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Mirrors > Home > MPE Home > Th. List > nbgrssvwo2 | Structured version Visualization version GIF version |
Description: The neighbors of a vertex ð form a subset of all vertices except the vertex ð itself and a class ð which is not a neighbor of ð. (Contributed by Alexander van der Vekens, 13-Jul-2018.) (Revised by AV, 3-Nov-2020.) (Revised by AV, 12-Feb-2022.) |
Ref | Expression |
---|---|
nbgrssovtx.v | ⢠ð = (Vtxâðº) |
Ref | Expression |
---|---|
nbgrssvwo2 | ⢠(ð â (ðº NeighbVtx ð) â (ðº NeighbVtx ð) â (ð â {ð, ð})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nbgrssovtx.v | . . . 4 ⢠ð = (Vtxâðº) | |
2 | 1 | nbgrssovtx 29218 | . . 3 ⢠(ðº NeighbVtx ð) â (ð â {ð}) |
3 | df-nel 3037 | . . . . 5 ⢠(ð â (ðº NeighbVtx ð) â ¬ ð â (ðº NeighbVtx ð)) | |
4 | disjsn 4711 | . . . . 5 ⢠(((ðº NeighbVtx ð) â© {ð}) = â â ¬ ð â (ðº NeighbVtx ð)) | |
5 | 3, 4 | sylbb2 237 | . . . 4 ⢠(ð â (ðº NeighbVtx ð) â ((ðº NeighbVtx ð) â© {ð}) = â ) |
6 | reldisj 4447 | . . . 4 ⢠((ðº NeighbVtx ð) â (ð â {ð}) â (((ðº NeighbVtx ð) â© {ð}) = â â (ðº NeighbVtx ð) â ((ð â {ð}) â {ð}))) | |
7 | 5, 6 | imbitrid 243 | . . 3 ⢠((ðº NeighbVtx ð) â (ð â {ð}) â (ð â (ðº NeighbVtx ð) â (ðº NeighbVtx ð) â ((ð â {ð}) â {ð}))) |
8 | 2, 7 | ax-mp 5 | . 2 ⢠(ð â (ðº NeighbVtx ð) â (ðº NeighbVtx ð) â ((ð â {ð}) â {ð})) |
9 | prcom 4732 | . . . 4 ⢠{ð, ð} = {ð, ð} | |
10 | 9 | difeq2i 4111 | . . 3 ⢠(ð â {ð, ð}) = (ð â {ð, ð}) |
11 | difpr 4802 | . . 3 ⢠(ð â {ð, ð}) = ((ð â {ð}) â {ð}) | |
12 | 10, 11 | eqtri 2753 | . 2 ⢠(ð â {ð, ð}) = ((ð â {ð}) â {ð}) |
13 | 8, 12 | sseqtrrdi 4024 | 1 ⢠(ð â (ðº NeighbVtx ð) â (ðº NeighbVtx ð) â (ð â {ð, ð})) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 â wi 4 = wceq 1533 â wcel 2098 â wnel 3036 â cdif 3936 â© cin 3938 â wss 3939 â c0 4318 {csn 4624 {cpr 4626 âcfv 6543 (class class class)co 7416 Vtxcvtx 28853 NeighbVtx cnbgr 29189 |
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 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pr 5423 ax-un 7738 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6495 df-fun 6545 df-fv 6551 df-ov 7419 df-oprab 7420 df-mpo 7421 df-1st 7991 df-2nd 7992 df-nbgr 29190 |
This theorem is referenced by: nbfusgrlevtxm2 29235 |
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