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Mirrors > Home > MPE Home > Th. List > nbupgrel | Structured version Visualization version GIF version |
Description: A neighbor of a vertex in a pseudograph. (Contributed by AV, 5-Nov-2020.) |
Ref | Expression |
---|---|
nbuhgr.v | ⊢ 𝑉 = (Vtx‘𝐺) |
nbuhgr.e | ⊢ 𝐸 = (Edg‘𝐺) |
Ref | Expression |
---|---|
nbupgrel | ⊢ (((𝐺 ∈ UPGraph ∧ 𝐾 ∈ 𝑉) ∧ (𝑁 ∈ 𝑉 ∧ 𝑁 ≠ 𝐾)) → (𝑁 ∈ (𝐺 NeighbVtx 𝐾) ↔ {𝑁, 𝐾} ∈ 𝐸)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nbuhgr.v | . . . . . 6 ⊢ 𝑉 = (Vtx‘𝐺) | |
2 | nbuhgr.e | . . . . . 6 ⊢ 𝐸 = (Edg‘𝐺) | |
3 | 1, 2 | nbupgr 29185 | . . . . 5 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐾 ∈ 𝑉) → (𝐺 NeighbVtx 𝐾) = {𝑛 ∈ (𝑉 ∖ {𝐾}) ∣ {𝐾, 𝑛} ∈ 𝐸}) |
4 | 3 | eleq2d 2815 | . . . 4 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐾 ∈ 𝑉) → (𝑁 ∈ (𝐺 NeighbVtx 𝐾) ↔ 𝑁 ∈ {𝑛 ∈ (𝑉 ∖ {𝐾}) ∣ {𝐾, 𝑛} ∈ 𝐸})) |
5 | preq2 4743 | . . . . . 6 ⊢ (𝑛 = 𝑁 → {𝐾, 𝑛} = {𝐾, 𝑁}) | |
6 | 5 | eleq1d 2814 | . . . . 5 ⊢ (𝑛 = 𝑁 → ({𝐾, 𝑛} ∈ 𝐸 ↔ {𝐾, 𝑁} ∈ 𝐸)) |
7 | 6 | elrab 3684 | . . . 4 ⊢ (𝑁 ∈ {𝑛 ∈ (𝑉 ∖ {𝐾}) ∣ {𝐾, 𝑛} ∈ 𝐸} ↔ (𝑁 ∈ (𝑉 ∖ {𝐾}) ∧ {𝐾, 𝑁} ∈ 𝐸)) |
8 | 4, 7 | bitrdi 286 | . . 3 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐾 ∈ 𝑉) → (𝑁 ∈ (𝐺 NeighbVtx 𝐾) ↔ (𝑁 ∈ (𝑉 ∖ {𝐾}) ∧ {𝐾, 𝑁} ∈ 𝐸))) |
9 | 8 | adantr 479 | . 2 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐾 ∈ 𝑉) ∧ (𝑁 ∈ 𝑉 ∧ 𝑁 ≠ 𝐾)) → (𝑁 ∈ (𝐺 NeighbVtx 𝐾) ↔ (𝑁 ∈ (𝑉 ∖ {𝐾}) ∧ {𝐾, 𝑁} ∈ 𝐸))) |
10 | eldifsn 4795 | . . . . 5 ⊢ (𝑁 ∈ (𝑉 ∖ {𝐾}) ↔ (𝑁 ∈ 𝑉 ∧ 𝑁 ≠ 𝐾)) | |
11 | 10 | biimpri 227 | . . . 4 ⊢ ((𝑁 ∈ 𝑉 ∧ 𝑁 ≠ 𝐾) → 𝑁 ∈ (𝑉 ∖ {𝐾})) |
12 | 11 | adantl 480 | . . 3 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐾 ∈ 𝑉) ∧ (𝑁 ∈ 𝑉 ∧ 𝑁 ≠ 𝐾)) → 𝑁 ∈ (𝑉 ∖ {𝐾})) |
13 | 12 | biantrurd 531 | . 2 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐾 ∈ 𝑉) ∧ (𝑁 ∈ 𝑉 ∧ 𝑁 ≠ 𝐾)) → ({𝐾, 𝑁} ∈ 𝐸 ↔ (𝑁 ∈ (𝑉 ∖ {𝐾}) ∧ {𝐾, 𝑁} ∈ 𝐸))) |
14 | prcom 4741 | . . . 4 ⊢ {𝐾, 𝑁} = {𝑁, 𝐾} | |
15 | 14 | eleq1i 2820 | . . 3 ⊢ ({𝐾, 𝑁} ∈ 𝐸 ↔ {𝑁, 𝐾} ∈ 𝐸) |
16 | 15 | a1i 11 | . 2 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐾 ∈ 𝑉) ∧ (𝑁 ∈ 𝑉 ∧ 𝑁 ≠ 𝐾)) → ({𝐾, 𝑁} ∈ 𝐸 ↔ {𝑁, 𝐾} ∈ 𝐸)) |
17 | 9, 13, 16 | 3bitr2d 306 | 1 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐾 ∈ 𝑉) ∧ (𝑁 ∈ 𝑉 ∧ 𝑁 ≠ 𝐾)) → (𝑁 ∈ (𝐺 NeighbVtx 𝐾) ↔ {𝑁, 𝐾} ∈ 𝐸)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ≠ wne 2937 {crab 3430 ∖ cdif 3946 {csn 4632 {cpr 4634 ‘cfv 6553 (class class class)co 7426 Vtxcvtx 28837 Edgcedg 28888 UPGraphcupgr 28921 NeighbVtx cnbgr 29173 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7748 ax-cnex 11204 ax-resscn 11205 ax-1cn 11206 ax-icn 11207 ax-addcl 11208 ax-addrcl 11209 ax-mulcl 11210 ax-mulrcl 11211 ax-mulcom 11212 ax-addass 11213 ax-mulass 11214 ax-distr 11215 ax-i2m1 11216 ax-1ne0 11217 ax-1rid 11218 ax-rnegex 11219 ax-rrecex 11220 ax-cnre 11221 ax-pre-lttri 11222 ax-pre-lttrn 11223 ax-pre-ltadd 11224 ax-pre-mulgt0 11225 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7879 df-1st 8001 df-2nd 8002 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-1o 8495 df-2o 8496 df-oadd 8499 df-er 8733 df-en 8973 df-dom 8974 df-sdom 8975 df-fin 8976 df-dju 9934 df-card 9972 df-pnf 11290 df-mnf 11291 df-xr 11292 df-ltxr 11293 df-le 11294 df-sub 11486 df-neg 11487 df-nn 12253 df-2 12315 df-n0 12513 df-xnn0 12585 df-z 12599 df-uz 12863 df-fz 13527 df-hash 14332 df-edg 28889 df-upgr 28923 df-nbgr 29174 |
This theorem is referenced by: nbupgrres 29205 cplgr3v 29276 |
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