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Mirrors > Home > MPE Home > Th. List > frgrancvvdeqlemA | Structured version Visualization version GIF version |
Description: Lemma A for frgrancvvdeq 26569. This corresponds to statement 1 in [Huneke] p. 1: "This common neighbor cannot be x, as x and y are not adjacent.". (Contributed by Alexander van der Vekens, 23-Dec-2017.) |
Ref | Expression |
---|---|
frgrancvvdeq.nx | ⊢ 𝐷 = (〈𝑉, 𝐸〉 Neighbors 𝑋) |
frgrancvvdeq.ny | ⊢ 𝑁 = (〈𝑉, 𝐸〉 Neighbors 𝑌) |
frgrancvvdeq.x | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
frgrancvvdeq.y | ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
frgrancvvdeq.ne | ⊢ (𝜑 → 𝑋 ≠ 𝑌) |
frgrancvvdeq.xy | ⊢ (𝜑 → 𝑌 ∉ 𝐷) |
frgrancvvdeq.f | ⊢ (𝜑 → 𝑉 FriendGrph 𝐸) |
frgrancvvdeq.a | ⊢ 𝐴 = (𝑥 ∈ 𝐷 ↦ (℩𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ ran 𝐸)) |
Ref | Expression |
---|---|
frgrancvvdeqlemA | ⊢ (𝜑 → ∀𝑥 ∈ 𝐷 (𝐴‘𝑥) ≠ 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frgrancvvdeq.nx | . . . 4 ⊢ 𝐷 = (〈𝑉, 𝐸〉 Neighbors 𝑋) | |
2 | frgrancvvdeq.ny | . . . 4 ⊢ 𝑁 = (〈𝑉, 𝐸〉 Neighbors 𝑌) | |
3 | frgrancvvdeq.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
4 | frgrancvvdeq.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑉) | |
5 | frgrancvvdeq.ne | . . . 4 ⊢ (𝜑 → 𝑋 ≠ 𝑌) | |
6 | frgrancvvdeq.xy | . . . 4 ⊢ (𝜑 → 𝑌 ∉ 𝐷) | |
7 | frgrancvvdeq.f | . . . 4 ⊢ (𝜑 → 𝑉 FriendGrph 𝐸) | |
8 | frgrancvvdeq.a | . . . 4 ⊢ 𝐴 = (𝑥 ∈ 𝐷 ↦ (℩𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ ran 𝐸)) | |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | frgrancvvdeqlem6 26562 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → {(𝐴‘𝑥)} = ((〈𝑉, 𝐸〉 Neighbors 𝑥) ∩ 𝑁)) |
10 | fvex 6113 | . . . . 5 ⊢ (𝐴‘𝑥) ∈ V | |
11 | 10 | snid 4155 | . . . 4 ⊢ (𝐴‘𝑥) ∈ {(𝐴‘𝑥)} |
12 | eleq2 2677 | . . . . . 6 ⊢ ({(𝐴‘𝑥)} = ((〈𝑉, 𝐸〉 Neighbors 𝑥) ∩ 𝑁) → ((𝐴‘𝑥) ∈ {(𝐴‘𝑥)} ↔ (𝐴‘𝑥) ∈ ((〈𝑉, 𝐸〉 Neighbors 𝑥) ∩ 𝑁))) | |
13 | 12 | biimpa 500 | . . . . 5 ⊢ (({(𝐴‘𝑥)} = ((〈𝑉, 𝐸〉 Neighbors 𝑥) ∩ 𝑁) ∧ (𝐴‘𝑥) ∈ {(𝐴‘𝑥)}) → (𝐴‘𝑥) ∈ ((〈𝑉, 𝐸〉 Neighbors 𝑥) ∩ 𝑁)) |
14 | elin 3758 | . . . . . 6 ⊢ ((𝐴‘𝑥) ∈ ((〈𝑉, 𝐸〉 Neighbors 𝑥) ∩ 𝑁) ↔ ((𝐴‘𝑥) ∈ (〈𝑉, 𝐸〉 Neighbors 𝑥) ∧ (𝐴‘𝑥) ∈ 𝑁)) | |
15 | 1, 2, 3, 4, 5, 6, 7, 8 | frgrancvvdeqlem2 26558 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∉ 𝑁) |
16 | df-nel 2783 | . . . . . . . . . . 11 ⊢ (𝑋 ∉ 𝑁 ↔ ¬ 𝑋 ∈ 𝑁) | |
17 | eleq1 2676 | . . . . . . . . . . . . . 14 ⊢ ((𝐴‘𝑥) = 𝑋 → ((𝐴‘𝑥) ∈ 𝑁 ↔ 𝑋 ∈ 𝑁)) | |
18 | 17 | biimpcd 238 | . . . . . . . . . . . . 13 ⊢ ((𝐴‘𝑥) ∈ 𝑁 → ((𝐴‘𝑥) = 𝑋 → 𝑋 ∈ 𝑁)) |
19 | 18 | con3rr3 150 | . . . . . . . . . . . 12 ⊢ (¬ 𝑋 ∈ 𝑁 → ((𝐴‘𝑥) ∈ 𝑁 → ¬ (𝐴‘𝑥) = 𝑋)) |
20 | df-ne 2782 | . . . . . . . . . . . 12 ⊢ ((𝐴‘𝑥) ≠ 𝑋 ↔ ¬ (𝐴‘𝑥) = 𝑋) | |
21 | 19, 20 | syl6ibr 241 | . . . . . . . . . . 11 ⊢ (¬ 𝑋 ∈ 𝑁 → ((𝐴‘𝑥) ∈ 𝑁 → (𝐴‘𝑥) ≠ 𝑋)) |
22 | 16, 21 | sylbi 206 | . . . . . . . . . 10 ⊢ (𝑋 ∉ 𝑁 → ((𝐴‘𝑥) ∈ 𝑁 → (𝐴‘𝑥) ≠ 𝑋)) |
23 | 15, 22 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → ((𝐴‘𝑥) ∈ 𝑁 → (𝐴‘𝑥) ≠ 𝑋)) |
24 | 23 | adantr 480 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((𝐴‘𝑥) ∈ 𝑁 → (𝐴‘𝑥) ≠ 𝑋)) |
25 | 24 | com12 32 | . . . . . . 7 ⊢ ((𝐴‘𝑥) ∈ 𝑁 → ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝐴‘𝑥) ≠ 𝑋)) |
26 | 25 | adantl 481 | . . . . . 6 ⊢ (((𝐴‘𝑥) ∈ (〈𝑉, 𝐸〉 Neighbors 𝑥) ∧ (𝐴‘𝑥) ∈ 𝑁) → ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝐴‘𝑥) ≠ 𝑋)) |
27 | 14, 26 | sylbi 206 | . . . . 5 ⊢ ((𝐴‘𝑥) ∈ ((〈𝑉, 𝐸〉 Neighbors 𝑥) ∩ 𝑁) → ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝐴‘𝑥) ≠ 𝑋)) |
28 | 13, 27 | syl 17 | . . . 4 ⊢ (({(𝐴‘𝑥)} = ((〈𝑉, 𝐸〉 Neighbors 𝑥) ∩ 𝑁) ∧ (𝐴‘𝑥) ∈ {(𝐴‘𝑥)}) → ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝐴‘𝑥) ≠ 𝑋)) |
29 | 11, 28 | mpan2 703 | . . 3 ⊢ ({(𝐴‘𝑥)} = ((〈𝑉, 𝐸〉 Neighbors 𝑥) ∩ 𝑁) → ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝐴‘𝑥) ≠ 𝑋)) |
30 | 9, 29 | mpcom 37 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝐴‘𝑥) ≠ 𝑋) |
31 | 30 | ralrimiva 2949 | 1 ⊢ (𝜑 → ∀𝑥 ∈ 𝐷 (𝐴‘𝑥) ≠ 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∉ wnel 2781 ∀wral 2896 ∩ cin 3539 {csn 4125 {cpr 4127 〈cop 4131 class class class wbr 4583 ↦ cmpt 4643 ran crn 5039 ‘cfv 5804 ℩crio 6510 (class class class)co 6549 Neighbors cnbgra 25946 FriendGrph cfrgra 26515 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-hash 12980 df-usgra 25862 df-nbgra 25949 df-frgra 26516 |
This theorem is referenced by: (None) |
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