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| Description: Lemma 4 for konigsberg 30276: Vertices 0, 1, 3 are vertices of odd degree. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by AV, 28-Feb-2021.) | 
| 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〉 | 
| Ref | Expression | 
|---|---|
| konigsberglem4 | ⊢ {0, 1, 3} ⊆ {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)} | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | 3nn0 12544 | . . . . . 6 ⊢ 3 ∈ ℕ0 | |
| 2 | 0elfz 13664 | . . . . . 6 ⊢ (3 ∈ ℕ0 → 0 ∈ (0...3)) | |
| 3 | 1, 2 | ax-mp 5 | . . . . 5 ⊢ 0 ∈ (0...3) | 
| 4 | konigsberg.v | . . . . 5 ⊢ 𝑉 = (0...3) | |
| 5 | 3, 4 | eleqtrri 2840 | . . . 4 ⊢ 0 ∈ 𝑉 | 
| 6 | n2dvds3 16408 | . . . . 5 ⊢ ¬ 2 ∥ 3 | |
| 7 | konigsberg.e | . . . . . . 7 ⊢ 𝐸 = 〈“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”〉 | |
| 8 | konigsberg.g | . . . . . . 7 ⊢ 𝐺 = 〈𝑉, 𝐸〉 | |
| 9 | 4, 7, 8 | konigsberglem1 30271 | . . . . . 6 ⊢ ((VtxDeg‘𝐺)‘0) = 3 | 
| 10 | 9 | breq2i 5151 | . . . . 5 ⊢ (2 ∥ ((VtxDeg‘𝐺)‘0) ↔ 2 ∥ 3) | 
| 11 | 6, 10 | mtbir 323 | . . . 4 ⊢ ¬ 2 ∥ ((VtxDeg‘𝐺)‘0) | 
| 12 | fveq2 6906 | . . . . . . 7 ⊢ (𝑥 = 0 → ((VtxDeg‘𝐺)‘𝑥) = ((VtxDeg‘𝐺)‘0)) | |
| 13 | 12 | breq2d 5155 | . . . . . 6 ⊢ (𝑥 = 0 → (2 ∥ ((VtxDeg‘𝐺)‘𝑥) ↔ 2 ∥ ((VtxDeg‘𝐺)‘0))) | 
| 14 | 13 | notbid 318 | . . . . 5 ⊢ (𝑥 = 0 → (¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥) ↔ ¬ 2 ∥ ((VtxDeg‘𝐺)‘0))) | 
| 15 | 14 | elrab 3692 | . . . 4 ⊢ (0 ∈ {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)} ↔ (0 ∈ 𝑉 ∧ ¬ 2 ∥ ((VtxDeg‘𝐺)‘0))) | 
| 16 | 5, 11, 15 | mpbir2an 711 | . . 3 ⊢ 0 ∈ {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)} | 
| 17 | 1nn0 12542 | . . . . . 6 ⊢ 1 ∈ ℕ0 | |
| 18 | 1le3 12478 | . . . . . 6 ⊢ 1 ≤ 3 | |
| 19 | elfz2nn0 13658 | . . . . . 6 ⊢ (1 ∈ (0...3) ↔ (1 ∈ ℕ0 ∧ 3 ∈ ℕ0 ∧ 1 ≤ 3)) | |
| 20 | 17, 1, 18, 19 | mpbir3an 1342 | . . . . 5 ⊢ 1 ∈ (0...3) | 
| 21 | 20, 4 | eleqtrri 2840 | . . . 4 ⊢ 1 ∈ 𝑉 | 
| 22 | 4, 7, 8 | konigsberglem2 30272 | . . . . . 6 ⊢ ((VtxDeg‘𝐺)‘1) = 3 | 
| 23 | 22 | breq2i 5151 | . . . . 5 ⊢ (2 ∥ ((VtxDeg‘𝐺)‘1) ↔ 2 ∥ 3) | 
| 24 | 6, 23 | mtbir 323 | . . . 4 ⊢ ¬ 2 ∥ ((VtxDeg‘𝐺)‘1) | 
| 25 | fveq2 6906 | . . . . . . 7 ⊢ (𝑥 = 1 → ((VtxDeg‘𝐺)‘𝑥) = ((VtxDeg‘𝐺)‘1)) | |
| 26 | 25 | breq2d 5155 | . . . . . 6 ⊢ (𝑥 = 1 → (2 ∥ ((VtxDeg‘𝐺)‘𝑥) ↔ 2 ∥ ((VtxDeg‘𝐺)‘1))) | 
| 27 | 26 | notbid 318 | . . . . 5 ⊢ (𝑥 = 1 → (¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥) ↔ ¬ 2 ∥ ((VtxDeg‘𝐺)‘1))) | 
| 28 | 27 | elrab 3692 | . . . 4 ⊢ (1 ∈ {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)} ↔ (1 ∈ 𝑉 ∧ ¬ 2 ∥ ((VtxDeg‘𝐺)‘1))) | 
| 29 | 21, 24, 28 | mpbir2an 711 | . . 3 ⊢ 1 ∈ {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)} | 
| 30 | 3re 12346 | . . . . . . 7 ⊢ 3 ∈ ℝ | |
| 31 | 30 | leidi 11797 | . . . . . 6 ⊢ 3 ≤ 3 | 
| 32 | elfz2nn0 13658 | . . . . . 6 ⊢ (3 ∈ (0...3) ↔ (3 ∈ ℕ0 ∧ 3 ∈ ℕ0 ∧ 3 ≤ 3)) | |
| 33 | 1, 1, 31, 32 | mpbir3an 1342 | . . . . 5 ⊢ 3 ∈ (0...3) | 
| 34 | 33, 4 | eleqtrri 2840 | . . . 4 ⊢ 3 ∈ 𝑉 | 
| 35 | 4, 7, 8 | konigsberglem3 30273 | . . . . . 6 ⊢ ((VtxDeg‘𝐺)‘3) = 3 | 
| 36 | 35 | breq2i 5151 | . . . . 5 ⊢ (2 ∥ ((VtxDeg‘𝐺)‘3) ↔ 2 ∥ 3) | 
| 37 | 6, 36 | mtbir 323 | . . . 4 ⊢ ¬ 2 ∥ ((VtxDeg‘𝐺)‘3) | 
| 38 | fveq2 6906 | . . . . . . 7 ⊢ (𝑥 = 3 → ((VtxDeg‘𝐺)‘𝑥) = ((VtxDeg‘𝐺)‘3)) | |
| 39 | 38 | breq2d 5155 | . . . . . 6 ⊢ (𝑥 = 3 → (2 ∥ ((VtxDeg‘𝐺)‘𝑥) ↔ 2 ∥ ((VtxDeg‘𝐺)‘3))) | 
| 40 | 39 | notbid 318 | . . . . 5 ⊢ (𝑥 = 3 → (¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥) ↔ ¬ 2 ∥ ((VtxDeg‘𝐺)‘3))) | 
| 41 | 40 | elrab 3692 | . . . 4 ⊢ (3 ∈ {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)} ↔ (3 ∈ 𝑉 ∧ ¬ 2 ∥ ((VtxDeg‘𝐺)‘3))) | 
| 42 | 34, 37, 41 | mpbir2an 711 | . . 3 ⊢ 3 ∈ {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)} | 
| 43 | 16, 29, 42 | 3pm3.2i 1340 | . 2 ⊢ (0 ∈ {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)} ∧ 1 ∈ {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)} ∧ 3 ∈ {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) | 
| 44 | c0ex 11255 | . . 3 ⊢ 0 ∈ V | |
| 45 | 1ex 11257 | . . 3 ⊢ 1 ∈ V | |
| 46 | 3ex 12348 | . . 3 ⊢ 3 ∈ V | |
| 47 | 44, 45, 46 | tpss 4837 | . 2 ⊢ ((0 ∈ {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)} ∧ 1 ∈ {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)} ∧ 3 ∈ {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) ↔ {0, 1, 3} ⊆ {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) | 
| 48 | 43, 47 | mpbi 230 | 1 ⊢ {0, 1, 3} ⊆ {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)} | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 {crab 3436 ⊆ wss 3951 {cpr 4628 {ctp 4630 〈cop 4632 class class class wbr 5143 ‘cfv 6561 (class class class)co 7431 0cc0 11155 1c1 11156 ≤ cle 11296 2c2 12321 3c3 12322 ℕ0cn0 12526 ...cfz 13547 〈“cs7 14885 ∥ cdvds 16290 VtxDegcvtxdg 29483 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-oadd 8510 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-dju 9941 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-n0 12527 df-xnn0 12600 df-z 12614 df-uz 12879 df-xadd 13155 df-fz 13548 df-fzo 13695 df-hash 14370 df-word 14553 df-concat 14609 df-s1 14634 df-s2 14887 df-s3 14888 df-s4 14889 df-s5 14890 df-s6 14891 df-s7 14892 df-dvds 16291 df-vtx 29015 df-iedg 29016 df-vtxdg 29484 | 
| This theorem is referenced by: konigsberglem5 30275 | 
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