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| Mirrors > Home > MPE Home > Th. List > umgredgnlp | Structured version Visualization version GIF version | ||
| Description: An edge of a multigraph is not a loop. (Contributed by AV, 9-Jan-2020.) (Revised by AV, 8-Jun-2021.) |
| Ref | Expression |
|---|---|
| umgredgnlp.e | ⊢ 𝐸 = (Edg‘𝐺) |
| Ref | Expression |
|---|---|
| umgredgnlp | ⊢ ((𝐺 ∈ UMGraph ∧ 𝐶 ∈ 𝐸) → ¬ ∃𝑣 𝐶 = {𝑣}) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | vex 3484 | . . . . . 6 ⊢ 𝑣 ∈ V | |
| 2 | hashsng 14408 | . . . . . 6 ⊢ (𝑣 ∈ V → (♯‘{𝑣}) = 1) | |
| 3 | 1ne2 12474 | . . . . . . . 8 ⊢ 1 ≠ 2 | |
| 4 | 3 | neii 2942 | . . . . . . 7 ⊢ ¬ 1 = 2 |
| 5 | eqeq1 2741 | . . . . . . 7 ⊢ ((♯‘{𝑣}) = 1 → ((♯‘{𝑣}) = 2 ↔ 1 = 2)) | |
| 6 | 4, 5 | mtbiri 327 | . . . . . 6 ⊢ ((♯‘{𝑣}) = 1 → ¬ (♯‘{𝑣}) = 2) |
| 7 | 1, 2, 6 | mp2b 10 | . . . . 5 ⊢ ¬ (♯‘{𝑣}) = 2 |
| 8 | fveqeq2 6915 | . . . . 5 ⊢ (𝐶 = {𝑣} → ((♯‘𝐶) = 2 ↔ (♯‘{𝑣}) = 2)) | |
| 9 | 7, 8 | mtbiri 327 | . . . 4 ⊢ (𝐶 = {𝑣} → ¬ (♯‘𝐶) = 2) |
| 10 | 9 | intnand 488 | . . 3 ⊢ (𝐶 = {𝑣} → ¬ (𝐶 ∈ 𝒫 (Vtx‘𝐺) ∧ (♯‘𝐶) = 2)) |
| 11 | umgredgnlp.e | . . . . 5 ⊢ 𝐸 = (Edg‘𝐺) | |
| 12 | 11 | eleq2i 2833 | . . . 4 ⊢ (𝐶 ∈ 𝐸 ↔ 𝐶 ∈ (Edg‘𝐺)) |
| 13 | edgumgr 29152 | . . . 4 ⊢ ((𝐺 ∈ UMGraph ∧ 𝐶 ∈ (Edg‘𝐺)) → (𝐶 ∈ 𝒫 (Vtx‘𝐺) ∧ (♯‘𝐶) = 2)) | |
| 14 | 12, 13 | sylan2b 594 | . . 3 ⊢ ((𝐺 ∈ UMGraph ∧ 𝐶 ∈ 𝐸) → (𝐶 ∈ 𝒫 (Vtx‘𝐺) ∧ (♯‘𝐶) = 2)) |
| 15 | 10, 14 | nsyl3 138 | . 2 ⊢ ((𝐺 ∈ UMGraph ∧ 𝐶 ∈ 𝐸) → ¬ 𝐶 = {𝑣}) |
| 16 | 15 | nexdv 1936 | 1 ⊢ ((𝐺 ∈ UMGraph ∧ 𝐶 ∈ 𝐸) → ¬ ∃𝑣 𝐶 = {𝑣}) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1540 ∃wex 1779 ∈ wcel 2108 Vcvv 3480 𝒫 cpw 4600 {csn 4626 ‘cfv 6561 1c1 11156 2c2 12321 ♯chash 14369 Vtxcvtx 29013 Edgcedg 29064 UMGraphcumgr 29098 |
| 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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-n0 12527 df-z 12614 df-uz 12879 df-fz 13548 df-hash 14370 df-edg 29065 df-umgr 29100 |
| This theorem is referenced by: (None) |
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