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Mirrors > Home > MPE Home > Th. List > Mathboxes > frgrwopreglem2 | Structured version Visualization version GIF version |
Description: Lemma 2 for frgrwopreg 41486. In a friendship graph with at least two vertices, the degree of a vertex must be at least 2. (Contributed by Alexander van der Vekens, 30-Dec-2017.) (Revised by AV, 10-May-2021.) |
Ref | Expression |
---|---|
frgrwopreg.v | ⊢ 𝑉 = (Vtx‘𝐺) |
frgrwopreg.d | ⊢ 𝐷 = (VtxDeg‘𝐺) |
frgrwopreg.a | ⊢ 𝐴 = {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾} |
frgrwopreg.b | ⊢ 𝐵 = (𝑉 ∖ 𝐴) |
Ref | Expression |
---|---|
frgrwopreglem2 | ⊢ ((𝐺 ∈ FriendGraph ∧ 1 < (#‘𝑉) ∧ 𝐴 ≠ ∅) → 1 < 𝐾) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | n0 3890 | . . 3 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴) | |
2 | frgrwopreg.a | . . . . . 6 ⊢ 𝐴 = {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾} | |
3 | 2 | rabeq2i 3170 | . . . . 5 ⊢ (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝑉 ∧ (𝐷‘𝑥) = 𝐾)) |
4 | 3 | exbii 1764 | . . . 4 ⊢ (∃𝑥 𝑥 ∈ 𝐴 ↔ ∃𝑥(𝑥 ∈ 𝑉 ∧ (𝐷‘𝑥) = 𝐾)) |
5 | frgrwopreg.v | . . . . . . . . . 10 ⊢ 𝑉 = (Vtx‘𝐺) | |
6 | 5 | vtxdgelxnn0 40687 | . . . . . . . . 9 ⊢ (𝑥 ∈ 𝑉 → ((VtxDeg‘𝐺)‘𝑥) ∈ ℕ0*) |
7 | 6 | ad2antrr 758 | . . . . . . . 8 ⊢ (((𝑥 ∈ 𝑉 ∧ (𝐷‘𝑥) = 𝐾) ∧ (1 < (#‘𝑉) ∧ 𝐺 ∈ FriendGraph )) → ((VtxDeg‘𝐺)‘𝑥) ∈ ℕ0*) |
8 | an3 864 | . . . . . . . . . 10 ⊢ (((𝑥 ∈ 𝑉 ∧ (𝐷‘𝑥) = 𝐾) ∧ (1 < (#‘𝑉) ∧ 𝐺 ∈ FriendGraph )) → (𝑥 ∈ 𝑉 ∧ 𝐺 ∈ FriendGraph )) | |
9 | 8 | ancomd 466 | . . . . . . . . 9 ⊢ (((𝑥 ∈ 𝑉 ∧ (𝐷‘𝑥) = 𝐾) ∧ (1 < (#‘𝑉) ∧ 𝐺 ∈ FriendGraph )) → (𝐺 ∈ FriendGraph ∧ 𝑥 ∈ 𝑉)) |
10 | simprl 790 | . . . . . . . . 9 ⊢ (((𝑥 ∈ 𝑉 ∧ (𝐷‘𝑥) = 𝐾) ∧ (1 < (#‘𝑉) ∧ 𝐺 ∈ FriendGraph )) → 1 < (#‘𝑉)) | |
11 | 5 | vdgfrgrgt2 41468 | . . . . . . . . 9 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑥 ∈ 𝑉) → (1 < (#‘𝑉) → 2 ≤ ((VtxDeg‘𝐺)‘𝑥))) |
12 | 9, 10, 11 | sylc 63 | . . . . . . . 8 ⊢ (((𝑥 ∈ 𝑉 ∧ (𝐷‘𝑥) = 𝐾) ∧ (1 < (#‘𝑉) ∧ 𝐺 ∈ FriendGraph )) → 2 ≤ ((VtxDeg‘𝐺)‘𝑥)) |
13 | xnn0xr 11245 | . . . . . . . . 9 ⊢ (((VtxDeg‘𝐺)‘𝑥) ∈ ℕ0* → ((VtxDeg‘𝐺)‘𝑥) ∈ ℝ*) | |
14 | 1lt2 11071 | . . . . . . . . . 10 ⊢ 1 < 2 | |
15 | 1re 9918 | . . . . . . . . . . . 12 ⊢ 1 ∈ ℝ | |
16 | 15 | rexri 9976 | . . . . . . . . . . 11 ⊢ 1 ∈ ℝ* |
17 | 2re 10967 | . . . . . . . . . . . 12 ⊢ 2 ∈ ℝ | |
18 | 17 | rexri 9976 | . . . . . . . . . . 11 ⊢ 2 ∈ ℝ* |
19 | xrltletr 11864 | . . . . . . . . . . 11 ⊢ ((1 ∈ ℝ* ∧ 2 ∈ ℝ* ∧ ((VtxDeg‘𝐺)‘𝑥) ∈ ℝ*) → ((1 < 2 ∧ 2 ≤ ((VtxDeg‘𝐺)‘𝑥)) → 1 < ((VtxDeg‘𝐺)‘𝑥))) | |
20 | 16, 18, 19 | mp3an12 1406 | . . . . . . . . . 10 ⊢ (((VtxDeg‘𝐺)‘𝑥) ∈ ℝ* → ((1 < 2 ∧ 2 ≤ ((VtxDeg‘𝐺)‘𝑥)) → 1 < ((VtxDeg‘𝐺)‘𝑥))) |
21 | 14, 20 | mpani 708 | . . . . . . . . 9 ⊢ (((VtxDeg‘𝐺)‘𝑥) ∈ ℝ* → (2 ≤ ((VtxDeg‘𝐺)‘𝑥) → 1 < ((VtxDeg‘𝐺)‘𝑥))) |
22 | 13, 21 | syl 17 | . . . . . . . 8 ⊢ (((VtxDeg‘𝐺)‘𝑥) ∈ ℕ0* → (2 ≤ ((VtxDeg‘𝐺)‘𝑥) → 1 < ((VtxDeg‘𝐺)‘𝑥))) |
23 | 7, 12, 22 | sylc 63 | . . . . . . 7 ⊢ (((𝑥 ∈ 𝑉 ∧ (𝐷‘𝑥) = 𝐾) ∧ (1 < (#‘𝑉) ∧ 𝐺 ∈ FriendGraph )) → 1 < ((VtxDeg‘𝐺)‘𝑥)) |
24 | frgrwopreg.d | . . . . . . . . . 10 ⊢ 𝐷 = (VtxDeg‘𝐺) | |
25 | 24 | fveq1i 6104 | . . . . . . . . 9 ⊢ (𝐷‘𝑥) = ((VtxDeg‘𝐺)‘𝑥) |
26 | simpr 476 | . . . . . . . . 9 ⊢ ((𝑥 ∈ 𝑉 ∧ (𝐷‘𝑥) = 𝐾) → (𝐷‘𝑥) = 𝐾) | |
27 | 25, 26 | syl5eqr 2658 | . . . . . . . 8 ⊢ ((𝑥 ∈ 𝑉 ∧ (𝐷‘𝑥) = 𝐾) → ((VtxDeg‘𝐺)‘𝑥) = 𝐾) |
28 | 27 | adantr 480 | . . . . . . 7 ⊢ (((𝑥 ∈ 𝑉 ∧ (𝐷‘𝑥) = 𝐾) ∧ (1 < (#‘𝑉) ∧ 𝐺 ∈ FriendGraph )) → ((VtxDeg‘𝐺)‘𝑥) = 𝐾) |
29 | 23, 28 | breqtrd 4609 | . . . . . 6 ⊢ (((𝑥 ∈ 𝑉 ∧ (𝐷‘𝑥) = 𝐾) ∧ (1 < (#‘𝑉) ∧ 𝐺 ∈ FriendGraph )) → 1 < 𝐾) |
30 | 29 | exp32 629 | . . . . 5 ⊢ ((𝑥 ∈ 𝑉 ∧ (𝐷‘𝑥) = 𝐾) → (1 < (#‘𝑉) → (𝐺 ∈ FriendGraph → 1 < 𝐾))) |
31 | 30 | exlimiv 1845 | . . . 4 ⊢ (∃𝑥(𝑥 ∈ 𝑉 ∧ (𝐷‘𝑥) = 𝐾) → (1 < (#‘𝑉) → (𝐺 ∈ FriendGraph → 1 < 𝐾))) |
32 | 4, 31 | sylbi 206 | . . 3 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (1 < (#‘𝑉) → (𝐺 ∈ FriendGraph → 1 < 𝐾))) |
33 | 1, 32 | sylbi 206 | . 2 ⊢ (𝐴 ≠ ∅ → (1 < (#‘𝑉) → (𝐺 ∈ FriendGraph → 1 < 𝐾))) |
34 | 33 | 3imp31 1250 | 1 ⊢ ((𝐺 ∈ FriendGraph ∧ 1 < (#‘𝑉) ∧ 𝐴 ≠ ∅) → 1 < 𝐾) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∃wex 1695 ∈ wcel 1977 ≠ wne 2780 {crab 2900 ∖ cdif 3537 ∅c0 3874 class class class wbr 4583 ‘cfv 5804 1c1 9816 ℝ*cxr 9952 < clt 9953 ≤ cle 9954 2c2 10947 ℕ0*cxnn0 11240 #chash 12979 Vtxcvtx 25673 VtxDegcvtxdg 40681 FriendGraph cfrgr 41428 |
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-ifp 1007 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-map 7746 df-pm 7747 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-3 10957 df-n0 11170 df-xnn0 11241 df-z 11255 df-uz 11564 df-xadd 11823 df-fz 12198 df-fzo 12335 df-hash 12980 df-word 13154 df-concat 13156 df-s1 13157 df-s2 13444 df-s3 13445 df-uhgr 25724 df-upgr 25749 df-umgr 25750 df-edga 25793 df-uspgr 40380 df-usgr 40381 df-vtxdg 40682 df-1wlks 40800 df-wlkson 40802 df-trls 40901 df-trlson 40902 df-pths 40923 df-spths 40924 df-pthson 40925 df-spthson 40926 df-conngr 41354 df-frgr 41429 |
This theorem is referenced by: (None) |
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