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Mirrors > Home > MPE Home > Th. List > frgrawopreg1 | Structured version Visualization version GIF version |
Description: According to statement 5 in [Huneke] p. 2: "If A ... is a singleton, then that singleton is a universal friend". (Contributed by Alexander van der Vekens, 1-Jan-2018.) |
Ref | Expression |
---|---|
frgrawopreg.a | ⊢ 𝐴 = {𝑥 ∈ 𝑉 ∣ ((𝑉 VDeg 𝐸)‘𝑥) = 𝐾} |
frgrawopreg.b | ⊢ 𝐵 = (𝑉 ∖ 𝐴) |
Ref | Expression |
---|---|
frgrawopreg1 | ⊢ ((𝑉 FriendGrph 𝐸 ∧ (#‘𝐴) = 1) → ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ ran 𝐸) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frgrawopreg.a | . . . . . 6 ⊢ 𝐴 = {𝑥 ∈ 𝑉 ∣ ((𝑉 VDeg 𝐸)‘𝑥) = 𝐾} | |
2 | frgrawopreg.b | . . . . . 6 ⊢ 𝐵 = (𝑉 ∖ 𝐴) | |
3 | 1, 2 | frgrawopreglem1 26571 | . . . . 5 ⊢ (𝑉 FriendGrph 𝐸 → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
4 | 3 | simpld 474 | . . . 4 ⊢ (𝑉 FriendGrph 𝐸 → 𝐴 ∈ V) |
5 | hash1snb 13068 | . . . 4 ⊢ (𝐴 ∈ V → ((#‘𝐴) = 1 ↔ ∃𝑣 𝐴 = {𝑣})) | |
6 | 4, 5 | syl 17 | . . 3 ⊢ (𝑉 FriendGrph 𝐸 → ((#‘𝐴) = 1 ↔ ∃𝑣 𝐴 = {𝑣})) |
7 | exsnrex 4168 | . . . . 5 ⊢ (∃𝑣 𝐴 = {𝑣} ↔ ∃𝑣 ∈ 𝐴 𝐴 = {𝑣}) | |
8 | ssrab2 3650 | . . . . . . . 8 ⊢ {𝑥 ∈ 𝑉 ∣ ((𝑉 VDeg 𝐸)‘𝑥) = 𝐾} ⊆ 𝑉 | |
9 | 1, 8 | eqsstri 3598 | . . . . . . 7 ⊢ 𝐴 ⊆ 𝑉 |
10 | ssrexv 3630 | . . . . . . 7 ⊢ (𝐴 ⊆ 𝑉 → (∃𝑣 ∈ 𝐴 𝐴 = {𝑣} → ∃𝑣 ∈ 𝑉 𝐴 = {𝑣})) | |
11 | 9, 10 | ax-mp 5 | . . . . . 6 ⊢ (∃𝑣 ∈ 𝐴 𝐴 = {𝑣} → ∃𝑣 ∈ 𝑉 𝐴 = {𝑣}) |
12 | 1, 2 | frgrawopreglem4 26574 | . . . . . . . 8 ⊢ (𝑉 FriendGrph 𝐸 → ∀𝑢 ∈ 𝐴 ∀𝑤 ∈ 𝐵 {𝑢, 𝑤} ∈ ran 𝐸) |
13 | vsnid 4156 | . . . . . . . . . . 11 ⊢ 𝑣 ∈ {𝑣} | |
14 | eleq2 2677 | . . . . . . . . . . 11 ⊢ (𝐴 = {𝑣} → (𝑣 ∈ 𝐴 ↔ 𝑣 ∈ {𝑣})) | |
15 | 13, 14 | mpbiri 247 | . . . . . . . . . 10 ⊢ (𝐴 = {𝑣} → 𝑣 ∈ 𝐴) |
16 | preq1 4212 | . . . . . . . . . . . . 13 ⊢ (𝑢 = 𝑣 → {𝑢, 𝑤} = {𝑣, 𝑤}) | |
17 | 16 | eleq1d 2672 | . . . . . . . . . . . 12 ⊢ (𝑢 = 𝑣 → ({𝑢, 𝑤} ∈ ran 𝐸 ↔ {𝑣, 𝑤} ∈ ran 𝐸)) |
18 | 17 | ralbidv 2969 | . . . . . . . . . . 11 ⊢ (𝑢 = 𝑣 → (∀𝑤 ∈ 𝐵 {𝑢, 𝑤} ∈ ran 𝐸 ↔ ∀𝑤 ∈ 𝐵 {𝑣, 𝑤} ∈ ran 𝐸)) |
19 | 18 | rspcv 3278 | . . . . . . . . . 10 ⊢ (𝑣 ∈ 𝐴 → (∀𝑢 ∈ 𝐴 ∀𝑤 ∈ 𝐵 {𝑢, 𝑤} ∈ ran 𝐸 → ∀𝑤 ∈ 𝐵 {𝑣, 𝑤} ∈ ran 𝐸)) |
20 | 15, 19 | syl 17 | . . . . . . . . 9 ⊢ (𝐴 = {𝑣} → (∀𝑢 ∈ 𝐴 ∀𝑤 ∈ 𝐵 {𝑢, 𝑤} ∈ ran 𝐸 → ∀𝑤 ∈ 𝐵 {𝑣, 𝑤} ∈ ran 𝐸)) |
21 | difeq2 3684 | . . . . . . . . . . 11 ⊢ (𝐴 = {𝑣} → (𝑉 ∖ 𝐴) = (𝑉 ∖ {𝑣})) | |
22 | 2, 21 | syl5eq 2656 | . . . . . . . . . 10 ⊢ (𝐴 = {𝑣} → 𝐵 = (𝑉 ∖ {𝑣})) |
23 | 22 | raleqdv 3121 | . . . . . . . . 9 ⊢ (𝐴 = {𝑣} → (∀𝑤 ∈ 𝐵 {𝑣, 𝑤} ∈ ran 𝐸 ↔ ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ ran 𝐸)) |
24 | 20, 23 | sylibd 228 | . . . . . . . 8 ⊢ (𝐴 = {𝑣} → (∀𝑢 ∈ 𝐴 ∀𝑤 ∈ 𝐵 {𝑢, 𝑤} ∈ ran 𝐸 → ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ ran 𝐸)) |
25 | 12, 24 | syl5com 31 | . . . . . . 7 ⊢ (𝑉 FriendGrph 𝐸 → (𝐴 = {𝑣} → ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ ran 𝐸)) |
26 | 25 | reximdv 2999 | . . . . . 6 ⊢ (𝑉 FriendGrph 𝐸 → (∃𝑣 ∈ 𝑉 𝐴 = {𝑣} → ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ ran 𝐸)) |
27 | 11, 26 | syl5com 31 | . . . . 5 ⊢ (∃𝑣 ∈ 𝐴 𝐴 = {𝑣} → (𝑉 FriendGrph 𝐸 → ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ ran 𝐸)) |
28 | 7, 27 | sylbi 206 | . . . 4 ⊢ (∃𝑣 𝐴 = {𝑣} → (𝑉 FriendGrph 𝐸 → ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ ran 𝐸)) |
29 | 28 | com12 32 | . . 3 ⊢ (𝑉 FriendGrph 𝐸 → (∃𝑣 𝐴 = {𝑣} → ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ ran 𝐸)) |
30 | 6, 29 | sylbid 229 | . 2 ⊢ (𝑉 FriendGrph 𝐸 → ((#‘𝐴) = 1 → ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ ran 𝐸)) |
31 | 30 | imp 444 | 1 ⊢ ((𝑉 FriendGrph 𝐸 ∧ (#‘𝐴) = 1) → ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ ran 𝐸) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∃wex 1695 ∈ wcel 1977 ∀wral 2896 ∃wrex 2897 {crab 2900 Vcvv 3173 ∖ cdif 3537 ⊆ wss 3540 {csn 4125 {cpr 4127 class class class wbr 4583 ran crn 5039 ‘cfv 5804 (class class class)co 6549 1c1 9816 #chash 12979 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-2o 7448 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-xnn0 11241 df-z 11255 df-uz 11564 df-xadd 11823 df-fz 12198 df-hash 12980 df-usgra 25862 df-nbgra 25949 df-vdgr 26421 df-frgra 26516 |
This theorem is referenced by: frgraregorufr0 26579 |
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