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Mirrors > Home > MPE Home > Th. List > Mathboxes > uvtx2vtx1edg | Structured version Visualization version GIF version |
Description: If a graph has two vertices, and there is an edge between the vertices, then each vertex is universal. (Contributed by AV, 1-Nov-2020.) (Revised by AV, 25-Mar-2021.) |
Ref | Expression |
---|---|
uvtxael.v | ⊢ 𝑉 = (Vtx‘𝐺) |
isuvtxa.e | ⊢ 𝐸 = (Edg‘𝐺) |
Ref | Expression |
---|---|
uvtx2vtx1edg | ⊢ (((#‘𝑉) = 2 ∧ 𝑉 ∈ 𝐸) → ∀𝑣 ∈ 𝑉 𝑣 ∈ (UnivVtx‘𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uvtxael.v | . . 3 ⊢ 𝑉 = (Vtx‘𝐺) | |
2 | isuvtxa.e | . . 3 ⊢ 𝐸 = (Edg‘𝐺) | |
3 | 1, 2 | nbgr2vtx1edg 40572 | . 2 ⊢ (((#‘𝑉) = 2 ∧ 𝑉 ∈ 𝐸) → ∀𝑣 ∈ 𝑉 ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)) |
4 | hash2prb 13111 | . . . . . . 7 ⊢ (𝑉 ∈ 𝐸 → ((#‘𝑉) = 2 ↔ ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑉 (𝑥 ≠ 𝑦 ∧ 𝑉 = {𝑥, 𝑦}))) | |
5 | 1 | 1vgrex 25679 | . . . . . . . . . 10 ⊢ (𝑥 ∈ 𝑉 → 𝐺 ∈ V) |
6 | 5 | a1d 25 | . . . . . . . . 9 ⊢ (𝑥 ∈ 𝑉 → ((𝑥 ≠ 𝑦 ∧ 𝑉 = {𝑥, 𝑦}) → 𝐺 ∈ V)) |
7 | 6 | adantr 480 | . . . . . . . 8 ⊢ ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → ((𝑥 ≠ 𝑦 ∧ 𝑉 = {𝑥, 𝑦}) → 𝐺 ∈ V)) |
8 | 7 | rexlimivv 3018 | . . . . . . 7 ⊢ (∃𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑉 (𝑥 ≠ 𝑦 ∧ 𝑉 = {𝑥, 𝑦}) → 𝐺 ∈ V) |
9 | 4, 8 | syl6bi 242 | . . . . . 6 ⊢ (𝑉 ∈ 𝐸 → ((#‘𝑉) = 2 → 𝐺 ∈ V)) |
10 | 9 | impcom 445 | . . . . 5 ⊢ (((#‘𝑉) = 2 ∧ 𝑉 ∈ 𝐸) → 𝐺 ∈ V) |
11 | 1 | uvtxael 40614 | . . . . 5 ⊢ (𝐺 ∈ V → (𝑣 ∈ (UnivVtx‘𝐺) ↔ (𝑣 ∈ 𝑉 ∧ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)))) |
12 | 10, 11 | syl 17 | . . . 4 ⊢ (((#‘𝑉) = 2 ∧ 𝑉 ∈ 𝐸) → (𝑣 ∈ (UnivVtx‘𝐺) ↔ (𝑣 ∈ 𝑉 ∧ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)))) |
13 | 12 | baibd 946 | . . 3 ⊢ ((((#‘𝑉) = 2 ∧ 𝑉 ∈ 𝐸) ∧ 𝑣 ∈ 𝑉) → (𝑣 ∈ (UnivVtx‘𝐺) ↔ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣))) |
14 | 13 | ralbidva 2968 | . 2 ⊢ (((#‘𝑉) = 2 ∧ 𝑉 ∈ 𝐸) → (∀𝑣 ∈ 𝑉 𝑣 ∈ (UnivVtx‘𝐺) ↔ ∀𝑣 ∈ 𝑉 ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣))) |
15 | 3, 14 | mpbird 246 | 1 ⊢ (((#‘𝑉) = 2 ∧ 𝑉 ∈ 𝐸) → ∀𝑣 ∈ 𝑉 𝑣 ∈ (UnivVtx‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 ∃wrex 2897 Vcvv 3173 ∖ cdif 3537 {csn 4125 {cpr 4127 ‘cfv 5804 (class class class)co 6549 2c2 10947 #chash 12979 Vtxcvtx 25673 Edgcedga 25792 NeighbVtx cnbgr 40550 UnivVtxcuvtxa 40551 |
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-fal 1481 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-2o 7448 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-z 11255 df-uz 11564 df-fz 12198 df-hash 12980 df-nbgr 40554 df-uvtxa 40556 |
This theorem is referenced by: (None) |
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