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Mirrors > Home > MPE Home > Th. List > ramtub | Structured version Visualization version GIF version |
Description: The Ramsey number is a lower bound on the set of all numbers with the Ramsey number property. (Contributed by Mario Carneiro, 20-Apr-2015.) (Revised by AV, 14-Sep-2020.) |
Ref | Expression |
---|---|
ramval.c | ⊢ 𝐶 = (𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖}) |
ramval.t | ⊢ 𝑇 = {𝑛 ∈ ℕ0 ∣ ∀𝑠(𝑛 ≤ (♯‘𝑠) → ∀𝑓 ∈ (𝑅 ↑𝑚 (𝑠𝐶𝑀))∃𝑐 ∈ 𝑅 ∃𝑥 ∈ 𝒫 𝑠((𝐹‘𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (◡𝑓 “ {𝑐})))} |
Ref | Expression |
---|---|
ramtub | ⊢ (((𝑀 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ0) ∧ 𝐴 ∈ 𝑇) → (𝑀 Ramsey 𝐹) ≤ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ramval.c | . . . 4 ⊢ 𝐶 = (𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖}) | |
2 | ramval.t | . . . 4 ⊢ 𝑇 = {𝑛 ∈ ℕ0 ∣ ∀𝑠(𝑛 ≤ (♯‘𝑠) → ∀𝑓 ∈ (𝑅 ↑𝑚 (𝑠𝐶𝑀))∃𝑐 ∈ 𝑅 ∃𝑥 ∈ 𝒫 𝑠((𝐹‘𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (◡𝑓 “ {𝑐})))} | |
3 | 1, 2 | ramcl2lem 16191 | . . 3 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ0) → (𝑀 Ramsey 𝐹) = if(𝑇 = ∅, +∞, inf(𝑇, ℝ, < ))) |
4 | n0i 4180 | . . . 4 ⊢ (𝐴 ∈ 𝑇 → ¬ 𝑇 = ∅) | |
5 | 4 | iffalsed 4355 | . . 3 ⊢ (𝐴 ∈ 𝑇 → if(𝑇 = ∅, +∞, inf(𝑇, ℝ, < )) = inf(𝑇, ℝ, < )) |
6 | 3, 5 | sylan9eq 2828 | . 2 ⊢ (((𝑀 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ0) ∧ 𝐴 ∈ 𝑇) → (𝑀 Ramsey 𝐹) = inf(𝑇, ℝ, < )) |
7 | 2 | ssrab3 3943 | . . . . 5 ⊢ 𝑇 ⊆ ℕ0 |
8 | nn0uz 12087 | . . . . 5 ⊢ ℕ0 = (ℤ≥‘0) | |
9 | 7, 8 | sseqtri 3889 | . . . 4 ⊢ 𝑇 ⊆ (ℤ≥‘0) |
10 | 9 | a1i 11 | . . 3 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ0) → 𝑇 ⊆ (ℤ≥‘0)) |
11 | infssuzle 12138 | . . 3 ⊢ ((𝑇 ⊆ (ℤ≥‘0) ∧ 𝐴 ∈ 𝑇) → inf(𝑇, ℝ, < ) ≤ 𝐴) | |
12 | 10, 11 | sylan 572 | . 2 ⊢ (((𝑀 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ0) ∧ 𝐴 ∈ 𝑇) → inf(𝑇, ℝ, < ) ≤ 𝐴) |
13 | 6, 12 | eqbrtrd 4945 | 1 ⊢ (((𝑀 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ0) ∧ 𝐴 ∈ 𝑇) → (𝑀 Ramsey 𝐹) ≤ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 ∧ w3a 1068 ∀wal 1505 = wceq 1507 ∈ wcel 2048 ∀wral 3082 ∃wrex 3083 {crab 3086 Vcvv 3409 ⊆ wss 3825 ∅c0 4173 ifcif 4344 𝒫 cpw 4416 {csn 4435 class class class wbr 4923 ◡ccnv 5399 “ cima 5403 ⟶wf 6178 ‘cfv 6182 (class class class)co 6970 ∈ cmpo 6972 ↑𝑚 cmap 8198 infcinf 8692 ℝcr 10326 0cc0 10327 +∞cpnf 10463 < clt 10466 ≤ cle 10467 ℕ0cn0 11700 ℤ≥cuz 12051 ♯chash 13498 Ramsey cram 16181 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2745 ax-rep 5043 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 ax-cnex 10383 ax-resscn 10384 ax-1cn 10385 ax-icn 10386 ax-addcl 10387 ax-addrcl 10388 ax-mulcl 10389 ax-mulrcl 10390 ax-mulcom 10391 ax-addass 10392 ax-mulass 10393 ax-distr 10394 ax-i2m1 10395 ax-1ne0 10396 ax-1rid 10397 ax-rnegex 10398 ax-rrecex 10399 ax-cnre 10400 ax-pre-lttri 10401 ax-pre-lttrn 10402 ax-pre-ltadd 10403 ax-pre-mulgt0 10404 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2014 df-mo 2544 df-eu 2580 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rmo 3090 df-rab 3091 df-v 3411 df-sbc 3678 df-csb 3783 df-dif 3828 df-un 3830 df-in 3832 df-ss 3839 df-pss 3841 df-nul 4174 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4707 df-iun 4788 df-br 4924 df-opab 4986 df-mpt 5003 df-tr 5025 df-id 5305 df-eprel 5310 df-po 5319 df-so 5320 df-fr 5359 df-we 5361 df-xp 5406 df-rel 5407 df-cnv 5408 df-co 5409 df-dm 5410 df-rn 5411 df-res 5412 df-ima 5413 df-pred 5980 df-ord 6026 df-on 6027 df-lim 6028 df-suc 6029 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-riota 6931 df-ov 6973 df-oprab 6974 df-mpo 6975 df-om 7391 df-1st 7494 df-2nd 7495 df-wrecs 7743 df-recs 7805 df-rdg 7843 df-er 8081 df-map 8200 df-en 8299 df-dom 8300 df-sdom 8301 df-sup 8693 df-inf 8694 df-pnf 10468 df-mnf 10469 df-xr 10470 df-ltxr 10471 df-le 10472 df-sub 10664 df-neg 10665 df-nn 11432 df-n0 11701 df-z 11787 df-uz 12052 df-ram 16183 |
This theorem is referenced by: ramub 16195 |
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