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Mirrors > Home > MPE Home > Th. List > ramtcl2 | Structured version Visualization version GIF version |
Description: The Ramsey number is an integer iff there is a number with the Ramsey number property. (Contributed by Mario Carneiro, 20-Apr-2015.) (Revised by AV, 14-Sep-2020.) |
Ref | Expression |
---|---|
ramval.c | β’ πΆ = (π β V, π β β0 β¦ {π β π« π β£ (β―βπ) = π}) |
ramval.t | 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})))} |
Ref | Expression |
---|---|
ramtcl2 | β’ ((π β β0 β§ π β π β§ πΉ:π βΆβ0) β ((π Ramsey πΉ) β β0 β π β β )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ramval.c | . . . . 5 β’ πΆ = (π β V, π β β0 β¦ {π β π« π β£ (β―βπ) = π}) | |
2 | ramval.t | . . . . 5 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})))} | |
3 | 1, 2 | ramcl2lem 16985 | . . . 4 β’ ((π β β0 β§ π β π β§ πΉ:π βΆβ0) β (π Ramsey πΉ) = if(π = β , +β, inf(π, β, < ))) |
4 | 3 | eleq1d 2814 | . . 3 β’ ((π β β0 β§ π β π β§ πΉ:π βΆβ0) β ((π Ramsey πΉ) β β0 β if(π = β , +β, inf(π, β, < )) β β0)) |
5 | pnfnre 11293 | . . . . . 6 β’ +β β β | |
6 | 5 | neli 3045 | . . . . 5 β’ Β¬ +β β β |
7 | iftrue 4538 | . . . . . . 7 β’ (π = β β if(π = β , +β, inf(π, β, < )) = +β) | |
8 | 7 | eleq1d 2814 | . . . . . 6 β’ (π = β β (if(π = β , +β, inf(π, β, < )) β β0 β +β β β0)) |
9 | nn0re 12519 | . . . . . 6 β’ (+β β β0 β +β β β) | |
10 | 8, 9 | biimtrdi 252 | . . . . 5 β’ (π = β β (if(π = β , +β, inf(π, β, < )) β β0 β +β β β)) |
11 | 6, 10 | mtoi 198 | . . . 4 β’ (π = β β Β¬ if(π = β , +β, inf(π, β, < )) β β0) |
12 | 11 | necon2ai 2967 | . . 3 β’ (if(π = β , +β, inf(π, β, < )) β β0 β π β β ) |
13 | 4, 12 | biimtrdi 252 | . 2 β’ ((π β β0 β§ π β π β§ πΉ:π βΆβ0) β ((π Ramsey πΉ) β β0 β π β β )) |
14 | 1, 2 | ramtcl 16986 | . . 3 β’ ((π β β0 β§ π β π β§ πΉ:π βΆβ0) β ((π Ramsey πΉ) β π β π β β )) |
15 | 2 | ssrab3 4080 | . . . 4 β’ π β β0 |
16 | 15 | sseli 3978 | . . 3 β’ ((π Ramsey πΉ) β π β (π Ramsey πΉ) β β0) |
17 | 14, 16 | syl6bir 253 | . 2 β’ ((π β β0 β§ π β π β§ πΉ:π βΆβ0) β (π β β β (π Ramsey πΉ) β β0)) |
18 | 13, 17 | impbid 211 | 1 β’ ((π β β0 β§ π β π β§ πΉ:π βΆβ0) β ((π Ramsey πΉ) β β0 β π β β )) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 394 β§ w3a 1084 βwal 1531 = wceq 1533 β wcel 2098 β wne 2937 βwral 3058 βwrex 3067 {crab 3430 Vcvv 3473 β wss 3949 β c0 4326 ifcif 4532 π« cpw 4606 {csn 4632 class class class wbr 5152 β‘ccnv 5681 β cima 5685 βΆwf 6549 βcfv 6553 (class class class)co 7426 β cmpo 7428 βm cmap 8851 infcinf 9472 βcr 11145 +βcpnf 11283 < clt 11286 β€ cle 11287 β0cn0 12510 β―chash 14329 Ramsey cram 16975 |
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 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 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 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-1st 7999 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-er 8731 df-map 8853 df-en 8971 df-dom 8972 df-sdom 8973 df-sup 9473 df-inf 9474 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-nn 12251 df-n0 12511 df-z 12597 df-uz 12861 df-ram 16977 |
This theorem is referenced by: rami 16991 ramcl2 16992 ramsey 17006 |
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