![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 29126 | . . 3 ⢠(ðº NeighbVtx ð) â (ð â {ð}) |
3 | df-nel 3041 | . . . . 5 ⢠(ð â (ðº NeighbVtx ð) â ¬ ð â (ðº NeighbVtx ð)) | |
4 | disjsn 4710 | . . . . 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 4446 | . . . 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 4731 | . . . 4 ⢠{ð, ð} = {ð, ð} | |
10 | 9 | difeq2i 4114 | . . 3 ⢠(ð â {ð, ð}) = (ð â {ð, ð}) |
11 | difpr 4801 | . . 3 ⢠(ð â {ð, ð}) = ((ð â {ð}) â {ð}) | |
12 | 10, 11 | eqtri 2754 | . 2 ⢠(ð â {ð, ð}) = ((ð â {ð}) â {ð}) |
13 | 8, 12 | sseqtrrdi 4028 | 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 3040 â cdif 3940 â© cin 3942 â wss 3943 â c0 4317 {csn 4623 {cpr 4625 âcfv 6537 (class class class)co 7405 Vtxcvtx 28764 NeighbVtx cnbgr 29097 |
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 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 ax-un 7722 |
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 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6489 df-fun 6539 df-fv 6545 df-ov 7408 df-oprab 7409 df-mpo 7410 df-1st 7974 df-2nd 7975 df-nbgr 29098 |
This theorem is referenced by: nbfusgrlevtxm2 29143 |
Copyright terms: Public domain | W3C validator |