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Theorem konigsbergiedgwen 16930

Description: The indexed edges of the Königsberg graph 𝐺 is a word over the pairs of vertices. (Contributed by AV, 28-Feb-2021.)

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

**Assertion**
Ref	Expression
konigsbergiedgwen	⊢ 𝐸 ∈ Word {𝑥 ∈ 𝒫 𝑉 ∣ 𝑥 ≈ 2_o}

Distinct variable group: 𝑥,𝑉 Allowed substitution hints: 𝐸(𝑥) 𝐺(𝑥)

**Proof of Theorem konigsbergiedgwen**
Step	Hyp	Ref	Expression
1		3nn0 9592	. . . . . . 7 ⊢ 3 ∈ ℕ₀
2		0elfz 10542	. . . . . . 7 ⊢ (3 ∈ ℕ₀ → 0 ∈ (0...3))
3	1, 2	ax-mp 5	. . . . . 6 ⊢ 0 ∈ (0...3)
4		1nn0 9590	. . . . . . 7 ⊢ 1 ∈ ℕ₀
5		1le3 9527	. . . . . . 7 ⊢ 1 ≤ 3
6		elfz2nn0 10536	. . . . . . 7 ⊢ (1 ∈ (0...3) ↔ (1 ∈ ℕ₀ ∧ 3 ∈ ℕ₀ ∧ 1 ≤ 3))
7	4, 1, 5, 6	mpbir3an 1210	. . . . . 6 ⊢ 1 ∈ (0...3)
8		0ne1 9380	. . . . . 6 ⊢ 0 ≠ 1
9	3, 7, 8	umgrbien 16556	. . . . 5 ⊢ {0, 1} ∈ {𝑥 ∈ 𝒫 (0...3) ∣ 𝑥 ≈ 2_o}
10	9	a1i 9	. . . 4 ⊢ (⊤ → {0, 1} ∈ {𝑥 ∈ 𝒫 (0...3) ∣ 𝑥 ≈ 2_o})
11		2nn0 9591	. . . . . . 7 ⊢ 2 ∈ ℕ₀
12		2re 9383	. . . . . . . 8 ⊢ 2 ∈ ℝ
13		3re 9387	. . . . . . . 8 ⊢ 3 ∈ ℝ
14		2lt3 9486	. . . . . . . 8 ⊢ 2 < 3
15	12, 13, 14	ltleii 8435	. . . . . . 7 ⊢ 2 ≤ 3
16		elfz2nn0 10536	. . . . . . 7 ⊢ (2 ∈ (0...3) ↔ (2 ∈ ℕ₀ ∧ 3 ∈ ℕ₀ ∧ 2 ≤ 3))
17	11, 1, 15, 16	mpbir3an 1210	. . . . . 6 ⊢ 2 ∈ (0...3)
18		0ne2 9521	. . . . . 6 ⊢ 0 ≠ 2
19	3, 17, 18	umgrbien 16556	. . . . 5 ⊢ {0, 2} ∈ {𝑥 ∈ 𝒫 (0...3) ∣ 𝑥 ≈ 2_o}
20	19	a1i 9	. . . 4 ⊢ (⊤ → {0, 2} ∈ {𝑥 ∈ 𝒫 (0...3) ∣ 𝑥 ≈ 2_o})
21		nn0fz0 10543	. . . . . . 7 ⊢ (3 ∈ ℕ₀ ↔ 3 ∈ (0...3))
22	1, 21	mpbi 145	. . . . . 6 ⊢ 3 ∈ (0...3)
23		3ne0 9408	. . . . . . 7 ⊢ 3 ≠ 0
24	23	necomi 2505	. . . . . 6 ⊢ 0 ≠ 3
25	3, 22, 24	umgrbien 16556	. . . . 5 ⊢ {0, 3} ∈ {𝑥 ∈ 𝒫 (0...3) ∣ 𝑥 ≈ 2_o}
26	25	a1i 9	. . . 4 ⊢ (⊤ → {0, 3} ∈ {𝑥 ∈ 𝒫 (0...3) ∣ 𝑥 ≈ 2_o})
27		1ne2 9522	. . . . . 6 ⊢ 1 ≠ 2
28	7, 17, 27	umgrbien 16556	. . . . 5 ⊢ {1, 2} ∈ {𝑥 ∈ 𝒫 (0...3) ∣ 𝑥 ≈ 2_o}
29	28	a1i 9	. . . 4 ⊢ (⊤ → {1, 2} ∈ {𝑥 ∈ 𝒫 (0...3) ∣ 𝑥 ≈ 2_o})
30	12, 14	ltneii 8429	. . . . . 6 ⊢ 2 ≠ 3
31	17, 22, 30	umgrbien 16556	. . . . 5 ⊢ {2, 3} ∈ {𝑥 ∈ 𝒫 (0...3) ∣ 𝑥 ≈ 2_o}
32	31	a1i 9	. . . 4 ⊢ (⊤ → {2, 3} ∈ {𝑥 ∈ 𝒫 (0...3) ∣ 𝑥 ≈ 2_o})
33	10, 20, 26, 29, 29, 32, 32	s7cld 11577	. . 3 ⊢ (⊤ → ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩ ∈ Word {𝑥 ∈ 𝒫 (0...3) ∣ 𝑥 ≈ 2_o})
34	33	mptru 1411	. 2 ⊢ ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩ ∈ Word {𝑥 ∈ 𝒫 (0...3) ∣ 𝑥 ≈ 2_o}
35		konigsberg.e	. 2 ⊢ 𝐸 = ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩
36		konigsberg.v	. . . . 5 ⊢ 𝑉 = (0...3)
37	36	pweqi 3692	. . . 4 ⊢ 𝒫 𝑉 = 𝒫 (0...3)
38	37	rabeqi 2814	. . 3 ⊢ {𝑥 ∈ 𝒫 𝑉 ∣ 𝑥 ≈ 2_o} = {𝑥 ∈ 𝒫 (0...3) ∣ 𝑥 ≈ 2_o}
39	38	wrdeqi 11349	. 2 ⊢ Word {𝑥 ∈ 𝒫 𝑉 ∣ 𝑥 ≈ 2_o} = Word {𝑥 ∈ 𝒫 (0...3) ∣ 𝑥 ≈ 2_o}
40	34, 35, 39	3eltr4i 2320	1 ⊢ 𝐸 ∈ Word {𝑥 ∈ 𝒫 𝑉 ∣ 𝑥 ≈ 2_o}

Colors of variables: wff set class

This proof depends on syntax axioms: = wceq 1402 ⊤wtru 1403 ∈ wcel 2209 {crab 2532 𝒫 cpw 3688 {cpr 3710 ⟨cop 3712 class class class wbr 4131 (class class class)co 6088 2_oc2o 6685 ≈ cen 7024 0cc0 8185 1c1 8186 ≤ cle 8367 2c2 9364 3c3 9365 ℕ₀cn0 9574 ...cfz 10428 Word cword 11326 ⟨“cs7 11548

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-coll 4248 ax-sep 4251 ax-nul 4261 ax-pow 4313 ax-pr 4348 ax-un 4580 ax-setind 4686 ax-iinf 4737 ax-cnex 8276 ax-resscn 8277 ax-1cn 8278 ax-1re 8279 ax-icn 8280 ax-addcl 8281 ax-addrcl 8282 ax-mulcl 8283 ax-addcom 8285 ax-addass 8287 ax-distr 8289 ax-i2m1 8290 ax-0lt1 8291 ax-0id 8293 ax-rnegex 8294 ax-cnre 8296 ax-pre-ltirr 8297 ax-pre-ltwlin 8298 ax-pre-lttrn 8299 ax-pre-apti 8300 ax-pre-ltadd 8301

This proof depends on definitions: df-bi 117 df-dc 847 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-reu 2535 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-nul 3521 df-if 3639 df-pw 3690 df-sn 3715 df-pr 3716 df-op 3718 df-uni 3936 df-int 3971 df-iun 4014 df-br 4132 df-opab 4194 df-mpt 4195 df-tr 4232 df-id 4440 df-iord 4513 df-on 4515 df-ilim 4516 df-suc 4518 df-iom 4740 df-xp 4782 df-rel 4783 df-cnv 4784 df-co 4785 df-dm 4786 df-rn 4787 df-res 4788 df-ima 4789 df-iota 5340 df-fun 5382 df-fn 5383 df-f 5384 df-f1 5385 df-fo 5386 df-f1o 5387 df-fv 5388 df-riota 6041 df-ov 6091 df-oprab 6092 df-mpo 6093 df-1st 6378 df-2nd 6379 df-recs 6580 df-frec 6666 df-1o 6691 df-2o 6692 df-er 6811 df-en 7027 df-dom 7028 df-fin 7029 df-pnf 8368 df-mnf 8369 df-xr 8370 df-ltxr 8371 df-le 8372 df-sub 8506 df-neg 8507 df-inn 9314 df-2 9372 df-3 9373 df-n0 9575 df-z 9656 df-uz 9938 df-fz 10429 df-fzo 10567 df-ihash 11237 df-word 11327 df-concat 11381 df-s1 11406 df-s2 11550 df-s3 11551 df-s4 11552 df-s5 11553 df-s6 11554 df-s7 11555

This theorem is used by: konigsbergssiedgwpren 16931 konigsbergumgr 16933

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