![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > 1egrvtxdg1r | Structured version Visualization version GIF version |
Description: The vertex degree of a one-edge graph, case 3: an edge from some other vertex to the given vertex contributes one to the vertex's degree. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 22-Dec-2017.) (Revised by AV, 21-Feb-2021.) |
Ref | Expression |
---|---|
1egrvtxdg1.v | ⊢ (𝜑 → (Vtx‘𝐺) = 𝑉) |
1egrvtxdg1.a | ⊢ (𝜑 → 𝐴 ∈ 𝑋) |
1egrvtxdg1.b | ⊢ (𝜑 → 𝐵 ∈ 𝑉) |
1egrvtxdg1.c | ⊢ (𝜑 → 𝐶 ∈ 𝑉) |
1egrvtxdg1.n | ⊢ (𝜑 → 𝐵 ≠ 𝐶) |
1egrvtxdg1.i | ⊢ (𝜑 → (iEdg‘𝐺) = {〈𝐴, {𝐵, 𝐶}〉}) |
Ref | Expression |
---|---|
1egrvtxdg1r | ⊢ (𝜑 → ((VtxDeg‘𝐺)‘𝐶) = 1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1egrvtxdg1.v | . 2 ⊢ (𝜑 → (Vtx‘𝐺) = 𝑉) | |
2 | 1egrvtxdg1.a | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑋) | |
3 | 1egrvtxdg1.c | . 2 ⊢ (𝜑 → 𝐶 ∈ 𝑉) | |
4 | 1egrvtxdg1.b | . 2 ⊢ (𝜑 → 𝐵 ∈ 𝑉) | |
5 | 1egrvtxdg1.n | . . 3 ⊢ (𝜑 → 𝐵 ≠ 𝐶) | |
6 | 5 | necomd 2991 | . 2 ⊢ (𝜑 → 𝐶 ≠ 𝐵) |
7 | 1egrvtxdg1.i | . . 3 ⊢ (𝜑 → (iEdg‘𝐺) = {〈𝐴, {𝐵, 𝐶}〉}) | |
8 | prcom 4732 | . . . . . 6 ⊢ {𝐵, 𝐶} = {𝐶, 𝐵} | |
9 | 8 | a1i 11 | . . . . 5 ⊢ (𝜑 → {𝐵, 𝐶} = {𝐶, 𝐵}) |
10 | 9 | opeq2d 4876 | . . . 4 ⊢ (𝜑 → 〈𝐴, {𝐵, 𝐶}〉 = 〈𝐴, {𝐶, 𝐵}〉) |
11 | 10 | sneqd 4636 | . . 3 ⊢ (𝜑 → {〈𝐴, {𝐵, 𝐶}〉} = {〈𝐴, {𝐶, 𝐵}〉}) |
12 | 7, 11 | eqtrd 2767 | . 2 ⊢ (𝜑 → (iEdg‘𝐺) = {〈𝐴, {𝐶, 𝐵}〉}) |
13 | 1, 2, 3, 4, 6, 12 | 1egrvtxdg1 29310 | 1 ⊢ (𝜑 → ((VtxDeg‘𝐺)‘𝐶) = 1) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 ≠ wne 2935 {csn 4624 {cpr 4626 〈cop 4630 ‘cfv 6542 1c1 11131 Vtxcvtx 28796 iEdgciedg 28797 VtxDegcvtxdg 29266 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11186 ax-resscn 11187 ax-1cn 11188 ax-icn 11189 ax-addcl 11190 ax-addrcl 11191 ax-mulcl 11192 ax-mulrcl 11193 ax-mulcom 11194 ax-addass 11195 ax-mulass 11196 ax-distr 11197 ax-i2m1 11198 ax-1ne0 11199 ax-1rid 11200 ax-rnegex 11201 ax-rrecex 11202 ax-cnre 11203 ax-pre-lttri 11204 ax-pre-lttrn 11205 ax-pre-ltadd 11206 ax-pre-mulgt0 11207 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 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 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-oadd 8484 df-er 8718 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-dju 9916 df-card 9954 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11468 df-neg 11469 df-nn 12235 df-2 12297 df-n0 12495 df-xnn0 12567 df-z 12581 df-uz 12845 df-xadd 13117 df-fz 13509 df-hash 14314 df-edg 28848 df-upgr 28882 df-umgr 28883 df-uspgr 28950 df-usgr 28951 df-vtxdg 29267 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |