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Theorem ramsey 14396
Description: Ramsey's theorem with the definition Ramsey eliminated. If 
M is an integer,  R is a specified finite set of colors, and  F : R --> NN0 is a set of lower bounds for each color, then there is an  n such that for every set  s of size greater than  n and every coloring  f of the set  ( s C M ) of all  M-element subsets of  s, there is a color  c and a subset  x  C_  s such that  x is larger than  F (
c ) and the  M-element subsets of  x are monochromatic with color  c. This is the hypergraph version of Ramsey's theorem; the version for simple graphs is the case  M  =  2. This is Metamath 100 proof #31. (Contributed by Mario Carneiro, 23-Apr-2015.)
Hypothesis
Ref Expression
ramsey.c  |-  C  =  ( a  e.  _V ,  i  e.  NN0  |->  { b  e.  ~P a  |  ( # `  b
)  =  i } )
Assertion
Ref Expression
ramsey  |-  ( ( M  e.  NN0  /\  R  e.  Fin  /\  F : R --> NN0 )  ->  E. n  e.  NN0  A. s ( n  <_  ( # `  s
)  ->  A. f  e.  ( R  ^m  (
s C M ) ) E. c  e.  R  E. x  e. 
~P  s ( ( F `  c )  <_  ( # `  x
)  /\  ( x C M )  C_  ( `' f " {
c } ) ) ) )
Distinct variable groups:    f, c, n, s, x, F    a,
b, c, f, i, n, s, x, M    R, c, f, n, s, x    C, c, f, x
Allowed substitution hints:    C( i, n, s, a, b)    R( i, a, b)    F( i, a, b)

Proof of Theorem ramsey
StepHypRef Expression
1 ramcl 14395 . . 3  |-  ( ( M  e.  NN0  /\  R  e.  Fin  /\  F : R --> NN0 )  ->  ( M Ramsey  F )  e.  NN0 )
2 ramsey.c . . . 4  |-  C  =  ( a  e.  _V ,  i  e.  NN0  |->  { b  e.  ~P a  |  ( # `  b
)  =  i } )
3 eqid 2460 . . . 4  |-  { n  e.  NN0  |  A. s
( n  <_  ( # `
 s )  ->  A. f  e.  ( R  ^m  ( s C M ) ) E. c  e.  R  E. x  e.  ~P  s
( ( F `  c )  <_  ( # `
 x )  /\  ( x C M )  C_  ( `' f " { c } ) ) ) }  =  { n  e. 
NN0  |  A. s
( n  <_  ( # `
 s )  ->  A. f  e.  ( R  ^m  ( s C M ) ) E. c  e.  R  E. x  e.  ~P  s
( ( F `  c )  <_  ( # `
 x )  /\  ( x C M )  C_  ( `' f " { c } ) ) ) }
42, 3ramtcl2 14377 . . 3  |-  ( ( M  e.  NN0  /\  R  e.  Fin  /\  F : R --> NN0 )  ->  (
( M Ramsey  F )  e.  NN0  <->  { n  e.  NN0  | 
A. s ( n  <_  ( # `  s
)  ->  A. f  e.  ( R  ^m  (
s C M ) ) E. c  e.  R  E. x  e. 
~P  s ( ( F `  c )  <_  ( # `  x
)  /\  ( x C M )  C_  ( `' f " {
c } ) ) ) }  =/=  (/) ) )
51, 4mpbid 210 . 2  |-  ( ( M  e.  NN0  /\  R  e.  Fin  /\  F : R --> NN0 )  ->  { n  e.  NN0  |  A. s
( n  <_  ( # `
 s )  ->  A. f  e.  ( R  ^m  ( s C M ) ) E. c  e.  R  E. x  e.  ~P  s
( ( F `  c )  <_  ( # `
 x )  /\  ( x C M )  C_  ( `' f " { c } ) ) ) }  =/=  (/) )
6 rabn0 3798 . 2  |-  ( { n  e.  NN0  |  A. s ( n  <_ 
( # `  s )  ->  A. f  e.  ( R  ^m  ( s C M ) ) E. c  e.  R  E. x  e.  ~P  s ( ( F `
 c )  <_ 
( # `  x )  /\  ( x C M )  C_  ( `' f " {
c } ) ) ) }  =/=  (/)  <->  E. n  e.  NN0  A. s ( n  <_  ( # `  s
)  ->  A. f  e.  ( R  ^m  (
s C M ) ) E. c  e.  R  E. x  e. 
~P  s ( ( F `  c )  <_  ( # `  x
)  /\  ( x C M )  C_  ( `' f " {
c } ) ) ) )
75, 6sylib 196 1  |-  ( ( M  e.  NN0  /\  R  e.  Fin  /\  F : R --> NN0 )  ->  E. n  e.  NN0  A. s ( n  <_  ( # `  s
)  ->  A. f  e.  ( R  ^m  (
s C M ) ) E. c  e.  R  E. x  e. 
~P  s ( ( F `  c )  <_  ( # `  x
)  /\  ( x C M )  C_  ( `' f " {
c } ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 968   A.wal 1372    = wceq 1374    e. wcel 1762    =/= wne 2655   A.wral 2807   E.wrex 2808   {crab 2811   _Vcvv 3106    C_ wss 3469   (/)c0 3778   ~Pcpw 4003   {csn 4020   class class class wbr 4440   `'ccnv 4991   "cima 4995   -->wf 5575   ` cfv 5579  (class class class)co 6275    |-> cmpt2 6277    ^m cmap 7410   Fincfn 7506    <_ cle 9618   NN0cn0 10784   #chash 12360   Ramsey cram 14365
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-inf2 8047  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-fal 1380  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-se 4832  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-isom 5588  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-recs 7032  df-rdg 7066  df-1o 7120  df-2o 7121  df-oadd 7124  df-er 7301  df-map 7412  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-sup 7890  df-oi 7924  df-card 8309  df-cda 8537  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-div 10196  df-nn 10526  df-2 10583  df-3 10584  df-n0 10785  df-z 10854  df-uz 11072  df-rp 11210  df-ico 11524  df-fz 11662  df-fzo 11782  df-seq 12064  df-exp 12123  df-fac 12309  df-bc 12336  df-hash 12361  df-cj 12882  df-re 12883  df-im 12884  df-sqr 13018  df-abs 13019  df-clim 13260  df-sum 13458  df-ram 14367
This theorem is referenced by: (None)
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