Theorem cusgredgex 34667

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 29220	. . . . . . . 8 ⊢ (𝐺 ∈ ComplUSGraph → 𝐺 ∈ ComplGraph)
2		cusgredgex.1	. . . . . . . . 9 ⊢ 𝑉 = (Vtx‘𝐺)
3		cusgredgex.2	. . . . . . . . 9 ⊢ 𝐸 = (Edg‘𝐺)
4	2, 3	cplgredgex 34666	. . . . . . . 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 3066	. . . . . 6 ⊢ (∃𝑒 ∈ 𝐸 {𝐴, 𝐵} ⊆ 𝑒 ↔ ∃𝑒(𝑒 ∈ 𝐸 ∧ {𝐴, 𝐵} ⊆ 𝑒))
8	6, 7	sylib 217	. . . . 5 ⊢ ((𝐺 ∈ ComplUSGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ (𝑉 ∖ {𝐴}))) → ∃𝑒(𝑒 ∈ 𝐸 ∧ {𝐴, 𝐵} ⊆ 𝑒))
9		eldifsni 4789	. . . . . . . . . . . . . . . 16 ⊢ (𝐵 ∈ (𝑉 ∖ {𝐴}) → 𝐵 ≠ 𝐴)
10	9	necomd 2991	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ∈ (𝑉 ∖ {𝐴}) → 𝐴 ≠ 𝐵)
11	10	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ (𝑉 ∖ {𝐴})) → 𝐴 ≠ 𝐵)
12		hashprg 14378	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ (𝑉 ∖ {𝐴})) → (𝐴 ≠ 𝐵 ↔ (♯‘{𝐴, 𝐵}) = 2))
13	11, 12	mpbid 231	. . . . . . . . . . . . 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 29219	. . . . . . . . . . . . . 14 ⊢ (𝐺 ∈ ComplUSGraph → 𝐺 ∈ USGraph)
16	3	usgredgppr 28996	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ USGraph ∧ 𝑒 ∈ 𝐸) → (♯‘𝑒) = 2)
17	15, 16	sylan 579	. . . . . . . . . . . . 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 2770	. . . . . . . . . . 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 3473	. . . . . . . . . . . . . . . 16 ⊢ 𝑒 ∈ V
22		2nn0 12511	. . . . . . . . . . . . . . . 16 ⊢ 2 ∈ ℕ0
23		hashvnfin 14343	. . . . . . . . . . . . . . . 16 ⊢ ((𝑒 ∈ V ∧ 2 ∈ ℕ0) → ((♯‘𝑒) = 2 → 𝑒 ∈ Fin))
24	21, 22, 23	mp2an 691	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝑒) = 2 → 𝑒 ∈ Fin)
25	17, 24	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑒 ∈ 𝐸) → 𝑒 ∈ Fin)
26		fisshasheq 34660	. . . . . . . . . . . . . 14 ⊢ ((𝑒 ∈ Fin ∧ {𝐴, 𝐵} ⊆ 𝑒 ∧ (♯‘{𝐴, 𝐵}) = (♯‘𝑒)) → {𝐴, 𝐵} = 𝑒)
27	25, 26	syl3an1 1161	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ ComplUSGraph ∧ 𝑒 ∈ 𝐸) ∧ {𝐴, 𝐵} ⊆ 𝑒 ∧ (♯‘{𝐴, 𝐵}) = (♯‘𝑒)) → {𝐴, 𝐵} = 𝑒)
28	27	3comr 1123	. . . . . . . . . . . 12 ⊢ (((♯‘{𝐴, 𝐵}) = (♯‘𝑒) ∧ (𝐺 ∈ ComplUSGraph ∧ 𝑒 ∈ 𝐸) ∧ {𝐴, 𝐵} ⊆ 𝑒) → {𝐴, 𝐵} = 𝑒)
29	28	3exp 1117	. . . . . . . . . . 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 1108	. . . . . . . . 9 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑒 ∈ 𝐸 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ (𝑉 ∖ {𝐴}))) → ({𝐴, 𝐵} ⊆ 𝑒 → {𝐴, 𝐵} = 𝑒))
32	31	3com23 1124	. . . . . . . 8 ⊢ ((𝐺 ∈ ComplUSGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ (𝑉 ∖ {𝐴})) ∧ 𝑒 ∈ 𝐸) → ({𝐴, 𝐵} ⊆ 𝑒 → {𝐴, 𝐵} = 𝑒))
33	32	3expia 1119	. . . . . . 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 1913	. . . . 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 2734	. . . . . . 7 ⊢ ({𝐴, 𝐵} = 𝑒 ↔ 𝑒 = {𝐴, 𝐵})
39	38	anbi1i 623	. . . . . 6 ⊢ (({𝐴, 𝐵} = 𝑒 ∧ 𝑒 ∈ 𝐸) ↔ (𝑒 = {𝐴, 𝐵} ∧ 𝑒 ∈ 𝐸))
40	37, 39	sylib 217	. . . . 5 ⊢ ((𝑒 ∈ 𝐸 ∧ {𝐴, 𝐵} = 𝑒) → (𝑒 = {𝐴, 𝐵} ∧ 𝑒 ∈ 𝐸))
41	40	eximi 1830	. . . 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 5428	. . . 4 ⊢ {𝐴, 𝐵} ∈ V
44		eleq1 2816	. . . 4 ⊢ (𝑒 = {𝐴, 𝐵} → (𝑒 ∈ 𝐸 ↔ {𝐴, 𝐵} ∈ 𝐸))
45	43, 44	ceqsexv 3521	. . 3 ⊢ (∃𝑒(𝑒 = {𝐴, 𝐵} ∧ 𝑒 ∈ 𝐸) ↔ {𝐴, 𝐵} ∈ 𝐸)
46	42, 45	sylib 217	. 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 1534  ∃wex 1774   ∈ wcel 2099   ≠ wne 2935  ∃wrex 3065  Vcvv 3469   ∖ cdif 3941   ⊆ wss 3944  {csn 4624  {cpr 4626  ‘cfv 6542  Fincfn 8955  2c2 12289  ℕ₀cn0 12494  ♯chash 14313  Vtxcvtx 28796  Edgcedg 28847  USGraphcusgr 28949  ComplGraphccplgr 29209  ComplUSGraphccusgr 29210

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734  ax-cnex 11186  ax-resscn 11187  ax-1cn 11188  ax-icn 11189  ax-addcl 11190  ax-addrcl 11191  ax-mulcl 11192  ax-mulrcl 11193  ax-mulcom 11194  ax-addass 11195  ax-mulass 11196  ax-distr 11197  ax-i2m1 11198  ax-1ne0 11199  ax-1rid 11200  ax-rnegex 11201  ax-rrecex 11202  ax-cnre 11203  ax-pre-lttri 11204  ax-pre-lttrn 11205  ax-pre-ltadd 11206  ax-pre-mulgt0 11207

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7865  df-1st 7987  df-2nd 7988  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-oadd 8484  df-er 8718  df-en 8956  df-dom 8957  df-sdom 8958  df-fin 8959  df-dju 9916  df-card 9954  df-pnf 11272  df-mnf 11273  df-xr 11274  df-ltxr 11275  df-le 11276  df-sub 11468  df-neg 11469  df-nn 12235  df-2 12297  df-n0 12495  df-z 12581  df-uz 12845  df-fz 13509  df-hash 14314  df-edg 28848  df-usgr 28951  df-nbgr 29133  df-uvtx 29186  df-cplgr 29211  df-cusgr 29212

This theorem is referenced by:  cusgredgex2  34668