Theorem konigsbergiedgwOLD 41417

Description: The indexed edges of the Königsberg graph 𝐺 is a word over the pairs of vertices. (Contributed by AV, 28-Feb-2021.) Obsolete version of konigsbergiedgw 41416 as of 9-Mar-2021. (New usage is discouraged.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
konigsberg-av.v	⊢ 𝑉 = (0...3)
konigsberg-av.e	⊢ 𝐸 = ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩
konigsberg-av.g	⊢ 𝐺 = ⟨𝑉, 𝐸⟩

Assertion

Ref	Expression
konigsbergiedgwOLD	⊢ 𝐸 ∈ Word {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2}

Distinct variable group:   𝑥,𝑉

Allowed substitution hints:   𝐸(𝑥)   𝐺(𝑥)

Proof of Theorem konigsbergiedgwOLD

Step	Hyp	Ref	Expression
1		3nn0 11187	. . 3 ⊢ 3 ∈ ℕ0
2		0elfz 12305	. . . . . . 7 ⊢ (3 ∈ ℕ0 → 0 ∈ (0...3))
3	1, 2	ax-mp 5	. . . . . 6 ⊢ 0 ∈ (0...3)
4		1nn0 11185	. . . . . . 7 ⊢ 1 ∈ ℕ0
5		1le3 11121	. . . . . . 7 ⊢ 1 ≤ 3
6		elfz2nn0 12300	. . . . . . 7 ⊢ (1 ∈ (0...3) ↔ (1 ∈ ℕ0 ∧ 3 ∈ ℕ0 ∧ 1 ≤ 3))
7	4, 1, 5, 6	mpbir3an 1237	. . . . . 6 ⊢ 1 ∈ (0...3)
8	3, 7	upgrbi 25760	. . . . 5 ⊢ {0, 1} ∈ {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (#‘𝑥) ≤ 2}
9	8	a1i 11	. . . 4 ⊢ (3 ∈ ℕ0 → {0, 1} ∈ {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (#‘𝑥) ≤ 2})
10		2nn0 11186	. . . . . . 7 ⊢ 2 ∈ ℕ0
11		2re 10967	. . . . . . . 8 ⊢ 2 ∈ ℝ
12		3re 10971	. . . . . . . 8 ⊢ 3 ∈ ℝ
13		2lt3 11072	. . . . . . . 8 ⊢ 2 < 3
14	11, 12, 13	ltleii 10039	. . . . . . 7 ⊢ 2 ≤ 3
15		elfz2nn0 12300	. . . . . . 7 ⊢ (2 ∈ (0...3) ↔ (2 ∈ ℕ0 ∧ 3 ∈ ℕ0 ∧ 2 ≤ 3))
16	10, 1, 14, 15	mpbir3an 1237	. . . . . 6 ⊢ 2 ∈ (0...3)
17	3, 16	upgrbi 25760	. . . . 5 ⊢ {0, 2} ∈ {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (#‘𝑥) ≤ 2}
18	17	a1i 11	. . . 4 ⊢ (3 ∈ ℕ0 → {0, 2} ∈ {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (#‘𝑥) ≤ 2})
19		nn0fz0 12306	. . . . . . 7 ⊢ (3 ∈ ℕ0 ↔ 3 ∈ (0...3))
20	1, 19	mpbi 219	. . . . . 6 ⊢ 3 ∈ (0...3)
21	3, 20	upgrbi 25760	. . . . 5 ⊢ {0, 3} ∈ {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (#‘𝑥) ≤ 2}
22	21	a1i 11	. . . 4 ⊢ (3 ∈ ℕ0 → {0, 3} ∈ {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (#‘𝑥) ≤ 2})
23	7, 16	upgrbi 25760	. . . . 5 ⊢ {1, 2} ∈ {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (#‘𝑥) ≤ 2}
24	23	a1i 11	. . . 4 ⊢ (3 ∈ ℕ0 → {1, 2} ∈ {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (#‘𝑥) ≤ 2})
25	16, 20	upgrbi 25760	. . . . 5 ⊢ {2, 3} ∈ {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (#‘𝑥) ≤ 2}
26	25	a1i 11	. . . 4 ⊢ (3 ∈ ℕ0 → {2, 3} ∈ {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (#‘𝑥) ≤ 2})
27	9, 18, 22, 24, 24, 26, 26	s7cld 13471	. . 3 ⊢ (3 ∈ ℕ0 → ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩ ∈ Word {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (#‘𝑥) ≤ 2})
28	1, 27	ax-mp 5	. 2 ⊢ ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩ ∈ Word {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (#‘𝑥) ≤ 2}
29		konigsberg-av.e	. 2 ⊢ 𝐸 = ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩
30		konigsberg-av.v	. . . . . 6 ⊢ 𝑉 = (0...3)
31	30	pweqi 4112	. . . . 5 ⊢ 𝒫 𝑉 = 𝒫 (0...3)
32	31	difeq1i 3686	. . . 4 ⊢ (𝒫 𝑉 ∖ {∅}) = (𝒫 (0...3) ∖ {∅})
33		rabeq 3166	. . . 4 ⊢ ((𝒫 𝑉 ∖ {∅}) = (𝒫 (0...3) ∖ {∅}) → {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2} = {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (#‘𝑥) ≤ 2})
34	32, 33	ax-mp 5	. . 3 ⊢ {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2} = {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (#‘𝑥) ≤ 2}
35		wrdeq 13182	. . 3 ⊢ ({𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2} = {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (#‘𝑥) ≤ 2} → Word {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2} = Word {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (#‘𝑥) ≤ 2})
36	34, 35	ax-mp 5	. 2 ⊢ Word {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2} = Word {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (#‘𝑥) ≤ 2}
37	28, 29, 36	3eltr4i 2701	1 ⊢ 𝐸 ∈ Word {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2}

Syntax hints:   = wceq 1475   ∈ wcel 1977  {crab 2900   ∖ cdif 3537  ∅c0 3874  𝒫 cpw 4108  {csn 4125  {cpr 4127  ⟨cop 4131   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549  0cc0 9815  1c1 9816   ≤ cle 9954  2c2 10947  3c3 10948  ℕ₀cn0 11169  ...cfz 12197  #chash 12979  Word cword 13146  ⟨“cs7 13442

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-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-3 10957  df-n0 11170  df-xnn0 11241  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-concat 13156  df-s1 13157  df-s2 13444  df-s3 13445  df-s4 13446  df-s5 13447  df-s6 13448  df-s7 13449

This theorem is referenced by:  konigsbergupgrOLD  41421