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Mirrors > Home > MPE Home > Th. List > usgr1vr | Structured version Visualization version GIF version |
Description: A simple graph with one vertex has no edges. (Contributed by AV, 18-Oct-2020.) (Revised by AV, 21-Mar-2021.) (Proof shortened by AV, 2-Apr-2021.) |
Ref | Expression |
---|---|
usgr1vr | ⊢ ((𝐴 ∈ 𝑋 ∧ (Vtx‘𝐺) = {𝐴}) → (𝐺 ∈ USGraph → (iEdg‘𝐺) = ∅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | usgruhgr 28992 | . . . . 5 ⊢ (𝐺 ∈ USGraph → 𝐺 ∈ UHGraph) | |
2 | 1 | adantl 481 | . . . 4 ⊢ (((𝐴 ∈ 𝑋 ∧ (Vtx‘𝐺) = {𝐴}) ∧ 𝐺 ∈ USGraph) → 𝐺 ∈ UHGraph) |
3 | fveq2 6891 | . . . . . 6 ⊢ ((Vtx‘𝐺) = {𝐴} → (♯‘(Vtx‘𝐺)) = (♯‘{𝐴})) | |
4 | hashsng 14354 | . . . . . 6 ⊢ (𝐴 ∈ 𝑋 → (♯‘{𝐴}) = 1) | |
5 | 3, 4 | sylan9eqr 2789 | . . . . 5 ⊢ ((𝐴 ∈ 𝑋 ∧ (Vtx‘𝐺) = {𝐴}) → (♯‘(Vtx‘𝐺)) = 1) |
6 | 5 | adantr 480 | . . . 4 ⊢ (((𝐴 ∈ 𝑋 ∧ (Vtx‘𝐺) = {𝐴}) ∧ 𝐺 ∈ USGraph) → (♯‘(Vtx‘𝐺)) = 1) |
7 | eqid 2727 | . . . . . . 7 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
8 | eqid 2727 | . . . . . . 7 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺) | |
9 | 7, 8 | usgrislfuspgr 28993 | . . . . . 6 ⊢ (𝐺 ∈ USGraph ↔ (𝐺 ∈ USPGraph ∧ (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ 2 ≤ (♯‘𝑥)})) |
10 | 9 | simprbi 496 | . . . . 5 ⊢ (𝐺 ∈ USGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ 2 ≤ (♯‘𝑥)}) |
11 | 10 | adantl 481 | . . . 4 ⊢ (((𝐴 ∈ 𝑋 ∧ (Vtx‘𝐺) = {𝐴}) ∧ 𝐺 ∈ USGraph) → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ 2 ≤ (♯‘𝑥)}) |
12 | eqid 2727 | . . . . 5 ⊢ {𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ 2 ≤ (♯‘𝑥)} = {𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ 2 ≤ (♯‘𝑥)} | |
13 | 7, 8, 12 | lfuhgr1v0e 29060 | . . . 4 ⊢ ((𝐺 ∈ UHGraph ∧ (♯‘(Vtx‘𝐺)) = 1 ∧ (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ 2 ≤ (♯‘𝑥)}) → (Edg‘𝐺) = ∅) |
14 | 2, 6, 11, 13 | syl3anc 1369 | . . 3 ⊢ (((𝐴 ∈ 𝑋 ∧ (Vtx‘𝐺) = {𝐴}) ∧ 𝐺 ∈ USGraph) → (Edg‘𝐺) = ∅) |
15 | uhgriedg0edg0 28933 | . . . . 5 ⊢ (𝐺 ∈ UHGraph → ((Edg‘𝐺) = ∅ ↔ (iEdg‘𝐺) = ∅)) | |
16 | 1, 15 | syl 17 | . . . 4 ⊢ (𝐺 ∈ USGraph → ((Edg‘𝐺) = ∅ ↔ (iEdg‘𝐺) = ∅)) |
17 | 16 | adantl 481 | . . 3 ⊢ (((𝐴 ∈ 𝑋 ∧ (Vtx‘𝐺) = {𝐴}) ∧ 𝐺 ∈ USGraph) → ((Edg‘𝐺) = ∅ ↔ (iEdg‘𝐺) = ∅)) |
18 | 14, 17 | mpbid 231 | . 2 ⊢ (((𝐴 ∈ 𝑋 ∧ (Vtx‘𝐺) = {𝐴}) ∧ 𝐺 ∈ USGraph) → (iEdg‘𝐺) = ∅) |
19 | 18 | ex 412 | 1 ⊢ ((𝐴 ∈ 𝑋 ∧ (Vtx‘𝐺) = {𝐴}) → (𝐺 ∈ USGraph → (iEdg‘𝐺) = ∅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1534 ∈ wcel 2099 {crab 3427 ∅c0 4318 𝒫 cpw 4598 {csn 4624 class class class wbr 5142 dom cdm 5672 ⟶wf 6538 ‘cfv 6542 1c1 11133 ≤ cle 11273 2c2 12291 ♯chash 14315 Vtxcvtx 28802 iEdgciedg 28803 Edgcedg 28853 UHGraphcuhgr 28862 USPGraphcuspgr 28954 USGraphcusgr 28955 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11188 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-pre-mulgt0 11209 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 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 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 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 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-oadd 8484 df-er 8718 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-dju 9918 df-card 9956 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-sub 11470 df-neg 11471 df-nn 12237 df-2 12299 df-n0 12497 df-xnn0 12569 df-z 12583 df-uz 12847 df-fz 13511 df-hash 14316 df-edg 28854 df-uhgr 28864 df-upgr 28888 df-uspgr 28956 df-usgr 28957 |
This theorem is referenced by: usgr1v 29062 usgr1v0e 29132 |
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