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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 16970) 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 |
---|---|---|---|
1 | ramcl 16998 | . . 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 2728 | . . . 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 16980 | . . 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 4386 | . 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 1085 βwal 1532 = wceq 1534 β wcel 2099 β wne 2937 βwral 3058 βwrex 3067 {crab 3429 Vcvv 3471 β wss 3947 β c0 4323 π« cpw 4603 {csn 4629 class class class wbr 5148 β‘ccnv 5677 β cima 5681 βΆwf 6544 βcfv 6548 (class class class)co 7420 β cmpo 7422 βm cmap 8845 Fincfn 8964 β€ cle 11280 β0cn0 12503 β―chash 14322 Ramsey cram 16968 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-inf2 9665 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 ax-pre-sup 11217 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-isom 6557 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-oadd 8491 df-er 8725 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-sup 9466 df-inf 9467 df-oi 9534 df-dju 9925 df-card 9963 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-div 11903 df-nn 12244 df-2 12306 df-3 12307 df-n0 12504 df-xnn0 12576 df-z 12590 df-uz 12854 df-rp 13008 df-ico 13363 df-fz 13518 df-fzo 13661 df-seq 14000 df-exp 14060 df-fac 14266 df-bc 14295 df-hash 14323 df-cj 15079 df-re 15080 df-im 15081 df-sqrt 15215 df-abs 15216 df-clim 15465 df-sum 15666 df-ram 16970 |
This theorem is referenced by: (None) |
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