![]() |
Mathbox for Alexander van der Vekens |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > 1hegrlfgr | Structured version Visualization version GIF version |
Description: A graph 𝐺 with one hyperedge joining at least two vertices is a loop-free graph. (Contributed by AV, 23-Feb-2021.) |
Ref | Expression |
---|---|
1hegrlfgr.a | ⊢ (𝜑 → 𝐴 ∈ 𝑋) |
1hegrlfgr.b | ⊢ (𝜑 → 𝐵 ∈ 𝑉) |
1hegrlfgr.c | ⊢ (𝜑 → 𝐶 ∈ 𝑉) |
1hegrlfgr.n | ⊢ (𝜑 → 𝐵 ≠ 𝐶) |
1hegrlfgr.x | ⊢ (𝜑 → 𝐸 ∈ 𝒫 𝑉) |
1hegrlfgr.i | ⊢ (𝜑 → (iEdg‘𝐺) = {〈𝐴, 𝐸〉}) |
1hegrlfgr.e | ⊢ (𝜑 → {𝐵, 𝐶} ⊆ 𝐸) |
Ref | Expression |
---|---|
1hegrlfgr | ⊢ (𝜑 → (iEdg‘𝐺):{𝐴}⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1hegrlfgr.a | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑋) | |
2 | 1hegrlfgr.x | . . . . 5 ⊢ (𝜑 → 𝐸 ∈ 𝒫 𝑉) | |
3 | f1osng 6486 | . . . . 5 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐸 ∈ 𝒫 𝑉) → {〈𝐴, 𝐸〉}:{𝐴}–1-1-onto→{𝐸}) | |
4 | 1, 2, 3 | syl2anc 576 | . . . 4 ⊢ (𝜑 → {〈𝐴, 𝐸〉}:{𝐴}–1-1-onto→{𝐸}) |
5 | f1of 6446 | . . . 4 ⊢ ({〈𝐴, 𝐸〉}:{𝐴}–1-1-onto→{𝐸} → {〈𝐴, 𝐸〉}:{𝐴}⟶{𝐸}) | |
6 | 4, 5 | syl 17 | . . 3 ⊢ (𝜑 → {〈𝐴, 𝐸〉}:{𝐴}⟶{𝐸}) |
7 | 1hegrlfgr.e | . . . . . . 7 ⊢ (𝜑 → {𝐵, 𝐶} ⊆ 𝐸) | |
8 | 1hegrlfgr.b | . . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ 𝑉) | |
9 | prid1g 4571 | . . . . . . . 8 ⊢ (𝐵 ∈ 𝑉 → 𝐵 ∈ {𝐵, 𝐶}) | |
10 | 8, 9 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ {𝐵, 𝐶}) |
11 | 7, 10 | sseldd 3861 | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝐸) |
12 | 1hegrlfgr.c | . . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ 𝑉) | |
13 | prid2g 4572 | . . . . . . . 8 ⊢ (𝐶 ∈ 𝑉 → 𝐶 ∈ {𝐵, 𝐶}) | |
14 | 12, 13 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ {𝐵, 𝐶}) |
15 | 7, 14 | sseldd 3861 | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ 𝐸) |
16 | 1hegrlfgr.n | . . . . . 6 ⊢ (𝜑 → 𝐵 ≠ 𝐶) | |
17 | 2, 11, 15, 16 | nehash2 13646 | . . . . 5 ⊢ (𝜑 → 2 ≤ (♯‘𝐸)) |
18 | fveq2 6501 | . . . . . . 7 ⊢ (𝑥 = 𝐸 → (♯‘𝑥) = (♯‘𝐸)) | |
19 | 18 | breq2d 4942 | . . . . . 6 ⊢ (𝑥 = 𝐸 → (2 ≤ (♯‘𝑥) ↔ 2 ≤ (♯‘𝐸))) |
20 | 19 | elrab 3595 | . . . . 5 ⊢ (𝐸 ∈ {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)} ↔ (𝐸 ∈ 𝒫 𝑉 ∧ 2 ≤ (♯‘𝐸))) |
21 | 2, 17, 20 | sylanbrc 575 | . . . 4 ⊢ (𝜑 → 𝐸 ∈ {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}) |
22 | 21 | snssd 4617 | . . 3 ⊢ (𝜑 → {𝐸} ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}) |
23 | 6, 22 | fssd 6360 | . 2 ⊢ (𝜑 → {〈𝐴, 𝐸〉}:{𝐴}⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}) |
24 | 1hegrlfgr.i | . . 3 ⊢ (𝜑 → (iEdg‘𝐺) = {〈𝐴, 𝐸〉}) | |
25 | 24 | feq1d 6331 | . 2 ⊢ (𝜑 → ((iEdg‘𝐺):{𝐴}⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)} ↔ {〈𝐴, 𝐸〉}:{𝐴}⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)})) |
26 | 23, 25 | mpbird 249 | 1 ⊢ (𝜑 → (iEdg‘𝐺):{𝐴}⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1507 ∈ wcel 2050 ≠ wne 2967 {crab 3092 ⊆ wss 3831 𝒫 cpw 4423 {csn 4442 {cpr 4444 〈cop 4448 class class class wbr 4930 ⟶wf 6186 –1-1-onto→wf1o 6189 ‘cfv 6190 ≤ cle 10477 2c2 11498 ♯chash 13508 iEdgciedg 26488 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-rep 5050 ax-sep 5061 ax-nul 5068 ax-pow 5120 ax-pr 5187 ax-un 7281 ax-cnex 10393 ax-resscn 10394 ax-1cn 10395 ax-icn 10396 ax-addcl 10397 ax-addrcl 10398 ax-mulcl 10399 ax-mulrcl 10400 ax-mulcom 10401 ax-addass 10402 ax-mulass 10403 ax-distr 10404 ax-i2m1 10405 ax-1ne0 10406 ax-1rid 10407 ax-rnegex 10408 ax-rrecex 10409 ax-cnre 10410 ax-pre-lttri 10411 ax-pre-lttrn 10412 ax-pre-ltadd 10413 ax-pre-mulgt0 10414 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2583 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-nel 3074 df-ral 3093 df-rex 3094 df-reu 3095 df-rmo 3096 df-rab 3097 df-v 3417 df-sbc 3684 df-csb 3789 df-dif 3834 df-un 3836 df-in 3838 df-ss 3845 df-pss 3847 df-nul 4181 df-if 4352 df-pw 4425 df-sn 4443 df-pr 4445 df-tp 4447 df-op 4449 df-uni 4714 df-int 4751 df-iun 4795 df-br 4931 df-opab 4993 df-mpt 5010 df-tr 5032 df-id 5313 df-eprel 5318 df-po 5327 df-so 5328 df-fr 5367 df-we 5369 df-xp 5414 df-rel 5415 df-cnv 5416 df-co 5417 df-dm 5418 df-rn 5419 df-res 5420 df-ima 5421 df-pred 5988 df-ord 6034 df-on 6035 df-lim 6036 df-suc 6037 df-iota 6154 df-fun 6192 df-fn 6193 df-f 6194 df-f1 6195 df-fo 6196 df-f1o 6197 df-fv 6198 df-riota 6939 df-ov 6981 df-oprab 6982 df-mpo 6983 df-om 7399 df-1st 7503 df-2nd 7504 df-wrecs 7752 df-recs 7814 df-rdg 7852 df-1o 7907 df-oadd 7911 df-er 8091 df-en 8309 df-dom 8310 df-sdom 8311 df-fin 8312 df-dju 9126 df-card 9164 df-pnf 10478 df-mnf 10479 df-xr 10480 df-ltxr 10481 df-le 10482 df-sub 10674 df-neg 10675 df-nn 11442 df-2 11506 df-n0 11711 df-xnn0 11783 df-z 11797 df-uz 12062 df-fz 12712 df-hash 13509 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |