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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 29109 | . . . . 5 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐾 ∈ 𝑉) → (𝐺 NeighbVtx 𝐾) = {𝑛 ∈ (𝑉 ∖ {𝐾}) ∣ {𝐾, 𝑛} ∈ 𝐸}) |
4 | 3 | eleq2d 2813 | . . . 4 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐾 ∈ 𝑉) → (𝑁 ∈ (𝐺 NeighbVtx 𝐾) ↔ 𝑁 ∈ {𝑛 ∈ (𝑉 ∖ {𝐾}) ∣ {𝐾, 𝑛} ∈ 𝐸})) |
5 | preq2 4733 | . . . . . 6 ⊢ (𝑛 = 𝑁 → {𝐾, 𝑛} = {𝐾, 𝑁}) | |
6 | 5 | eleq1d 2812 | . . . . 5 ⊢ (𝑛 = 𝑁 → ({𝐾, 𝑛} ∈ 𝐸 ↔ {𝐾, 𝑁} ∈ 𝐸)) |
7 | 6 | elrab 3678 | . . . 4 ⊢ (𝑁 ∈ {𝑛 ∈ (𝑉 ∖ {𝐾}) ∣ {𝐾, 𝑛} ∈ 𝐸} ↔ (𝑁 ∈ (𝑉 ∖ {𝐾}) ∧ {𝐾, 𝑁} ∈ 𝐸)) |
8 | 4, 7 | bitrdi 287 | . . 3 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐾 ∈ 𝑉) → (𝑁 ∈ (𝐺 NeighbVtx 𝐾) ↔ (𝑁 ∈ (𝑉 ∖ {𝐾}) ∧ {𝐾, 𝑁} ∈ 𝐸))) |
9 | 8 | adantr 480 | . 2 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐾 ∈ 𝑉) ∧ (𝑁 ∈ 𝑉 ∧ 𝑁 ≠ 𝐾)) → (𝑁 ∈ (𝐺 NeighbVtx 𝐾) ↔ (𝑁 ∈ (𝑉 ∖ {𝐾}) ∧ {𝐾, 𝑁} ∈ 𝐸))) |
10 | eldifsn 4785 | . . . . 5 ⊢ (𝑁 ∈ (𝑉 ∖ {𝐾}) ↔ (𝑁 ∈ 𝑉 ∧ 𝑁 ≠ 𝐾)) | |
11 | 10 | biimpri 227 | . . . 4 ⊢ ((𝑁 ∈ 𝑉 ∧ 𝑁 ≠ 𝐾) → 𝑁 ∈ (𝑉 ∖ {𝐾})) |
12 | 11 | adantl 481 | . . 3 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐾 ∈ 𝑉) ∧ (𝑁 ∈ 𝑉 ∧ 𝑁 ≠ 𝐾)) → 𝑁 ∈ (𝑉 ∖ {𝐾})) |
13 | 12 | biantrurd 532 | . 2 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐾 ∈ 𝑉) ∧ (𝑁 ∈ 𝑉 ∧ 𝑁 ≠ 𝐾)) → ({𝐾, 𝑁} ∈ 𝐸 ↔ (𝑁 ∈ (𝑉 ∖ {𝐾}) ∧ {𝐾, 𝑁} ∈ 𝐸))) |
14 | prcom 4731 | . . . 4 ⊢ {𝐾, 𝑁} = {𝑁, 𝐾} | |
15 | 14 | eleq1i 2818 | . . 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 307 | 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 395 = wceq 1533 ∈ wcel 2098 ≠ wne 2934 {crab 3426 ∖ cdif 3940 {csn 4623 {cpr 4625 ‘cfv 6537 (class class class)co 7405 Vtxcvtx 28764 Edgcedg 28815 UPGraphcupgr 28848 NeighbVtx cnbgr 29097 |
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 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-2o 8468 df-oadd 8471 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-dju 9898 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-n0 12477 df-xnn0 12549 df-z 12563 df-uz 12827 df-fz 13491 df-hash 14296 df-edg 28816 df-upgr 28850 df-nbgr 29098 |
This theorem is referenced by: nbupgrres 29129 cplgr3v 29200 |
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