nbupgrel - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > nbupgrel		Structured version Visualization version GIF version

Theorem nbupgrel 29272

Description: A neighbor of a vertex in a pseudograph. (Contributed by AV, 5-Nov-2020.)

**Hypotheses**
Ref	Expression
nbuhgr.v	⊢ 𝑉 = (Vtx‘𝐺)
nbuhgr.e	⊢ 𝐸 = (Edg‘𝐺)

**Assertion**
Ref	Expression
nbupgrel	⊢ (((𝐺 ∈ UPGraph ∧ 𝐾 ∈ 𝑉) ∧ (𝑁 ∈ 𝑉 ∧ 𝑁 ≠ 𝐾)) → (𝑁 ∈ (𝐺 NeighbVtx 𝐾) ↔ {𝑁, 𝐾} ∈ 𝐸))

Proof of Theorem nbupgrel Dummy variable 𝑛 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nbuhgr.v	. . . . . 6 ⊢ 𝑉 = (Vtx‘𝐺)
2		nbuhgr.e	. . . . . 6 ⊢ 𝐸 = (Edg‘𝐺)
3	1, 2	nbupgr 29271	. . . . 5 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐾 ∈ 𝑉) → (𝐺 NeighbVtx 𝐾) = {𝑛 ∈ (𝑉 ∖ {𝐾}) ∣ {𝐾, 𝑛} ∈ 𝐸})
4	3	eleq2d 2814	. . . 4 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐾 ∈ 𝑉) → (𝑁 ∈ (𝐺 NeighbVtx 𝐾) ↔ 𝑁 ∈ {𝑛 ∈ (𝑉 ∖ {𝐾}) ∣ {𝐾, 𝑛} ∈ 𝐸}))
5		preq2 4698	. . . . . 6 ⊢ (𝑛 = 𝑁 → {𝐾, 𝑛} = {𝐾, 𝑁})
6	5	eleq1d 2813	. . . . 5 ⊢ (𝑛 = 𝑁 → ({𝐾, 𝑛} ∈ 𝐸 ↔ {𝐾, 𝑁} ∈ 𝐸))
7	6	elrab 3659	. . . 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 4750	. . . . 5 ⊢ (𝑁 ∈ (𝑉 ∖ {𝐾}) ↔ (𝑁 ∈ 𝑉 ∧ 𝑁 ≠ 𝐾))
11	10	biimpri 228	. . . 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 4696	. . . 4 ⊢ {𝐾, 𝑁} = {𝑁, 𝐾}
15	14	eleq1i 2819	. . 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 206 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 {crab 3405 ∖ cdif 3911 {csn 4589 {cpr 4591 ‘cfv 6511 (class class class)co 7387 Vtxcvtx 28923 Edgcedg 28974 UPGraphcupgr 29007 NeighbVtx cnbgr 29259

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-int 4911 df-iun 4957 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-om 7843 df-1st 7968 df-2nd 7969 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-2o 8435 df-oadd 8438 df-er 8671 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-dju 9854 df-card 9892 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-nn 12187 df-2 12249 df-n0 12443 df-xnn0 12516 df-z 12530 df-uz 12794 df-fz 13469 df-hash 14296 df-edg 28975 df-upgr 29009 df-nbgr 29260

This theorem is referenced by: nbupgrres 29291 cplgr3v 29362

W3C validator