Theorem cusgredgex 35297

Description: Any two (distinct) vertices in a complete simple graph are connected to each other by an edge. (Contributed by BTernaryTau, 3-Oct-2023.)

Hypotheses

Ref	Expression
cusgredgex.1	⊢ 𝑉 = (Vtx‘𝐺)
cusgredgex.2	⊢ 𝐸 = (Edg‘𝐺)

Assertion

Ref	Expression
cusgredgex	⊢ (𝐺 ∈ ComplUSGraph → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ (𝑉 ∖ {𝐴})) → {𝐴, 𝐵} ∈ 𝐸))

Proof of Theorem cusgredgex

Dummy variable 𝑒 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cusgrcplgr 29476	. . . . . . . 8 ⊢ (𝐺 ∈ ComplUSGraph → 𝐺 ∈ ComplGraph)
2		cusgredgex.1	. . . . . . . . 9 ⊢ 𝑉 = (Vtx‘𝐺)
3		cusgredgex.2	. . . . . . . . 9 ⊢ 𝐸 = (Edg‘𝐺)
4	2, 3	cplgredgex 35296	. . . . . . . 8 ⊢ (𝐺 ∈ ComplGraph → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ (𝑉 ∖ {𝐴})) → ∃𝑒 ∈ 𝐸 {𝐴, 𝐵} ⊆ 𝑒))
5	1, 4	syl 17	. . . . . . 7 ⊢ (𝐺 ∈ ComplUSGraph → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ (𝑉 ∖ {𝐴})) → ∃𝑒 ∈ 𝐸 {𝐴, 𝐵} ⊆ 𝑒))
6	5	imp 406	. . . . . 6 ⊢ ((𝐺 ∈ ComplUSGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ (𝑉 ∖ {𝐴}))) → ∃𝑒 ∈ 𝐸 {𝐴, 𝐵} ⊆ 𝑒)
7		df-rex 3062	. . . . . 6 ⊢ (∃𝑒 ∈ 𝐸 {𝐴, 𝐵} ⊆ 𝑒 ↔ ∃𝑒(𝑒 ∈ 𝐸 ∧ {𝐴, 𝐵} ⊆ 𝑒))
8	6, 7	sylib 218	. . . . 5 ⊢ ((𝐺 ∈ ComplUSGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ (𝑉 ∖ {𝐴}))) → ∃𝑒(𝑒 ∈ 𝐸 ∧ {𝐴, 𝐵} ⊆ 𝑒))
9		eldifsni 4747	. . . . . . . . . . . . . . . 16 ⊢ (𝐵 ∈ (𝑉 ∖ {𝐴}) → 𝐵 ≠ 𝐴)
10	9	necomd 2988	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ∈ (𝑉 ∖ {𝐴}) → 𝐴 ≠ 𝐵)
11	10	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ (𝑉 ∖ {𝐴})) → 𝐴 ≠ 𝐵)
12		hashprg 14322	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ (𝑉 ∖ {𝐴})) → (𝐴 ≠ 𝐵 ↔ (♯‘{𝐴, 𝐵}) = 2))
13	11, 12	mpbid 232	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ (𝑉 ∖ {𝐴})) → (♯‘{𝐴, 𝐵}) = 2)
14	13	adantl 481	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ ComplUSGraph ∧ 𝑒 ∈ 𝐸) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ (𝑉 ∖ {𝐴}))) → (♯‘{𝐴, 𝐵}) = 2)
15		cusgrusgr 29475	. . . . . . . . . . . . . 14 ⊢ (𝐺 ∈ ComplUSGraph → 𝐺 ∈ USGraph)
16	3	usgredgppr 29252	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ USGraph ∧ 𝑒 ∈ 𝐸) → (♯‘𝑒) = 2)
17	15, 16	sylan 581	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑒 ∈ 𝐸) → (♯‘𝑒) = 2)
18	17	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ ComplUSGraph ∧ 𝑒 ∈ 𝐸) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ (𝑉 ∖ {𝐴}))) → (♯‘𝑒) = 2)
19	14, 18	eqtr4d 2775	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ ComplUSGraph ∧ 𝑒 ∈ 𝐸) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ (𝑉 ∖ {𝐴}))) → (♯‘{𝐴, 𝐵}) = (♯‘𝑒))
20		simpl 482	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ ComplUSGraph ∧ 𝑒 ∈ 𝐸) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ (𝑉 ∖ {𝐴}))) → (𝐺 ∈ ComplUSGraph ∧ 𝑒 ∈ 𝐸))
21		vex 3445	. . . . . . . . . . . . . . . 16 ⊢ 𝑒 ∈ V
22		2nn0 12422	. . . . . . . . . . . . . . . 16 ⊢ 2 ∈ ℕ0
23		hashvnfin 14287	. . . . . . . . . . . . . . . 16 ⊢ ((𝑒 ∈ V ∧ 2 ∈ ℕ0) → ((♯‘𝑒) = 2 → 𝑒 ∈ Fin))
24	21, 22, 23	mp2an 693	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝑒) = 2 → 𝑒 ∈ Fin)
25	17, 24	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑒 ∈ 𝐸) → 𝑒 ∈ Fin)
26		fisshasheq 35290	. . . . . . . . . . . . . 14 ⊢ ((𝑒 ∈ Fin ∧ {𝐴, 𝐵} ⊆ 𝑒 ∧ (♯‘{𝐴, 𝐵}) = (♯‘𝑒)) → {𝐴, 𝐵} = 𝑒)
27	25, 26	syl3an1 1164	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ ComplUSGraph ∧ 𝑒 ∈ 𝐸) ∧ {𝐴, 𝐵} ⊆ 𝑒 ∧ (♯‘{𝐴, 𝐵}) = (♯‘𝑒)) → {𝐴, 𝐵} = 𝑒)
28	27	3comr 1126	. . . . . . . . . . . 12 ⊢ (((♯‘{𝐴, 𝐵}) = (♯‘𝑒) ∧ (𝐺 ∈ ComplUSGraph ∧ 𝑒 ∈ 𝐸) ∧ {𝐴, 𝐵} ⊆ 𝑒) → {𝐴, 𝐵} = 𝑒)
29	28	3exp 1120	. . . . . . . . . . 11 ⊢ ((♯‘{𝐴, 𝐵}) = (♯‘𝑒) → ((𝐺 ∈ ComplUSGraph ∧ 𝑒 ∈ 𝐸) → ({𝐴, 𝐵} ⊆ 𝑒 → {𝐴, 𝐵} = 𝑒)))
30	19, 20, 29	sylc 65	. . . . . . . . . 10 ⊢ (((𝐺 ∈ ComplUSGraph ∧ 𝑒 ∈ 𝐸) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ (𝑉 ∖ {𝐴}))) → ({𝐴, 𝐵} ⊆ 𝑒 → {𝐴, 𝐵} = 𝑒))
31	30	3impa 1110	. . . . . . . . 9 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑒 ∈ 𝐸 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ (𝑉 ∖ {𝐴}))) → ({𝐴, 𝐵} ⊆ 𝑒 → {𝐴, 𝐵} = 𝑒))
32	31	3com23 1127	. . . . . . . 8 ⊢ ((𝐺 ∈ ComplUSGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ (𝑉 ∖ {𝐴})) ∧ 𝑒 ∈ 𝐸) → ({𝐴, 𝐵} ⊆ 𝑒 → {𝐴, 𝐵} = 𝑒))
33	32	3expia 1122	. . . . . . 7 ⊢ ((𝐺 ∈ ComplUSGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ (𝑉 ∖ {𝐴}))) → (𝑒 ∈ 𝐸 → ({𝐴, 𝐵} ⊆ 𝑒 → {𝐴, 𝐵} = 𝑒)))
34	33	imdistand 570	. . . . . 6 ⊢ ((𝐺 ∈ ComplUSGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ (𝑉 ∖ {𝐴}))) → ((𝑒 ∈ 𝐸 ∧ {𝐴, 𝐵} ⊆ 𝑒) → (𝑒 ∈ 𝐸 ∧ {𝐴, 𝐵} = 𝑒)))
35	34	eximdv 1919	. . . . 5 ⊢ ((𝐺 ∈ ComplUSGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ (𝑉 ∖ {𝐴}))) → (∃𝑒(𝑒 ∈ 𝐸 ∧ {𝐴, 𝐵} ⊆ 𝑒) → ∃𝑒(𝑒 ∈ 𝐸 ∧ {𝐴, 𝐵} = 𝑒)))
36	8, 35	mpd 15	. . . 4 ⊢ ((𝐺 ∈ ComplUSGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ (𝑉 ∖ {𝐴}))) → ∃𝑒(𝑒 ∈ 𝐸 ∧ {𝐴, 𝐵} = 𝑒))
37		pm3.22 459	. . . . . 6 ⊢ ((𝑒 ∈ 𝐸 ∧ {𝐴, 𝐵} = 𝑒) → ({𝐴, 𝐵} = 𝑒 ∧ 𝑒 ∈ 𝐸))
38		eqcom 2744	. . . . . . 7 ⊢ ({𝐴, 𝐵} = 𝑒 ↔ 𝑒 = {𝐴, 𝐵})
39	38	anbi1i 625	. . . . . 6 ⊢ (({𝐴, 𝐵} = 𝑒 ∧ 𝑒 ∈ 𝐸) ↔ (𝑒 = {𝐴, 𝐵} ∧ 𝑒 ∈ 𝐸))
40	37, 39	sylib 218	. . . . 5 ⊢ ((𝑒 ∈ 𝐸 ∧ {𝐴, 𝐵} = 𝑒) → (𝑒 = {𝐴, 𝐵} ∧ 𝑒 ∈ 𝐸))
41	40	eximi 1837	. . . 4 ⊢ (∃𝑒(𝑒 ∈ 𝐸 ∧ {𝐴, 𝐵} = 𝑒) → ∃𝑒(𝑒 = {𝐴, 𝐵} ∧ 𝑒 ∈ 𝐸))
42	36, 41	syl 17	. . 3 ⊢ ((𝐺 ∈ ComplUSGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ (𝑉 ∖ {𝐴}))) → ∃𝑒(𝑒 = {𝐴, 𝐵} ∧ 𝑒 ∈ 𝐸))
43		prex 5383	. . . 4 ⊢ {𝐴, 𝐵} ∈ V
44		eleq1 2825	. . . 4 ⊢ (𝑒 = {𝐴, 𝐵} → (𝑒 ∈ 𝐸 ↔ {𝐴, 𝐵} ∈ 𝐸))
45	43, 44	ceqsexv 3491	. . 3 ⊢ (∃𝑒(𝑒 = {𝐴, 𝐵} ∧ 𝑒 ∈ 𝐸) ↔ {𝐴, 𝐵} ∈ 𝐸)
46	42, 45	sylib 218	. 2 ⊢ ((𝐺 ∈ ComplUSGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ (𝑉 ∖ {𝐴}))) → {𝐴, 𝐵} ∈ 𝐸)
47	46	ex 412	1 ⊢ (𝐺 ∈ ComplUSGraph → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ (𝑉 ∖ {𝐴})) → {𝐴, 𝐵} ∈ 𝐸))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542  ∃wex 1781   ∈ wcel 2114   ≠ wne 2933  ∃wrex 3061  Vcvv 3441   ∖ cdif 3899   ⊆ wss 3902  {csn 4581  {cpr 4583  ‘cfv 6493  Fincfn 8887  2c2 12204  ℕ₀cn0 12405  ♯chash 14257  Vtxcvtx 29052  Edgcedg 29103  USGraphcusgr 29205  ComplGraphccplgr 29465  ComplUSGraphccusgr 29466

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682  ax-cnex 11086  ax-resscn 11087  ax-1cn 11088  ax-icn 11089  ax-addcl 11090  ax-addrcl 11091  ax-mulcl 11092  ax-mulrcl 11093  ax-mulcom 11094  ax-addass 11095  ax-mulass 11096  ax-distr 11097  ax-i2m1 11098  ax-1ne0 11099  ax-1rid 11100  ax-rnegex 11101  ax-rrecex 11102  ax-cnre 11103  ax-pre-lttri 11104  ax-pre-lttrn 11105  ax-pre-ltadd 11106  ax-pre-mulgt0 11107

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3062  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-int 4904  df-iun 4949  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-1o 8399  df-oadd 8403  df-er 8637  df-en 8888  df-dom 8889  df-sdom 8890  df-fin 8891  df-dju 9817  df-card 9855  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-nn 12150  df-2 12212  df-n0 12406  df-z 12493  df-uz 12756  df-fz 13428  df-hash 14258  df-edg 29104  df-usgr 29207  df-nbgr 29389  df-uvtx 29442  df-cplgr 29467  df-cusgr 29468

This theorem is referenced by:  cusgredgex2  35298