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Mirrors > Home > MPE Home > Th. List > uspgredg2vtxeu | Structured version Visualization version GIF version |
Description: For a vertex incident to an edge there is exactly one other vertex incident to the edge in a simple pseudograph. (Contributed by AV, 18-Oct-2020.) (Revised by AV, 6-Dec-2020.) |
Ref | Expression |
---|---|
uspgredg2vtxeu | ⊢ ((𝐺 ∈ USPGraph ∧ 𝐸 ∈ (Edg‘𝐺) ∧ 𝑌 ∈ 𝐸) → ∃!𝑦 ∈ (Vtx‘𝐺)𝐸 = {𝑌, 𝑦}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uspgrupgr 29210 | . . 3 ⊢ (𝐺 ∈ USPGraph → 𝐺 ∈ UPGraph) | |
2 | eqid 2735 | . . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
3 | eqid 2735 | . . . 4 ⊢ (Edg‘𝐺) = (Edg‘𝐺) | |
4 | 2, 3 | upgredg2vtx 29173 | . . 3 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐸 ∈ (Edg‘𝐺) ∧ 𝑌 ∈ 𝐸) → ∃𝑦 ∈ (Vtx‘𝐺)𝐸 = {𝑌, 𝑦}) |
5 | 1, 4 | syl3an1 1162 | . 2 ⊢ ((𝐺 ∈ USPGraph ∧ 𝐸 ∈ (Edg‘𝐺) ∧ 𝑌 ∈ 𝐸) → ∃𝑦 ∈ (Vtx‘𝐺)𝐸 = {𝑌, 𝑦}) |
6 | eqtr2 2759 | . . . . 5 ⊢ ((𝐸 = {𝑌, 𝑦} ∧ 𝐸 = {𝑌, 𝑥}) → {𝑌, 𝑦} = {𝑌, 𝑥}) | |
7 | vex 3482 | . . . . . 6 ⊢ 𝑦 ∈ V | |
8 | vex 3482 | . . . . . 6 ⊢ 𝑥 ∈ V | |
9 | 7, 8 | preqr2 4854 | . . . . 5 ⊢ ({𝑌, 𝑦} = {𝑌, 𝑥} → 𝑦 = 𝑥) |
10 | 6, 9 | syl 17 | . . . 4 ⊢ ((𝐸 = {𝑌, 𝑦} ∧ 𝐸 = {𝑌, 𝑥}) → 𝑦 = 𝑥) |
11 | 10 | a1i 11 | . . 3 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐸 ∈ (Edg‘𝐺) ∧ 𝑌 ∈ 𝐸) ∧ (𝑦 ∈ (Vtx‘𝐺) ∧ 𝑥 ∈ (Vtx‘𝐺))) → ((𝐸 = {𝑌, 𝑦} ∧ 𝐸 = {𝑌, 𝑥}) → 𝑦 = 𝑥)) |
12 | 11 | ralrimivva 3200 | . 2 ⊢ ((𝐺 ∈ USPGraph ∧ 𝐸 ∈ (Edg‘𝐺) ∧ 𝑌 ∈ 𝐸) → ∀𝑦 ∈ (Vtx‘𝐺)∀𝑥 ∈ (Vtx‘𝐺)((𝐸 = {𝑌, 𝑦} ∧ 𝐸 = {𝑌, 𝑥}) → 𝑦 = 𝑥)) |
13 | preq2 4739 | . . . 4 ⊢ (𝑦 = 𝑥 → {𝑌, 𝑦} = {𝑌, 𝑥}) | |
14 | 13 | eqeq2d 2746 | . . 3 ⊢ (𝑦 = 𝑥 → (𝐸 = {𝑌, 𝑦} ↔ 𝐸 = {𝑌, 𝑥})) |
15 | 14 | reu4 3740 | . 2 ⊢ (∃!𝑦 ∈ (Vtx‘𝐺)𝐸 = {𝑌, 𝑦} ↔ (∃𝑦 ∈ (Vtx‘𝐺)𝐸 = {𝑌, 𝑦} ∧ ∀𝑦 ∈ (Vtx‘𝐺)∀𝑥 ∈ (Vtx‘𝐺)((𝐸 = {𝑌, 𝑦} ∧ 𝐸 = {𝑌, 𝑥}) → 𝑦 = 𝑥))) |
16 | 5, 12, 15 | sylanbrc 583 | 1 ⊢ ((𝐺 ∈ USPGraph ∧ 𝐸 ∈ (Edg‘𝐺) ∧ 𝑌 ∈ 𝐸) → ∃!𝑦 ∈ (Vtx‘𝐺)𝐸 = {𝑌, 𝑦}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ∀wral 3059 ∃wrex 3068 ∃!wreu 3376 {cpr 4633 ‘cfv 6563 Vtxcvtx 29028 Edgcedg 29079 UPGraphcupgr 29112 USPGraphcuspgr 29180 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-oadd 8509 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-dju 9939 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-n0 12525 df-xnn0 12598 df-z 12612 df-uz 12877 df-fz 13545 df-hash 14367 df-edg 29080 df-upgr 29114 df-uspgr 29182 |
This theorem is referenced by: usgredg2vtxeu 29253 uspgredg2vlem 29255 uspgredg2v 29256 |
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