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Mirrors > Home > MPE Home > Th. List > wwlksnon0 | Structured version Visualization version GIF version |
Description: Sufficient conditions for a set of walks of a fixed length between two vertices to be empty. (Contributed by AV, 15-May-2021.) (Proof shortened by AV, 21-May-2021.) |
Ref | Expression |
---|---|
wwlksnon0.v | ⢠ð = (Vtxâðº) |
Ref | Expression |
---|---|
wwlksnon0 | ⢠(¬ ((ð â â0 â§ ðº â V) â§ (ðŽ â ð â§ ðµ â ð)) â (ðŽ(ð WWalksNOn ðº)ðµ) = â ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-wwlksnon 29521 | . 2 ⢠WWalksNOn = (ð â â0, ð â V ⊠(ð â (Vtxâð), ð â (Vtxâð) ⊠{ð€ â (ð WWalksN ð) ⣠((ð€â0) = ð â§ (ð€âð) = ð)})) | |
2 | wwlksnon0.v | . . 3 ⢠ð = (Vtxâðº) | |
3 | 2 | wwlksnon 29540 | . 2 ⢠((ð â â0 â§ ðº â V) â (ð WWalksNOn ðº) = (ð â ð, ð â ð ⊠{ð€ â (ð WWalksN ðº) ⣠((ð€â0) = ð â§ (ð€âð) = ð)})) |
4 | 1, 3 | 2mpo0 7648 | 1 ⢠(¬ ((ð â â0 â§ ðº â V) â§ (ðŽ â ð â§ ðµ â ð)) â (ðŽ(ð WWalksNOn ðº)ðµ) = â ) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 â wi 4 â§ wa 395 = wceq 1533 â wcel 2098 {crab 3424 Vcvv 3466 â c0 4314 âcfv 6533 (class class class)co 7401 â cmpo 7403 0cc0 11105 â0cn0 12468 Vtxcvtx 28691 WWalksN cwwlksn 29515 WWalksNOn cwwlksnon 29516 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-id 5564 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-ov 7404 df-oprab 7405 df-mpo 7406 df-1st 7968 df-2nd 7969 df-wwlksnon 29521 |
This theorem is referenced by: iswspthsnon 29545 |
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