Theorem cusgrafi 26010

Description: If the size of a complete simple graph is finite, then also its order is finite. (Contributed by Alexander van der Vekens, 13-Jan-2018.)

Assertion

Ref	Expression
cusgrafi	⊢ ((𝑉 ComplUSGrph 𝐸 ∧ 𝐸 ∈ Fin) → 𝑉 ∈ Fin)

Proof of Theorem cusgrafi

Dummy variables 𝑒 𝑛 𝑝 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfielex 8074	. . . . 5 ⊢ (¬ 𝑉 ∈ Fin → ∃𝑛 𝑛 ∈ 𝑉)
2		cusisusgra 25987	. . . . . . . 8 ⊢ (𝑉 ComplUSGrph 𝐸 → 𝑉 USGrph 𝐸)
3		usgrav 25867	. . . . . . . 8 ⊢ (𝑉 USGrph 𝐸 → (𝑉 ∈ V ∧ 𝐸 ∈ V))
4	2, 3	syl 17	. . . . . . 7 ⊢ (𝑉 ComplUSGrph 𝐸 → (𝑉 ∈ V ∧ 𝐸 ∈ V))
5		eqeq1 2614	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑒 = 𝑝 → (𝑒 = {𝑣, 𝑛} ↔ 𝑝 = {𝑣, 𝑛}))
6	5	anbi2d 736	. . . . . . . . . . . . . . . . 17 ⊢ (𝑒 = 𝑝 → ((𝑣 ≠ 𝑛 ∧ 𝑒 = {𝑣, 𝑛}) ↔ (𝑣 ≠ 𝑛 ∧ 𝑝 = {𝑣, 𝑛})))
7	6	rexbidv 3034	. . . . . . . . . . . . . . . 16 ⊢ (𝑒 = 𝑝 → (∃𝑣 ∈ 𝑉 (𝑣 ≠ 𝑛 ∧ 𝑒 = {𝑣, 𝑛}) ↔ ∃𝑣 ∈ 𝑉 (𝑣 ≠ 𝑛 ∧ 𝑝 = {𝑣, 𝑛})))
8	7	cbvrabv 3172	. . . . . . . . . . . . . . 15 ⊢ {𝑒 ∈ 𝒫 𝑉 ∣ ∃𝑣 ∈ 𝑉 (𝑣 ≠ 𝑛 ∧ 𝑒 = {𝑣, 𝑛})} = {𝑝 ∈ 𝒫 𝑉 ∣ ∃𝑣 ∈ 𝑉 (𝑣 ≠ 𝑛 ∧ 𝑝 = {𝑣, 𝑛})}
9		eqid 2610	. . . . . . . . . . . . . . 15 ⊢ (𝑝 ∈ (𝑉 ∖ {𝑛}) ↦ {𝑝, 𝑛}) = (𝑝 ∈ (𝑉 ∖ {𝑛}) ↦ {𝑝, 𝑛})
10	8, 9	cusgrafilem3 26009	. . . . . . . . . . . . . 14 ⊢ ((𝑉 ∈ V ∧ 𝑛 ∈ 𝑉) → (¬ 𝑉 ∈ Fin → ¬ {𝑒 ∈ 𝒫 𝑉 ∣ ∃𝑣 ∈ 𝑉 (𝑣 ≠ 𝑛 ∧ 𝑒 = {𝑣, 𝑛})} ∈ Fin))
11	10	ex 449	. . . . . . . . . . . . 13 ⊢ (𝑉 ∈ V → (𝑛 ∈ 𝑉 → (¬ 𝑉 ∈ Fin → ¬ {𝑒 ∈ 𝒫 𝑉 ∣ ∃𝑣 ∈ 𝑉 (𝑣 ≠ 𝑛 ∧ 𝑒 = {𝑣, 𝑛})} ∈ Fin)))
12	11	com13 86	. . . . . . . . . . . 12 ⊢ (¬ 𝑉 ∈ Fin → (𝑛 ∈ 𝑉 → (𝑉 ∈ V → ¬ {𝑒 ∈ 𝒫 𝑉 ∣ ∃𝑣 ∈ 𝑉 (𝑣 ≠ 𝑛 ∧ 𝑒 = {𝑣, 𝑛})} ∈ Fin)))
13	12	imp 444	. . . . . . . . . . 11 ⊢ ((¬ 𝑉 ∈ Fin ∧ 𝑛 ∈ 𝑉) → (𝑉 ∈ V → ¬ {𝑒 ∈ 𝒫 𝑉 ∣ ∃𝑣 ∈ 𝑉 (𝑣 ≠ 𝑛 ∧ 𝑒 = {𝑣, 𝑛})} ∈ Fin))
14	13	com12 32	. . . . . . . . . 10 ⊢ (𝑉 ∈ V → ((¬ 𝑉 ∈ Fin ∧ 𝑛 ∈ 𝑉) → ¬ {𝑒 ∈ 𝒫 𝑉 ∣ ∃𝑣 ∈ 𝑉 (𝑣 ≠ 𝑛 ∧ 𝑒 = {𝑣, 𝑛})} ∈ Fin))
15	8	cusgrafilem1 26007	. . . . . . . . . . . . . 14 ⊢ ((𝑉 ComplUSGrph 𝐸 ∧ 𝑛 ∈ 𝑉) → {𝑒 ∈ 𝒫 𝑉 ∣ ∃𝑣 ∈ 𝑉 (𝑣 ≠ 𝑛 ∧ 𝑒 = {𝑣, 𝑛})} ⊆ ran 𝐸)
16		ssfi 8065	. . . . . . . . . . . . . . . . 17 ⊢ ((ran 𝐸 ∈ Fin ∧ {𝑒 ∈ 𝒫 𝑉 ∣ ∃𝑣 ∈ 𝑉 (𝑣 ≠ 𝑛 ∧ 𝑒 = {𝑣, 𝑛})} ⊆ ran 𝐸) → {𝑒 ∈ 𝒫 𝑉 ∣ ∃𝑣 ∈ 𝑉 (𝑣 ≠ 𝑛 ∧ 𝑒 = {𝑣, 𝑛})} ∈ Fin)
17	16	expcom 450	. . . . . . . . . . . . . . . 16 ⊢ ({𝑒 ∈ 𝒫 𝑉 ∣ ∃𝑣 ∈ 𝑉 (𝑣 ≠ 𝑛 ∧ 𝑒 = {𝑣, 𝑛})} ⊆ ran 𝐸 → (ran 𝐸 ∈ Fin → {𝑒 ∈ 𝒫 𝑉 ∣ ∃𝑣 ∈ 𝑉 (𝑣 ≠ 𝑛 ∧ 𝑒 = {𝑣, 𝑛})} ∈ Fin))
18	17	con3d 147	. . . . . . . . . . . . . . 15 ⊢ ({𝑒 ∈ 𝒫 𝑉 ∣ ∃𝑣 ∈ 𝑉 (𝑣 ≠ 𝑛 ∧ 𝑒 = {𝑣, 𝑛})} ⊆ ran 𝐸 → (¬ {𝑒 ∈ 𝒫 𝑉 ∣ ∃𝑣 ∈ 𝑉 (𝑣 ≠ 𝑛 ∧ 𝑒 = {𝑣, 𝑛})} ∈ Fin → ¬ ran 𝐸 ∈ Fin))
19		rnfi 8132	. . . . . . . . . . . . . . . 16 ⊢ (𝐸 ∈ Fin → ran 𝐸 ∈ Fin)
20	19	con3i 149	. . . . . . . . . . . . . . 15 ⊢ (¬ ran 𝐸 ∈ Fin → ¬ 𝐸 ∈ Fin)
21	18, 20	syl6 34	. . . . . . . . . . . . . 14 ⊢ ({𝑒 ∈ 𝒫 𝑉 ∣ ∃𝑣 ∈ 𝑉 (𝑣 ≠ 𝑛 ∧ 𝑒 = {𝑣, 𝑛})} ⊆ ran 𝐸 → (¬ {𝑒 ∈ 𝒫 𝑉 ∣ ∃𝑣 ∈ 𝑉 (𝑣 ≠ 𝑛 ∧ 𝑒 = {𝑣, 𝑛})} ∈ Fin → ¬ 𝐸 ∈ Fin))
22	15, 21	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝑉 ComplUSGrph 𝐸 ∧ 𝑛 ∈ 𝑉) → (¬ {𝑒 ∈ 𝒫 𝑉 ∣ ∃𝑣 ∈ 𝑉 (𝑣 ≠ 𝑛 ∧ 𝑒 = {𝑣, 𝑛})} ∈ Fin → ¬ 𝐸 ∈ Fin))
23	22	expcom 450	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ 𝑉 → (𝑉 ComplUSGrph 𝐸 → (¬ {𝑒 ∈ 𝒫 𝑉 ∣ ∃𝑣 ∈ 𝑉 (𝑣 ≠ 𝑛 ∧ 𝑒 = {𝑣, 𝑛})} ∈ Fin → ¬ 𝐸 ∈ Fin)))
24	23	com23 84	. . . . . . . . . . 11 ⊢ (𝑛 ∈ 𝑉 → (¬ {𝑒 ∈ 𝒫 𝑉 ∣ ∃𝑣 ∈ 𝑉 (𝑣 ≠ 𝑛 ∧ 𝑒 = {𝑣, 𝑛})} ∈ Fin → (𝑉 ComplUSGrph 𝐸 → ¬ 𝐸 ∈ Fin)))
25	24	adantl 481	. . . . . . . . . 10 ⊢ ((¬ 𝑉 ∈ Fin ∧ 𝑛 ∈ 𝑉) → (¬ {𝑒 ∈ 𝒫 𝑉 ∣ ∃𝑣 ∈ 𝑉 (𝑣 ≠ 𝑛 ∧ 𝑒 = {𝑣, 𝑛})} ∈ Fin → (𝑉 ComplUSGrph 𝐸 → ¬ 𝐸 ∈ Fin)))
26	14, 25	sylcom 30	. . . . . . . . 9 ⊢ (𝑉 ∈ V → ((¬ 𝑉 ∈ Fin ∧ 𝑛 ∈ 𝑉) → (𝑉 ComplUSGrph 𝐸 → ¬ 𝐸 ∈ Fin)))
27	26	com23 84	. . . . . . . 8 ⊢ (𝑉 ∈ V → (𝑉 ComplUSGrph 𝐸 → ((¬ 𝑉 ∈ Fin ∧ 𝑛 ∈ 𝑉) → ¬ 𝐸 ∈ Fin)))
28	27	adantr 480	. . . . . . 7 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑉 ComplUSGrph 𝐸 → ((¬ 𝑉 ∈ Fin ∧ 𝑛 ∈ 𝑉) → ¬ 𝐸 ∈ Fin)))
29	4, 28	mpcom 37	. . . . . 6 ⊢ (𝑉 ComplUSGrph 𝐸 → ((¬ 𝑉 ∈ Fin ∧ 𝑛 ∈ 𝑉) → ¬ 𝐸 ∈ Fin))
30	29	com12 32	. . . . 5 ⊢ ((¬ 𝑉 ∈ Fin ∧ 𝑛 ∈ 𝑉) → (𝑉 ComplUSGrph 𝐸 → ¬ 𝐸 ∈ Fin))
31	1, 30	exlimddv 1850	. . . 4 ⊢ (¬ 𝑉 ∈ Fin → (𝑉 ComplUSGrph 𝐸 → ¬ 𝐸 ∈ Fin))
32	31	com12 32	. . 3 ⊢ (𝑉 ComplUSGrph 𝐸 → (¬ 𝑉 ∈ Fin → ¬ 𝐸 ∈ Fin))
33	32	con4d 113	. 2 ⊢ (𝑉 ComplUSGrph 𝐸 → (𝐸 ∈ Fin → 𝑉 ∈ Fin))
34	33	imp 444	1 ⊢ ((𝑉 ComplUSGrph 𝐸 ∧ 𝐸 ∈ Fin) → 𝑉 ∈ Fin)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∃wrex 2897  {crab 2900  Vcvv 3173   ∖ cdif 3537   ⊆ wss 3540  𝒫 cpw 4108  {csn 4125  {cpr 4127   class class class wbr 4583   ↦ cmpt 4643  ran crn 5039  Fincfn 7841   USGrph cusg 25859   ComplUSGrph ccusgra 25947

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-z 11255  df-uz 11564  df-fz 12198  df-hash 12980  df-usgra 25862  df-cusgra 25950

This theorem is referenced by:  sizeusglecusglem2  26013