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Mirrors > Home > MPE Home > Th. List > frgrancvvdeqlem2 | Structured version Visualization version GIF version |
Description: Lemma 2 for frgrancvvdeq 26569. (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 |
---|---|
frgrancvvdeqlem2 | ⊢ (𝜑 → 𝑋 ∉ 𝑁) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frgrancvvdeq.f | . . . . . 6 ⊢ (𝜑 → 𝑉 FriendGrph 𝐸) | |
2 | frisusgra 26519 | . . . . . 6 ⊢ (𝑉 FriendGrph 𝐸 → 𝑉 USGrph 𝐸) | |
3 | nbgraeledg 25959 | . . . . . 6 ⊢ (𝑉 USGrph 𝐸 → (𝑋 ∈ (〈𝑉, 𝐸〉 Neighbors 𝑌) ↔ {𝑋, 𝑌} ∈ ran 𝐸)) | |
4 | 1, 2, 3 | 3syl 18 | . . . . 5 ⊢ (𝜑 → (𝑋 ∈ (〈𝑉, 𝐸〉 Neighbors 𝑌) ↔ {𝑋, 𝑌} ∈ ran 𝐸)) |
5 | frgrancvvdeq.xy | . . . . . 6 ⊢ (𝜑 → 𝑌 ∉ 𝐷) | |
6 | df-nel 2783 | . . . . . . . . 9 ⊢ (𝑌 ∉ 𝐷 ↔ ¬ 𝑌 ∈ 𝐷) | |
7 | frgrancvvdeq.nx | . . . . . . . . . 10 ⊢ 𝐷 = (〈𝑉, 𝐸〉 Neighbors 𝑋) | |
8 | 7 | eleq2i 2680 | . . . . . . . . 9 ⊢ (𝑌 ∈ 𝐷 ↔ 𝑌 ∈ (〈𝑉, 𝐸〉 Neighbors 𝑋)) |
9 | 6, 8 | xchbinx 323 | . . . . . . . 8 ⊢ (𝑌 ∉ 𝐷 ↔ ¬ 𝑌 ∈ (〈𝑉, 𝐸〉 Neighbors 𝑋)) |
10 | nbgraeledg 25959 | . . . . . . . . . 10 ⊢ (𝑉 USGrph 𝐸 → (𝑌 ∈ (〈𝑉, 𝐸〉 Neighbors 𝑋) ↔ {𝑌, 𝑋} ∈ ran 𝐸)) | |
11 | 1, 2, 10 | 3syl 18 | . . . . . . . . 9 ⊢ (𝜑 → (𝑌 ∈ (〈𝑉, 𝐸〉 Neighbors 𝑋) ↔ {𝑌, 𝑋} ∈ ran 𝐸)) |
12 | 11 | notbid 307 | . . . . . . . 8 ⊢ (𝜑 → (¬ 𝑌 ∈ (〈𝑉, 𝐸〉 Neighbors 𝑋) ↔ ¬ {𝑌, 𝑋} ∈ ran 𝐸)) |
13 | 9, 12 | syl5bb 271 | . . . . . . 7 ⊢ (𝜑 → (𝑌 ∉ 𝐷 ↔ ¬ {𝑌, 𝑋} ∈ ran 𝐸)) |
14 | prcom 4211 | . . . . . . . . 9 ⊢ {𝑌, 𝑋} = {𝑋, 𝑌} | |
15 | 14 | eleq1i 2679 | . . . . . . . 8 ⊢ ({𝑌, 𝑋} ∈ ran 𝐸 ↔ {𝑋, 𝑌} ∈ ran 𝐸) |
16 | pm2.21 119 | . . . . . . . 8 ⊢ (¬ {𝑋, 𝑌} ∈ ran 𝐸 → ({𝑋, 𝑌} ∈ ran 𝐸 → 𝑋 ∉ (〈𝑉, 𝐸〉 Neighbors 𝑌))) | |
17 | 15, 16 | sylnbi 319 | . . . . . . 7 ⊢ (¬ {𝑌, 𝑋} ∈ ran 𝐸 → ({𝑋, 𝑌} ∈ ran 𝐸 → 𝑋 ∉ (〈𝑉, 𝐸〉 Neighbors 𝑌))) |
18 | 13, 17 | syl6bi 242 | . . . . . 6 ⊢ (𝜑 → (𝑌 ∉ 𝐷 → ({𝑋, 𝑌} ∈ ran 𝐸 → 𝑋 ∉ (〈𝑉, 𝐸〉 Neighbors 𝑌)))) |
19 | 5, 18 | mpd 15 | . . . . 5 ⊢ (𝜑 → ({𝑋, 𝑌} ∈ ran 𝐸 → 𝑋 ∉ (〈𝑉, 𝐸〉 Neighbors 𝑌))) |
20 | 4, 19 | sylbid 229 | . . . 4 ⊢ (𝜑 → (𝑋 ∈ (〈𝑉, 𝐸〉 Neighbors 𝑌) → 𝑋 ∉ (〈𝑉, 𝐸〉 Neighbors 𝑌))) |
21 | 20 | com12 32 | . . 3 ⊢ (𝑋 ∈ (〈𝑉, 𝐸〉 Neighbors 𝑌) → (𝜑 → 𝑋 ∉ (〈𝑉, 𝐸〉 Neighbors 𝑌))) |
22 | df-nel 2783 | . . . 4 ⊢ (𝑋 ∉ (〈𝑉, 𝐸〉 Neighbors 𝑌) ↔ ¬ 𝑋 ∈ (〈𝑉, 𝐸〉 Neighbors 𝑌)) | |
23 | ax-1 6 | . . . 4 ⊢ (𝑋 ∉ (〈𝑉, 𝐸〉 Neighbors 𝑌) → (𝜑 → 𝑋 ∉ (〈𝑉, 𝐸〉 Neighbors 𝑌))) | |
24 | 22, 23 | sylbir 224 | . . 3 ⊢ (¬ 𝑋 ∈ (〈𝑉, 𝐸〉 Neighbors 𝑌) → (𝜑 → 𝑋 ∉ (〈𝑉, 𝐸〉 Neighbors 𝑌))) |
25 | 21, 24 | pm2.61i 175 | . 2 ⊢ (𝜑 → 𝑋 ∉ (〈𝑉, 𝐸〉 Neighbors 𝑌)) |
26 | eqidd 2611 | . . 3 ⊢ (𝜑 → 𝑋 = 𝑋) | |
27 | frgrancvvdeq.ny | . . . 4 ⊢ 𝑁 = (〈𝑉, 𝐸〉 Neighbors 𝑌) | |
28 | 27 | a1i 11 | . . 3 ⊢ (𝜑 → 𝑁 = (〈𝑉, 𝐸〉 Neighbors 𝑌)) |
29 | 26, 28 | neleq12d 2887 | . 2 ⊢ (𝜑 → (𝑋 ∉ 𝑁 ↔ 𝑋 ∉ (〈𝑉, 𝐸〉 Neighbors 𝑌))) |
30 | 25, 29 | mpbird 246 | 1 ⊢ (𝜑 → 𝑋 ∉ 𝑁) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∉ wnel 2781 {cpr 4127 〈cop 4131 class class class wbr 4583 ↦ cmpt 4643 ran crn 5039 ℩crio 6510 (class class class)co 6549 USGrph cusg 25859 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: frgrancvvdeqlemA 26564 frgrancvvdeqlemB 26565 frgrancvvdeqlemC 26566 |
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