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Theorem konigsbergiedgwen 16639

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 9560	. . . . . . 7 ⊢ 3 ∈ ℕ₀
2		0elfz 10503	. . . . . . 7 ⊢ (3 ∈ ℕ₀ → 0 ∈ (0...3))
3	1, 2	ax-mp 5	. . . . . 6 ⊢ 0 ∈ (0...3)
4		1nn0 9558	. . . . . . 7 ⊢ 1 ∈ ℕ₀
5		1le3 9495	. . . . . . 7 ⊢ 1 ≤ 3
6		elfz2nn0 10497	. . . . . . 7 ⊢ (1 ∈ (0...3) ↔ (1 ∈ ℕ₀ ∧ 3 ∈ ℕ₀ ∧ 1 ≤ 3))
7	4, 1, 5, 6	mpbir3an 1210	. . . . . 6 ⊢ 1 ∈ (0...3)
8		0ne1 9350	. . . . . 6 ⊢ 0 ≠ 1
9	3, 7, 8	umgrbien 16265	. . . . 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 9559	. . . . . . 7 ⊢ 2 ∈ ℕ₀
12		2re 9353	. . . . . . . 8 ⊢ 2 ∈ ℝ
13		3re 9357	. . . . . . . 8 ⊢ 3 ∈ ℝ
14		2lt3 9454	. . . . . . . 8 ⊢ 2 < 3
15	12, 13, 14	ltleii 8418	. . . . . . 7 ⊢ 2 ≤ 3
16		elfz2nn0 10497	. . . . . . 7 ⊢ (2 ∈ (0...3) ↔ (2 ∈ ℕ₀ ∧ 3 ∈ ℕ₀ ∧ 2 ≤ 3))
17	11, 1, 15, 16	mpbir3an 1210	. . . . . 6 ⊢ 2 ∈ (0...3)
18		0ne2 9489	. . . . . 6 ⊢ 0 ≠ 2
19	3, 17, 18	umgrbien 16265	. . . . 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 10504	. . . . . . 7 ⊢ (3 ∈ ℕ₀ ↔ 3 ∈ (0...3))
22	1, 21	mpbi 145	. . . . . 6 ⊢ 3 ∈ (0...3)
23		3ne0 9378	. . . . . . 7 ⊢ 3 ≠ 0
24	23	necomi 2505	. . . . . 6 ⊢ 0 ≠ 3
25	3, 22, 24	umgrbien 16265	. . . . 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 9490	. . . . . 6 ⊢ 1 ≠ 2
28	7, 17, 27	umgrbien 16265	. . . . 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 8412	. . . . . 6 ⊢ 2 ≠ 3
31	17, 22, 30	umgrbien 16265	. . . . 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 11533	. . 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 3689	. . . 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 11305	. 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

Syntax hints: = wceq 1402 ⊤wtru 1403 ∈ wcel 2209 {crab 2532 𝒫 cpw 3685 {cpr 3706 ⟨cop 3708 class class class wbr 4125 (class class class)co 6075 2_oc2o 6671 ≈ cen 7010 0cc0 8169 1c1 8170 ≤ cle 8351 2c2 9334 3c3 9335 ℕ₀cn0 9542 ...cfz 10390 Word cword 11282 ⟨“cs7 11504

This theorem was proved from 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 4241 ax-sep 4244 ax-nul 4254 ax-pow 4306 ax-pr 4341 ax-un 4573 ax-setind 4679 ax-iinf 4730 ax-cnex 8260 ax-resscn 8261 ax-1cn 8262 ax-1re 8263 ax-icn 8264 ax-addcl 8265 ax-addrcl 8266 ax-mulcl 8267 ax-addcom 8269 ax-addass 8271 ax-distr 8273 ax-i2m1 8274 ax-0lt1 8275 ax-0id 8277 ax-rnegex 8278 ax-cnre 8280 ax-pre-ltirr 8281 ax-pre-ltwlin 8282 ax-pre-lttrn 8283 ax-pre-apti 8284 ax-pre-ltadd 8285

This theorem 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 3636 df-pw 3687 df-sn 3711 df-pr 3712 df-op 3714 df-uni 3931 df-int 3966 df-iun 4009 df-br 4126 df-opab 4188 df-mpt 4189 df-tr 4225 df-id 4433 df-iord 4506 df-on 4508 df-ilim 4509 df-suc 4511 df-iom 4733 df-xp 4775 df-rel 4776 df-cnv 4777 df-co 4778 df-dm 4779 df-rn 4780 df-res 4781 df-ima 4782 df-iota 5332 df-fun 5374 df-fn 5375 df-f 5376 df-f1 5377 df-fo 5378 df-f1o 5379 df-fv 5380 df-riota 6028 df-ov 6078 df-oprab 6079 df-mpo 6080 df-1st 6364 df-2nd 6365 df-recs 6566 df-frec 6652 df-1o 6677 df-2o 6678 df-er 6797 df-en 7013 df-dom 7014 df-fin 7015 df-pnf 8352 df-mnf 8353 df-xr 8354 df-ltxr 8355 df-le 8356 df-sub 8489 df-neg 8490 df-inn 9284 df-2 9342 df-3 9343 df-n0 9543 df-z 9624 df-uz 9901 df-fz 10391 df-fzo 10528 df-ihash 11193 df-word 11283 df-concat 11337 df-s1 11362 df-s2 11506 df-s3 11507 df-s4 11508 df-s5 11509 df-s6 11510 df-s7 11511

This theorem is referenced by: konigsbergssiedgwpren 16640 konigsbergumgr 16642

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