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Mirrors > Home > MPE Home > Th. List > vdfrgra0 | Structured version Visualization version GIF version |
Description: A vertex in a friendship graph has degree 0 if the graph consists of only one vertex. (Contributed by Alexander van der Vekens, 6-Dec-2017.) |
Ref | Expression |
---|---|
vdfrgra0 | ⊢ ((𝑉 FriendGrph 𝐸 ∧ 𝑁 ∈ 𝑉 ∧ (#‘𝑉) = 1) → ((𝑉 VDeg 𝐸)‘𝑁) = 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frisusgra 26519 | . . . . . . 7 ⊢ (𝑉 FriendGrph 𝐸 → 𝑉 USGrph 𝐸) | |
2 | usgrav 25867 | . . . . . . 7 ⊢ (𝑉 USGrph 𝐸 → (𝑉 ∈ V ∧ 𝐸 ∈ V)) | |
3 | 1, 2 | syl 17 | . . . . . 6 ⊢ (𝑉 FriendGrph 𝐸 → (𝑉 ∈ V ∧ 𝐸 ∈ V)) |
4 | hash1snb 13068 | . . . . . . . . 9 ⊢ (𝑉 ∈ V → ((#‘𝑉) = 1 ↔ ∃𝑎 𝑉 = {𝑎})) | |
5 | 4 | adantr 480 | . . . . . . . 8 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → ((#‘𝑉) = 1 ↔ ∃𝑎 𝑉 = {𝑎})) |
6 | breq1 4586 | . . . . . . . . . . 11 ⊢ (𝑉 = {𝑎} → (𝑉 FriendGrph 𝐸 ↔ {𝑎} FriendGrph 𝐸)) | |
7 | frisusgra 26519 | . . . . . . . . . . . 12 ⊢ ({𝑎} FriendGrph 𝐸 → {𝑎} USGrph 𝐸) | |
8 | usgra1v 25919 | . . . . . . . . . . . 12 ⊢ ({𝑎} USGrph 𝐸 ↔ 𝐸 = ∅) | |
9 | 7, 8 | sylib 207 | . . . . . . . . . . 11 ⊢ ({𝑎} FriendGrph 𝐸 → 𝐸 = ∅) |
10 | 6, 9 | syl6bi 242 | . . . . . . . . . 10 ⊢ (𝑉 = {𝑎} → (𝑉 FriendGrph 𝐸 → 𝐸 = ∅)) |
11 | 10 | a1d 25 | . . . . . . . . 9 ⊢ (𝑉 = {𝑎} → (𝑁 ∈ 𝑉 → (𝑉 FriendGrph 𝐸 → 𝐸 = ∅))) |
12 | 11 | exlimiv 1845 | . . . . . . . 8 ⊢ (∃𝑎 𝑉 = {𝑎} → (𝑁 ∈ 𝑉 → (𝑉 FriendGrph 𝐸 → 𝐸 = ∅))) |
13 | 5, 12 | syl6bi 242 | . . . . . . 7 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → ((#‘𝑉) = 1 → (𝑁 ∈ 𝑉 → (𝑉 FriendGrph 𝐸 → 𝐸 = ∅)))) |
14 | 13 | com24 93 | . . . . . 6 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑉 FriendGrph 𝐸 → (𝑁 ∈ 𝑉 → ((#‘𝑉) = 1 → 𝐸 = ∅)))) |
15 | 3, 14 | mpcom 37 | . . . . 5 ⊢ (𝑉 FriendGrph 𝐸 → (𝑁 ∈ 𝑉 → ((#‘𝑉) = 1 → 𝐸 = ∅))) |
16 | 15 | 3imp 1249 | . . . 4 ⊢ ((𝑉 FriendGrph 𝐸 ∧ 𝑁 ∈ 𝑉 ∧ (#‘𝑉) = 1) → 𝐸 = ∅) |
17 | 16 | oveq2d 6565 | . . 3 ⊢ ((𝑉 FriendGrph 𝐸 ∧ 𝑁 ∈ 𝑉 ∧ (#‘𝑉) = 1) → (𝑉 VDeg 𝐸) = (𝑉 VDeg ∅)) |
18 | 17 | fveq1d 6105 | . 2 ⊢ ((𝑉 FriendGrph 𝐸 ∧ 𝑁 ∈ 𝑉 ∧ (#‘𝑉) = 1) → ((𝑉 VDeg 𝐸)‘𝑁) = ((𝑉 VDeg ∅)‘𝑁)) |
19 | 3 | simpld 474 | . . . . 5 ⊢ (𝑉 FriendGrph 𝐸 → 𝑉 ∈ V) |
20 | 19 | anim1i 590 | . . . 4 ⊢ ((𝑉 FriendGrph 𝐸 ∧ 𝑁 ∈ 𝑉) → (𝑉 ∈ V ∧ 𝑁 ∈ 𝑉)) |
21 | 20 | 3adant3 1074 | . . 3 ⊢ ((𝑉 FriendGrph 𝐸 ∧ 𝑁 ∈ 𝑉 ∧ (#‘𝑉) = 1) → (𝑉 ∈ V ∧ 𝑁 ∈ 𝑉)) |
22 | vdgr0 26427 | . . 3 ⊢ ((𝑉 ∈ V ∧ 𝑁 ∈ 𝑉) → ((𝑉 VDeg ∅)‘𝑁) = 0) | |
23 | 21, 22 | syl 17 | . 2 ⊢ ((𝑉 FriendGrph 𝐸 ∧ 𝑁 ∈ 𝑉 ∧ (#‘𝑉) = 1) → ((𝑉 VDeg ∅)‘𝑁) = 0) |
24 | 18, 23 | eqtrd 2644 | 1 ⊢ ((𝑉 FriendGrph 𝐸 ∧ 𝑁 ∈ 𝑉 ∧ (#‘𝑉) = 1) → ((𝑉 VDeg 𝐸)‘𝑁) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∃wex 1695 ∈ wcel 1977 Vcvv 3173 ∅c0 3874 {csn 4125 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 0cc0 9815 1c1 9816 #chash 12979 USGrph cusg 25859 VDeg cvdg 26420 FriendGrph cfrgra 26515 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-n0 11170 df-z 11255 df-uz 11564 df-xadd 11823 df-fz 12198 df-hash 12980 df-usgra 25862 df-vdgr 26421 df-frgra 26516 |
This theorem is referenced by: (None) |
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