Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > cplgr1v | Structured version Visualization version GIF version |
Description: A graph with one vertex is complete. (Contributed by Alexander van der Vekens, 13-Oct-2017.) (Revised by AV, 1-Nov-2020.) (Revised by AV, 23-Mar-2021.) |
Ref | Expression |
---|---|
cplgr0v.v | ⊢ 𝑉 = (Vtx‘𝐺) |
Ref | Expression |
---|---|
cplgr1v | ⊢ ((#‘𝑉) = 1 → 𝐺 ∈ ComplGraph) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 476 | . . . 4 ⊢ (((#‘𝑉) = 1 ∧ 𝑣 ∈ 𝑉) → 𝑣 ∈ 𝑉) | |
2 | ral0 4028 | . . . . 5 ⊢ ∀𝑛 ∈ ∅ 𝑛 ∈ (𝐺 NeighbVtx 𝑣) | |
3 | cplgr0v.v | . . . . . . . . . 10 ⊢ 𝑉 = (Vtx‘𝐺) | |
4 | fvex 6113 | . . . . . . . . . 10 ⊢ (Vtx‘𝐺) ∈ V | |
5 | 3, 4 | eqeltri 2684 | . . . . . . . . 9 ⊢ 𝑉 ∈ V |
6 | hash1snb 13068 | . . . . . . . . 9 ⊢ (𝑉 ∈ V → ((#‘𝑉) = 1 ↔ ∃𝑛 𝑉 = {𝑛})) | |
7 | 5, 6 | ax-mp 5 | . . . . . . . 8 ⊢ ((#‘𝑉) = 1 ↔ ∃𝑛 𝑉 = {𝑛}) |
8 | velsn 4141 | . . . . . . . . . . . 12 ⊢ (𝑣 ∈ {𝑛} ↔ 𝑣 = 𝑛) | |
9 | sneq 4135 | . . . . . . . . . . . . . 14 ⊢ (𝑣 = 𝑛 → {𝑣} = {𝑛}) | |
10 | 9 | difeq2d 3690 | . . . . . . . . . . . . 13 ⊢ (𝑣 = 𝑛 → ({𝑛} ∖ {𝑣}) = ({𝑛} ∖ {𝑛})) |
11 | difid 3902 | . . . . . . . . . . . . 13 ⊢ ({𝑛} ∖ {𝑛}) = ∅ | |
12 | 10, 11 | syl6eq 2660 | . . . . . . . . . . . 12 ⊢ (𝑣 = 𝑛 → ({𝑛} ∖ {𝑣}) = ∅) |
13 | 8, 12 | sylbi 206 | . . . . . . . . . . 11 ⊢ (𝑣 ∈ {𝑛} → ({𝑛} ∖ {𝑣}) = ∅) |
14 | 13 | a1i 11 | . . . . . . . . . 10 ⊢ (𝑉 = {𝑛} → (𝑣 ∈ {𝑛} → ({𝑛} ∖ {𝑣}) = ∅)) |
15 | eleq2 2677 | . . . . . . . . . 10 ⊢ (𝑉 = {𝑛} → (𝑣 ∈ 𝑉 ↔ 𝑣 ∈ {𝑛})) | |
16 | difeq1 3683 | . . . . . . . . . . 11 ⊢ (𝑉 = {𝑛} → (𝑉 ∖ {𝑣}) = ({𝑛} ∖ {𝑣})) | |
17 | 16 | eqeq1d 2612 | . . . . . . . . . 10 ⊢ (𝑉 = {𝑛} → ((𝑉 ∖ {𝑣}) = ∅ ↔ ({𝑛} ∖ {𝑣}) = ∅)) |
18 | 14, 15, 17 | 3imtr4d 282 | . . . . . . . . 9 ⊢ (𝑉 = {𝑛} → (𝑣 ∈ 𝑉 → (𝑉 ∖ {𝑣}) = ∅)) |
19 | 18 | exlimiv 1845 | . . . . . . . 8 ⊢ (∃𝑛 𝑉 = {𝑛} → (𝑣 ∈ 𝑉 → (𝑉 ∖ {𝑣}) = ∅)) |
20 | 7, 19 | sylbi 206 | . . . . . . 7 ⊢ ((#‘𝑉) = 1 → (𝑣 ∈ 𝑉 → (𝑉 ∖ {𝑣}) = ∅)) |
21 | 20 | imp 444 | . . . . . 6 ⊢ (((#‘𝑉) = 1 ∧ 𝑣 ∈ 𝑉) → (𝑉 ∖ {𝑣}) = ∅) |
22 | 21 | raleqdv 3121 | . . . . 5 ⊢ (((#‘𝑉) = 1 ∧ 𝑣 ∈ 𝑉) → (∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ ∀𝑛 ∈ ∅ 𝑛 ∈ (𝐺 NeighbVtx 𝑣))) |
23 | 2, 22 | mpbiri 247 | . . . 4 ⊢ (((#‘𝑉) = 1 ∧ 𝑣 ∈ 𝑉) → ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)) |
24 | 3 | cplgr1vlem 40651 | . . . . . 6 ⊢ ((#‘𝑉) = 1 → 𝐺 ∈ V) |
25 | 3 | uvtxael 40614 | . . . . . 6 ⊢ (𝐺 ∈ V → (𝑣 ∈ (UnivVtx‘𝐺) ↔ (𝑣 ∈ 𝑉 ∧ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)))) |
26 | 24, 25 | syl 17 | . . . . 5 ⊢ ((#‘𝑉) = 1 → (𝑣 ∈ (UnivVtx‘𝐺) ↔ (𝑣 ∈ 𝑉 ∧ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)))) |
27 | 26 | adantr 480 | . . . 4 ⊢ (((#‘𝑉) = 1 ∧ 𝑣 ∈ 𝑉) → (𝑣 ∈ (UnivVtx‘𝐺) ↔ (𝑣 ∈ 𝑉 ∧ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)))) |
28 | 1, 23, 27 | mpbir2and 959 | . . 3 ⊢ (((#‘𝑉) = 1 ∧ 𝑣 ∈ 𝑉) → 𝑣 ∈ (UnivVtx‘𝐺)) |
29 | 28 | ralrimiva 2949 | . 2 ⊢ ((#‘𝑉) = 1 → ∀𝑣 ∈ 𝑉 𝑣 ∈ (UnivVtx‘𝐺)) |
30 | 3 | iscplgr 40636 | . . 3 ⊢ (𝐺 ∈ V → (𝐺 ∈ ComplGraph ↔ ∀𝑣 ∈ 𝑉 𝑣 ∈ (UnivVtx‘𝐺))) |
31 | 24, 30 | syl 17 | . 2 ⊢ ((#‘𝑉) = 1 → (𝐺 ∈ ComplGraph ↔ ∀𝑣 ∈ 𝑉 𝑣 ∈ (UnivVtx‘𝐺))) |
32 | 29, 31 | mpbird 246 | 1 ⊢ ((#‘𝑉) = 1 → 𝐺 ∈ ComplGraph) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∃wex 1695 ∈ wcel 1977 ∀wral 2896 Vcvv 3173 ∖ cdif 3537 ∅c0 3874 {csn 4125 ‘cfv 5804 (class class class)co 6549 1c1 9816 #chash 12979 Vtxcvtx 25673 NeighbVtx cnbgr 40550 UnivVtxcuvtxa 40551 ComplGraphccplgr 40552 |
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-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-hash 12980 df-uvtxa 40556 df-cplgr 40557 |
This theorem is referenced by: cusgr1v 40653 |
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