Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > usgr1v0e | Structured version Visualization version GIF version |
Description: The size of a (finite) simple graph with 1 vertex is 0. (Contributed by Alexander van der Vekens, 5-Jan-2018.) (Revised by AV, 22-Oct-2020.) |
Ref | Expression |
---|---|
fusgredgfi.v | ⊢ 𝑉 = (Vtx‘𝐺) |
fusgredgfi.e | ⊢ 𝐸 = (Edg‘𝐺) |
Ref | Expression |
---|---|
usgr1v0e | ⊢ ((𝐺 ∈ USGraph ∧ (#‘𝑉) = 1) → (#‘𝐸) = 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 472 | . . . . . . 7 ⊢ ((𝐺 ∈ USGraph ∧ 𝑉 = {𝑣}) → 𝐺 ∈ USGraph ) | |
2 | vex 3176 | . . . . . . . . 9 ⊢ 𝑣 ∈ V | |
3 | 2 | a1i 11 | . . . . . . . 8 ⊢ ((𝐺 ∈ USGraph ∧ 𝑉 = {𝑣}) → 𝑣 ∈ V) |
4 | fusgredgfi.v | . . . . . . . . . . 11 ⊢ 𝑉 = (Vtx‘𝐺) | |
5 | 4 | eqeq1i 2615 | . . . . . . . . . 10 ⊢ (𝑉 = {𝑣} ↔ (Vtx‘𝐺) = {𝑣}) |
6 | 5 | biimpi 205 | . . . . . . . . 9 ⊢ (𝑉 = {𝑣} → (Vtx‘𝐺) = {𝑣}) |
7 | 6 | adantl 481 | . . . . . . . 8 ⊢ ((𝐺 ∈ USGraph ∧ 𝑉 = {𝑣}) → (Vtx‘𝐺) = {𝑣}) |
8 | usgr1vr 40481 | . . . . . . . . 9 ⊢ ((𝑣 ∈ V ∧ (Vtx‘𝐺) = {𝑣}) → (𝐺 ∈ USGraph → (iEdg‘𝐺) = ∅)) | |
9 | 8 | 3adant1 1072 | . . . . . . . 8 ⊢ ((𝐺 ∈ USGraph ∧ 𝑣 ∈ V ∧ (Vtx‘𝐺) = {𝑣}) → (𝐺 ∈ USGraph → (iEdg‘𝐺) = ∅)) |
10 | 1, 3, 7, 9 | syl3anc 1318 | . . . . . . 7 ⊢ ((𝐺 ∈ USGraph ∧ 𝑉 = {𝑣}) → (𝐺 ∈ USGraph → (iEdg‘𝐺) = ∅)) |
11 | 1, 10 | mpd 15 | . . . . . 6 ⊢ ((𝐺 ∈ USGraph ∧ 𝑉 = {𝑣}) → (iEdg‘𝐺) = ∅) |
12 | fusgredgfi.e | . . . . . . . 8 ⊢ 𝐸 = (Edg‘𝐺) | |
13 | 12 | eqeq1i 2615 | . . . . . . 7 ⊢ (𝐸 = ∅ ↔ (Edg‘𝐺) = ∅) |
14 | usgruhgr 40413 | . . . . . . . . 9 ⊢ (𝐺 ∈ USGraph → 𝐺 ∈ UHGraph ) | |
15 | uhgriedg0edg0 25801 | . . . . . . . . 9 ⊢ (𝐺 ∈ UHGraph → ((Edg‘𝐺) = ∅ ↔ (iEdg‘𝐺) = ∅)) | |
16 | 14, 15 | syl 17 | . . . . . . . 8 ⊢ (𝐺 ∈ USGraph → ((Edg‘𝐺) = ∅ ↔ (iEdg‘𝐺) = ∅)) |
17 | 16 | adantr 480 | . . . . . . 7 ⊢ ((𝐺 ∈ USGraph ∧ 𝑉 = {𝑣}) → ((Edg‘𝐺) = ∅ ↔ (iEdg‘𝐺) = ∅)) |
18 | 13, 17 | syl5bb 271 | . . . . . 6 ⊢ ((𝐺 ∈ USGraph ∧ 𝑉 = {𝑣}) → (𝐸 = ∅ ↔ (iEdg‘𝐺) = ∅)) |
19 | 11, 18 | mpbird 246 | . . . . 5 ⊢ ((𝐺 ∈ USGraph ∧ 𝑉 = {𝑣}) → 𝐸 = ∅) |
20 | 19 | ex 449 | . . . 4 ⊢ (𝐺 ∈ USGraph → (𝑉 = {𝑣} → 𝐸 = ∅)) |
21 | 20 | exlimdv 1848 | . . 3 ⊢ (𝐺 ∈ USGraph → (∃𝑣 𝑉 = {𝑣} → 𝐸 = ∅)) |
22 | fvex 6113 | . . . . 5 ⊢ (Vtx‘𝐺) ∈ V | |
23 | 4, 22 | eqeltri 2684 | . . . 4 ⊢ 𝑉 ∈ V |
24 | hash1snb 13068 | . . . 4 ⊢ (𝑉 ∈ V → ((#‘𝑉) = 1 ↔ ∃𝑣 𝑉 = {𝑣})) | |
25 | 23, 24 | mp1i 13 | . . 3 ⊢ (𝐺 ∈ USGraph → ((#‘𝑉) = 1 ↔ ∃𝑣 𝑉 = {𝑣})) |
26 | fvex 6113 | . . . . 5 ⊢ (Edg‘𝐺) ∈ V | |
27 | 12, 26 | eqeltri 2684 | . . . 4 ⊢ 𝐸 ∈ V |
28 | hasheq0 13015 | . . . 4 ⊢ (𝐸 ∈ V → ((#‘𝐸) = 0 ↔ 𝐸 = ∅)) | |
29 | 27, 28 | mp1i 13 | . . 3 ⊢ (𝐺 ∈ USGraph → ((#‘𝐸) = 0 ↔ 𝐸 = ∅)) |
30 | 21, 25, 29 | 3imtr4d 282 | . 2 ⊢ (𝐺 ∈ USGraph → ((#‘𝑉) = 1 → (#‘𝐸) = 0)) |
31 | 30 | imp 444 | 1 ⊢ ((𝐺 ∈ USGraph ∧ (#‘𝑉) = 1) → (#‘𝐸) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∃wex 1695 ∈ wcel 1977 Vcvv 3173 ∅c0 3874 {csn 4125 ‘cfv 5804 0cc0 9815 1c1 9816 #chash 12979 Vtxcvtx 25673 iEdgciedg 25674 UHGraph cuhgr 25722 Edgcedga 25792 USGraph cusgr 40379 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-n0 11170 df-xnn0 11241 df-z 11255 df-uz 11564 df-fz 12198 df-hash 12980 df-uhgr 25724 df-upgr 25749 df-edga 25793 df-uspgr 40380 df-usgr 40381 |
This theorem is referenced by: cusgrsizeindb1 40666 |
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