 13547 ndxcnx 17230 Basecbs 17247 .efcedgf 29003 StarGrcstgr 47918 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-n0 12527 df-z 12614 df-uz 12879 df-fz 13548 df-stgr 47919 |
| This theorem is referenced by: (None) |
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