Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  konigsbergiedgwOLD Structured version   Visualization version   GIF version

Theorem konigsbergiedgwOLD 41412
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 41411 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
StepHypRef Expression
1 3nn0 11157 . . 3 3 ∈ ℕ0
2 0elfz 12260 . . . . . . 7 (3 ∈ ℕ0 → 0 ∈ (0...3))
31, 2ax-mp 5 . . . . . 6 0 ∈ (0...3)
4 1nn0 11155 . . . . . . 7 1 ∈ ℕ0
5 1le3 11091 . . . . . . 7 1 ≤ 3
6 elfz2nn0 12255 . . . . . . 7 (1 ∈ (0...3) ↔ (1 ∈ ℕ0 ∧ 3 ∈ ℕ0 ∧ 1 ≤ 3))
74, 1, 5, 6mpbir3an 1236 . . . . . 6 1 ∈ (0...3)
83, 7upgrbi 40314 . . . . 5 {0, 1} ∈ {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (#‘𝑥) ≤ 2}
98a1i 11 . . . 4 (3 ∈ ℕ0 → {0, 1} ∈ {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (#‘𝑥) ≤ 2})
10 2nn0 11156 . . . . . . 7 2 ∈ ℕ0
11 2re 10937 . . . . . . . 8 2 ∈ ℝ
12 3re 10941 . . . . . . . 8 3 ∈ ℝ
13 2lt3 11042 . . . . . . . 8 2 < 3
1411, 12, 13ltleii 10011 . . . . . . 7 2 ≤ 3
15 elfz2nn0 12255 . . . . . . 7 (2 ∈ (0...3) ↔ (2 ∈ ℕ0 ∧ 3 ∈ ℕ0 ∧ 2 ≤ 3))
1610, 1, 14, 15mpbir3an 1236 . . . . . 6 2 ∈ (0...3)
173, 16upgrbi 40314 . . . . 5 {0, 2} ∈ {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (#‘𝑥) ≤ 2}
1817a1i 11 . . . 4 (3 ∈ ℕ0 → {0, 2} ∈ {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (#‘𝑥) ≤ 2})
19 nn0fz0 12261 . . . . . . 7 (3 ∈ ℕ0 ↔ 3 ∈ (0...3))
201, 19mpbi 218 . . . . . 6 3 ∈ (0...3)
213, 20upgrbi 40314 . . . . 5 {0, 3} ∈ {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (#‘𝑥) ≤ 2}
2221a1i 11 . . . 4 (3 ∈ ℕ0 → {0, 3} ∈ {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (#‘𝑥) ≤ 2})
237, 16upgrbi 40314 . . . . 5 {1, 2} ∈ {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (#‘𝑥) ≤ 2}
2423a1i 11 . . . 4 (3 ∈ ℕ0 → {1, 2} ∈ {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (#‘𝑥) ≤ 2})
2516, 20upgrbi 40314 . . . . 5 {2, 3} ∈ {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (#‘𝑥) ≤ 2}
2625a1i 11 . . . 4 (3 ∈ ℕ0 → {2, 3} ∈ {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (#‘𝑥) ≤ 2})
279, 18, 22, 24, 24, 26, 26s7cld 13417 . . 3 (3 ∈ ℕ0 → ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩ ∈ Word {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (#‘𝑥) ≤ 2})
281, 27ax-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)
3130pweqi 4111 . . . . 5 𝒫 𝑉 = 𝒫 (0...3)
3231difeq1i 3685 . . . 4 (𝒫 𝑉 ∖ {∅}) = (𝒫 (0...3) ∖ {∅})
33 rabeq 3165 . . . 4 ((𝒫 𝑉 ∖ {∅}) = (𝒫 (0...3) ∖ {∅}) → {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2} = {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (#‘𝑥) ≤ 2})
3432, 33ax-mp 5 . . 3 {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2} = {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (#‘𝑥) ≤ 2}
35 wrdeq 13128 . . 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})
3634, 35ax-mp 5 . 2 Word {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2} = Word {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (#‘𝑥) ≤ 2}
3728, 29, 363eltr4i 2700 1 𝐸 ∈ Word {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1474  wcel 1976  {crab 2899  cdif 3536  c0 3873  𝒫 cpw 4107  {csn 4124  {cpr 4126  cop 4130   class class class wbr 4577  cfv 5790  (class class class)co 6527  0cc0 9792  1c1 9793  cle 9931  2c2 10917  3c3 10918  0cn0 11139  ...cfz 12152  #chash 12934  Word cword 13092  ⟨“cs7 13388
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824  ax-cnex 9848  ax-resscn 9849  ax-1cn 9850  ax-icn 9851  ax-addcl 9852  ax-addrcl 9853  ax-mulcl 9854  ax-mulrcl 9855  ax-mulcom 9856  ax-addass 9857  ax-mulass 9858  ax-distr 9859  ax-i2m1 9860  ax-1ne0 9861  ax-1rid 9862  ax-rnegex 9863  ax-rrecex 9864  ax-cnre 9865  ax-pre-lttri 9866  ax-pre-lttrn 9867  ax-pre-ltadd 9868  ax-pre-mulgt0 9869
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-om 6935  df-1st 7036  df-2nd 7037  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-1o 7424  df-oadd 7428  df-er 7606  df-en 7819  df-dom 7820  df-sdom 7821  df-fin 7822  df-card 8625  df-cda 8850  df-pnf 9932  df-mnf 9933  df-xr 9934  df-ltxr 9935  df-le 9936  df-sub 10119  df-neg 10120  df-nn 10868  df-2 10926  df-3 10927  df-n0 11140  df-z 11211  df-uz 11520  df-fz 12153  df-fzo 12290  df-hash 12935  df-word 13100  df-concat 13102  df-s1 13103  df-s2 13390  df-s3 13391  df-s4 13392  df-s5 13393  df-s6 13394  df-s7 13395
This theorem is referenced by:  konigsbergupgrOLD  41416
  Copyright terms: Public domain W3C validator