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Mirrors > Home > MPE Home > Th. List > g0wlk0 | Structured version Visualization version GIF version |
Description: There is no walk in a null graph (a class without vertices). (Contributed by Alexander van der Vekens, 2-Sep-2018.) (Revised by AV, 5-Mar-2021.) |
Ref | Expression |
---|---|
g0wlk0 | ⊢ ((Vtx‘𝐺) = ∅ → (Walks‘𝐺) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-1 6 | . 2 ⊢ ((Walks‘𝐺) = ∅ → ((Vtx‘𝐺) = ∅ → (Walks‘𝐺) = ∅)) | |
2 | neq0 4246 | . . 3 ⊢ (¬ (Walks‘𝐺) = ∅ ↔ ∃𝑤 𝑤 ∈ (Walks‘𝐺)) | |
3 | wlkv0 27692 | . . . . . 6 ⊢ (((Vtx‘𝐺) = ∅ ∧ 𝑤 ∈ (Walks‘𝐺)) → ((1st ‘𝑤) = ∅ ∧ (2nd ‘𝑤) = ∅)) | |
4 | wlkcpr 27670 | . . . . . . . 8 ⊢ (𝑤 ∈ (Walks‘𝐺) ↔ (1st ‘𝑤)(Walks‘𝐺)(2nd ‘𝑤)) | |
5 | wlkn0 27662 | . . . . . . . . 9 ⊢ ((1st ‘𝑤)(Walks‘𝐺)(2nd ‘𝑤) → (2nd ‘𝑤) ≠ ∅) | |
6 | eqneqall 2943 | . . . . . . . . . 10 ⊢ ((2nd ‘𝑤) = ∅ → ((2nd ‘𝑤) ≠ ∅ → (Walks‘𝐺) = ∅)) | |
7 | 6 | adantl 485 | . . . . . . . . 9 ⊢ (((1st ‘𝑤) = ∅ ∧ (2nd ‘𝑤) = ∅) → ((2nd ‘𝑤) ≠ ∅ → (Walks‘𝐺) = ∅)) |
8 | 5, 7 | syl5com 31 | . . . . . . . 8 ⊢ ((1st ‘𝑤)(Walks‘𝐺)(2nd ‘𝑤) → (((1st ‘𝑤) = ∅ ∧ (2nd ‘𝑤) = ∅) → (Walks‘𝐺) = ∅)) |
9 | 4, 8 | sylbi 220 | . . . . . . 7 ⊢ (𝑤 ∈ (Walks‘𝐺) → (((1st ‘𝑤) = ∅ ∧ (2nd ‘𝑤) = ∅) → (Walks‘𝐺) = ∅)) |
10 | 9 | adantl 485 | . . . . . 6 ⊢ (((Vtx‘𝐺) = ∅ ∧ 𝑤 ∈ (Walks‘𝐺)) → (((1st ‘𝑤) = ∅ ∧ (2nd ‘𝑤) = ∅) → (Walks‘𝐺) = ∅)) |
11 | 3, 10 | mpd 15 | . . . . 5 ⊢ (((Vtx‘𝐺) = ∅ ∧ 𝑤 ∈ (Walks‘𝐺)) → (Walks‘𝐺) = ∅) |
12 | 11 | expcom 417 | . . . 4 ⊢ (𝑤 ∈ (Walks‘𝐺) → ((Vtx‘𝐺) = ∅ → (Walks‘𝐺) = ∅)) |
13 | 12 | exlimiv 1938 | . . 3 ⊢ (∃𝑤 𝑤 ∈ (Walks‘𝐺) → ((Vtx‘𝐺) = ∅ → (Walks‘𝐺) = ∅)) |
14 | 2, 13 | sylbi 220 | . 2 ⊢ (¬ (Walks‘𝐺) = ∅ → ((Vtx‘𝐺) = ∅ → (Walks‘𝐺) = ∅)) |
15 | 1, 14 | pm2.61i 185 | 1 ⊢ ((Vtx‘𝐺) = ∅ → (Walks‘𝐺) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 = wceq 1543 ∃wex 1787 ∈ wcel 2112 ≠ wne 2932 ∅c0 4223 class class class wbr 5039 ‘cfv 6358 1st c1st 7737 2nd c2nd 7738 Vtxcvtx 27041 Walkscwlks 27638 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-ifp 1064 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-int 4846 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-1st 7739 df-2nd 7740 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-1o 8180 df-er 8369 df-map 8488 df-en 8605 df-dom 8606 df-sdom 8607 df-fin 8608 df-card 9520 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-nn 11796 df-n0 12056 df-z 12142 df-uz 12404 df-fz 13061 df-fzo 13204 df-hash 13862 df-word 14035 df-wlks 27641 |
This theorem is referenced by: 0wlk0 27694 wlk0prc 27695 acycgr0v 32777 prclisacycgr 32780 |
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