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Mirrors > Home > MPE Home > Th. List > ramsey | Structured version Visualization version GIF version |
Description: Ramsey's theorem with the definition of Ramsey (df-ram 16941) eliminated. If π is an integer, π is a specified finite set of colors, and πΉ:π βΆβ0 is a set of lower bounds for each color, then there is an π such that for every set π of size greater than π and every coloring π of the set (π πΆπ) of all π-element subsets of π , there is a color π and a subset π₯ β π such that π₯ is larger than πΉ(π) and the π -element subsets of π₯ are monochromatic with color π. This is the hypergraph version of Ramsey's theorem; the version for simple graphs is the case π = 2. This is Metamath 100 proof #31. (Contributed by Mario Carneiro, 23-Apr-2015.) |
Ref | Expression |
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ramsey.c | β’ πΆ = (π β V, π β β0 β¦ {π β π« π β£ (β―βπ) = π}) |
Ref | Expression |
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ramsey | β’ ((π β β0 β§ π β Fin β§ πΉ:π βΆβ0) β βπ β β0 βπ (π β€ (β―βπ ) β βπ β (π βm (π πΆπ))βπ β π βπ₯ β π« π ((πΉβπ) β€ (β―βπ₯) β§ (π₯πΆπ) β (β‘π β {π})))) |
Step | Hyp | Ref | Expression |
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1 | ramcl 16969 | . . 3 β’ ((π β β0 β§ π β Fin β§ πΉ:π βΆβ0) β (π Ramsey πΉ) β β0) | |
2 | ramsey.c | . . . 4 β’ πΆ = (π β V, π β β0 β¦ {π β π« π β£ (β―βπ) = π}) | |
3 | eqid 2726 | . . . 4 β’ {π β β0 β£ βπ (π β€ (β―βπ ) β βπ β (π βm (π πΆπ))βπ β π βπ₯ β π« π ((πΉβπ) β€ (β―βπ₯) β§ (π₯πΆπ) β (β‘π β {π})))} = {π β β0 β£ βπ (π β€ (β―βπ ) β βπ β (π βm (π πΆπ))βπ β π βπ₯ β π« π ((πΉβπ) β€ (β―βπ₯) β§ (π₯πΆπ) β (β‘π β {π})))} | |
4 | 2, 3 | ramtcl2 16951 | . . 3 β’ ((π β β0 β§ π β Fin β§ πΉ:π βΆβ0) β ((π Ramsey πΉ) β β0 β {π β β0 β£ βπ (π β€ (β―βπ ) β βπ β (π βm (π πΆπ))βπ β π βπ₯ β π« π ((πΉβπ) β€ (β―βπ₯) β§ (π₯πΆπ) β (β‘π β {π})))} β β )) |
5 | 1, 4 | mpbid 231 | . 2 β’ ((π β β0 β§ π β Fin β§ πΉ:π βΆβ0) β {π β β0 β£ βπ (π β€ (β―βπ ) β βπ β (π βm (π πΆπ))βπ β π βπ₯ β π« π ((πΉβπ) β€ (β―βπ₯) β§ (π₯πΆπ) β (β‘π β {π})))} β β ) |
6 | rabn0 4380 | . 2 β’ ({π β β0 β£ βπ (π β€ (β―βπ ) β βπ β (π βm (π πΆπ))βπ β π βπ₯ β π« π ((πΉβπ) β€ (β―βπ₯) β§ (π₯πΆπ) β (β‘π β {π})))} β β β βπ β β0 βπ (π β€ (β―βπ ) β βπ β (π βm (π πΆπ))βπ β π βπ₯ β π« π ((πΉβπ) β€ (β―βπ₯) β§ (π₯πΆπ) β (β‘π β {π})))) | |
7 | 5, 6 | sylib 217 | 1 β’ ((π β β0 β§ π β Fin β§ πΉ:π βΆβ0) β βπ β β0 βπ (π β€ (β―βπ ) β βπ β (π βm (π πΆπ))βπ β π βπ₯ β π« π ((πΉβπ) β€ (β―βπ₯) β§ (π₯πΆπ) β (β‘π β {π})))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 β§ w3a 1084 βwal 1531 = wceq 1533 β wcel 2098 β wne 2934 βwral 3055 βwrex 3064 {crab 3426 Vcvv 3468 β wss 3943 β c0 4317 π« cpw 4597 {csn 4623 class class class wbr 5141 β‘ccnv 5668 β cima 5672 βΆwf 6532 βcfv 6536 (class class class)co 7404 β cmpo 7406 βm cmap 8819 Fincfn 8938 β€ cle 11250 β0cn0 12473 β―chash 14293 Ramsey cram 16939 |
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-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-inf2 9635 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-pre-sup 11187 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 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-rmo 3370 df-reu 3371 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-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 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-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-oadd 8468 df-er 8702 df-map 8821 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-sup 9436 df-inf 9437 df-oi 9504 df-dju 9895 df-card 9933 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-div 11873 df-nn 12214 df-2 12276 df-3 12277 df-n0 12474 df-xnn0 12546 df-z 12560 df-uz 12824 df-rp 12978 df-ico 13333 df-fz 13488 df-fzo 13631 df-seq 13970 df-exp 14031 df-fac 14237 df-bc 14266 df-hash 14294 df-cj 15050 df-re 15051 df-im 15052 df-sqrt 15186 df-abs 15187 df-clim 15436 df-sum 15637 df-ram 16941 |
This theorem is referenced by: (None) |
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