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Mirrors > Home > MPE Home > Th. List > umgrpredgv | Structured version Visualization version GIF version |
Description: An edge of a multigraph always connects two vertices. Analogue of umgredgprv 28634. This theorem does not hold for arbitrary pseudographs: if either 𝑀 or 𝑁 is a proper class, then {𝑀, 𝑁} ∈ 𝐸 could still hold ({𝑀, 𝑁} would be either {𝑀} or {𝑁}, see prprc1 4768 or prprc2 4769, i.e. a loop), but 𝑀 ∈ 𝑉 or 𝑁 ∈ 𝑉 would not be true. (Contributed by AV, 27-Nov-2020.) |
Ref | Expression |
---|---|
upgredg.v | ⊢ 𝑉 = (Vtx‘𝐺) |
upgredg.e | ⊢ 𝐸 = (Edg‘𝐺) |
Ref | Expression |
---|---|
umgrpredgv | ⊢ ((𝐺 ∈ UMGraph ∧ {𝑀, 𝑁} ∈ 𝐸) → (𝑀 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | upgredg.e | . . . 4 ⊢ 𝐸 = (Edg‘𝐺) | |
2 | 1 | eleq2i 2823 | . . 3 ⊢ ({𝑀, 𝑁} ∈ 𝐸 ↔ {𝑀, 𝑁} ∈ (Edg‘𝐺)) |
3 | edgumgr 28662 | . . 3 ⊢ ((𝐺 ∈ UMGraph ∧ {𝑀, 𝑁} ∈ (Edg‘𝐺)) → ({𝑀, 𝑁} ∈ 𝒫 (Vtx‘𝐺) ∧ (♯‘{𝑀, 𝑁}) = 2)) | |
4 | 2, 3 | sylan2b 592 | . 2 ⊢ ((𝐺 ∈ UMGraph ∧ {𝑀, 𝑁} ∈ 𝐸) → ({𝑀, 𝑁} ∈ 𝒫 (Vtx‘𝐺) ∧ (♯‘{𝑀, 𝑁}) = 2)) |
5 | eqid 2730 | . . . . 5 ⊢ {𝑀, 𝑁} = {𝑀, 𝑁} | |
6 | 5 | hashprdifel 14362 | . . . 4 ⊢ ((♯‘{𝑀, 𝑁}) = 2 → (𝑀 ∈ {𝑀, 𝑁} ∧ 𝑁 ∈ {𝑀, 𝑁} ∧ 𝑀 ≠ 𝑁)) |
7 | upgredg.v | . . . . . . . . 9 ⊢ 𝑉 = (Vtx‘𝐺) | |
8 | 7 | eqcomi 2739 | . . . . . . . 8 ⊢ (Vtx‘𝐺) = 𝑉 |
9 | 8 | pweqi 4617 | . . . . . . 7 ⊢ 𝒫 (Vtx‘𝐺) = 𝒫 𝑉 |
10 | 9 | eleq2i 2823 | . . . . . 6 ⊢ ({𝑀, 𝑁} ∈ 𝒫 (Vtx‘𝐺) ↔ {𝑀, 𝑁} ∈ 𝒫 𝑉) |
11 | prelpw 5445 | . . . . . . 7 ⊢ ((𝑀 ∈ {𝑀, 𝑁} ∧ 𝑁 ∈ {𝑀, 𝑁}) → ((𝑀 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉) ↔ {𝑀, 𝑁} ∈ 𝒫 𝑉)) | |
12 | 11 | biimprd 247 | . . . . . 6 ⊢ ((𝑀 ∈ {𝑀, 𝑁} ∧ 𝑁 ∈ {𝑀, 𝑁}) → ({𝑀, 𝑁} ∈ 𝒫 𝑉 → (𝑀 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉))) |
13 | 10, 12 | biimtrid 241 | . . . . 5 ⊢ ((𝑀 ∈ {𝑀, 𝑁} ∧ 𝑁 ∈ {𝑀, 𝑁}) → ({𝑀, 𝑁} ∈ 𝒫 (Vtx‘𝐺) → (𝑀 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉))) |
14 | 13 | 3adant3 1130 | . . . 4 ⊢ ((𝑀 ∈ {𝑀, 𝑁} ∧ 𝑁 ∈ {𝑀, 𝑁} ∧ 𝑀 ≠ 𝑁) → ({𝑀, 𝑁} ∈ 𝒫 (Vtx‘𝐺) → (𝑀 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉))) |
15 | 6, 14 | syl 17 | . . 3 ⊢ ((♯‘{𝑀, 𝑁}) = 2 → ({𝑀, 𝑁} ∈ 𝒫 (Vtx‘𝐺) → (𝑀 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉))) |
16 | 15 | impcom 406 | . 2 ⊢ (({𝑀, 𝑁} ∈ 𝒫 (Vtx‘𝐺) ∧ (♯‘{𝑀, 𝑁}) = 2) → (𝑀 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉)) |
17 | 4, 16 | syl 17 | 1 ⊢ ((𝐺 ∈ UMGraph ∧ {𝑀, 𝑁} ∈ 𝐸) → (𝑀 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∧ w3a 1085 = wceq 1539 ∈ wcel 2104 ≠ wne 2938 𝒫 cpw 4601 {cpr 4629 ‘cfv 6542 2c2 12271 ♯chash 14294 Vtxcvtx 28523 Edgcedg 28574 UMGraphcumgr 28608 |
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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 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 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-oadd 8472 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-z 12563 df-uz 12827 df-fz 13489 df-hash 14295 df-edg 28575 df-umgr 28610 |
This theorem is referenced by: umgrnloop2 28673 usgrpredgv 28721 umgr2edg 28733 umgrvad2edg 28737 nbumgr 28871 umgr2adedgwlklem 29465 umgr2adedgspth 29469 frgrncvvdeqlem2 29820 fusgr2wsp2nb 29854 |
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