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Mirrors > Home > MPE Home > Th. List > konigsberglem4 | Structured version Visualization version GIF version |
Description: Lemma 4 for konigsberg 30286: 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 12542 | . . . . . 6 ⊢ 3 ∈ ℕ0 | |
2 | 0elfz 13661 | . . . . . 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 2838 | . . . 4 ⊢ 0 ∈ 𝑉 |
6 | n2dvds3 16405 | . . . . 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 30281 | . . . . . 6 ⊢ ((VtxDeg‘𝐺)‘0) = 3 |
10 | 9 | breq2i 5156 | . . . . 5 ⊢ (2 ∥ ((VtxDeg‘𝐺)‘0) ↔ 2 ∥ 3) |
11 | 6, 10 | mtbir 323 | . . . 4 ⊢ ¬ 2 ∥ ((VtxDeg‘𝐺)‘0) |
12 | fveq2 6907 | . . . . . . 7 ⊢ (𝑥 = 0 → ((VtxDeg‘𝐺)‘𝑥) = ((VtxDeg‘𝐺)‘0)) | |
13 | 12 | breq2d 5160 | . . . . . 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 3695 | . . . 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 12540 | . . . . . 6 ⊢ 1 ∈ ℕ0 | |
18 | 1le3 12476 | . . . . . 6 ⊢ 1 ≤ 3 | |
19 | elfz2nn0 13655 | . . . . . 6 ⊢ (1 ∈ (0...3) ↔ (1 ∈ ℕ0 ∧ 3 ∈ ℕ0 ∧ 1 ≤ 3)) | |
20 | 17, 1, 18, 19 | mpbir3an 1340 | . . . . 5 ⊢ 1 ∈ (0...3) |
21 | 20, 4 | eleqtrri 2838 | . . . 4 ⊢ 1 ∈ 𝑉 |
22 | 4, 7, 8 | konigsberglem2 30282 | . . . . . 6 ⊢ ((VtxDeg‘𝐺)‘1) = 3 |
23 | 22 | breq2i 5156 | . . . . 5 ⊢ (2 ∥ ((VtxDeg‘𝐺)‘1) ↔ 2 ∥ 3) |
24 | 6, 23 | mtbir 323 | . . . 4 ⊢ ¬ 2 ∥ ((VtxDeg‘𝐺)‘1) |
25 | fveq2 6907 | . . . . . . 7 ⊢ (𝑥 = 1 → ((VtxDeg‘𝐺)‘𝑥) = ((VtxDeg‘𝐺)‘1)) | |
26 | 25 | breq2d 5160 | . . . . . 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 3695 | . . . 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 12344 | . . . . . . 7 ⊢ 3 ∈ ℝ | |
31 | 30 | leidi 11795 | . . . . . 6 ⊢ 3 ≤ 3 |
32 | elfz2nn0 13655 | . . . . . 6 ⊢ (3 ∈ (0...3) ↔ (3 ∈ ℕ0 ∧ 3 ∈ ℕ0 ∧ 3 ≤ 3)) | |
33 | 1, 1, 31, 32 | mpbir3an 1340 | . . . . 5 ⊢ 3 ∈ (0...3) |
34 | 33, 4 | eleqtrri 2838 | . . . 4 ⊢ 3 ∈ 𝑉 |
35 | 4, 7, 8 | konigsberglem3 30283 | . . . . . 6 ⊢ ((VtxDeg‘𝐺)‘3) = 3 |
36 | 35 | breq2i 5156 | . . . . 5 ⊢ (2 ∥ ((VtxDeg‘𝐺)‘3) ↔ 2 ∥ 3) |
37 | 6, 36 | mtbir 323 | . . . 4 ⊢ ¬ 2 ∥ ((VtxDeg‘𝐺)‘3) |
38 | fveq2 6907 | . . . . . . 7 ⊢ (𝑥 = 3 → ((VtxDeg‘𝐺)‘𝑥) = ((VtxDeg‘𝐺)‘3)) | |
39 | 38 | breq2d 5160 | . . . . . 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 3695 | . . . 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 1338 | . 2 ⊢ (0 ∈ {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)} ∧ 1 ∈ {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)} ∧ 3 ∈ {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) |
44 | c0ex 11253 | . . 3 ⊢ 0 ∈ V | |
45 | 1ex 11255 | . . 3 ⊢ 1 ∈ V | |
46 | 3ex 12346 | . . 3 ⊢ 3 ∈ V | |
47 | 44, 45, 46 | tpss 4842 | . 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 1086 = wceq 1537 ∈ wcel 2106 {crab 3433 ⊆ wss 3963 {cpr 4633 {ctp 4635 〈cop 4637 class class class wbr 5148 ‘cfv 6563 (class class class)co 7431 0cc0 11153 1c1 11154 ≤ cle 11294 2c2 12319 3c3 12320 ℕ0cn0 12524 ...cfz 13544 〈“cs7 14882 ∥ cdvds 16287 VtxDegcvtxdg 29498 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-oadd 8509 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-dju 9939 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-n0 12525 df-xnn0 12598 df-z 12612 df-uz 12877 df-xadd 13153 df-fz 13545 df-fzo 13692 df-hash 14367 df-word 14550 df-concat 14606 df-s1 14631 df-s2 14884 df-s3 14885 df-s4 14886 df-s5 14887 df-s6 14888 df-s7 14889 df-dvds 16288 df-vtx 29030 df-iedg 29031 df-vtxdg 29499 |
This theorem is referenced by: konigsberglem5 30285 |
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