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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 16562 | . . 3 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ0) → (𝑀 Ramsey 𝐹) = if(𝑇 = ∅, +∞, inf(𝑇, ℝ, < ))) |
4 | n0i 4248 | . . . 4 ⊢ (𝐴 ∈ 𝑇 → ¬ 𝑇 = ∅) | |
5 | 4 | iffalsed 4450 | . . 3 ⊢ (𝐴 ∈ 𝑇 → if(𝑇 = ∅, +∞, inf(𝑇, ℝ, < )) = inf(𝑇, ℝ, < )) |
6 | 3, 5 | sylan9eq 2798 | . 2 ⊢ (((𝑀 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ0) ∧ 𝐴 ∈ 𝑇) → (𝑀 Ramsey 𝐹) = inf(𝑇, ℝ, < )) |
7 | 2 | ssrab3 3995 | . . . . 5 ⊢ 𝑇 ⊆ ℕ0 |
8 | nn0uz 12476 | . . . . 5 ⊢ ℕ0 = (ℤ≥‘0) | |
9 | 7, 8 | sseqtri 3937 | . . . 4 ⊢ 𝑇 ⊆ (ℤ≥‘0) |
10 | 9 | a1i 11 | . . 3 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ0) → 𝑇 ⊆ (ℤ≥‘0)) |
11 | infssuzle 12527 | . . 3 ⊢ ((𝑇 ⊆ (ℤ≥‘0) ∧ 𝐴 ∈ 𝑇) → inf(𝑇, ℝ, < ) ≤ 𝐴) | |
12 | 10, 11 | sylan 583 | . 2 ⊢ (((𝑀 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ0) ∧ 𝐴 ∈ 𝑇) → inf(𝑇, ℝ, < ) ≤ 𝐴) |
13 | 6, 12 | eqbrtrd 5075 | 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 399 ∧ w3a 1089 ∀wal 1541 = wceq 1543 ∈ wcel 2110 ∀wral 3061 ∃wrex 3062 {crab 3065 Vcvv 3408 ⊆ wss 3866 ∅c0 4237 ifcif 4439 𝒫 cpw 4513 {csn 4541 class class class wbr 5053 ◡ccnv 5550 “ cima 5554 ⟶wf 6376 ‘cfv 6380 (class class class)co 7213 ∈ cmpo 7215 ↑m cmap 8508 infcinf 9057 ℝcr 10728 0cc0 10729 +∞cpnf 10864 < clt 10867 ≤ cle 10868 ℕ0cn0 12090 ℤ≥cuz 12438 ♯chash 13896 Ramsey cram 16552 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-om 7645 df-1st 7761 df-2nd 7762 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-er 8391 df-map 8510 df-en 8627 df-dom 8628 df-sdom 8629 df-sup 9058 df-inf 9059 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-nn 11831 df-n0 12091 df-z 12177 df-uz 12439 df-ram 16554 |
This theorem is referenced by: ramub 16566 |
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