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Mirrors > Home > MPE Home > Th. List > ramtcl | Structured version Visualization version GIF version |
Description: The Ramsey number has the Ramsey number property if any number does. (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 |
---|---|
ramtcl | ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ0) → ((𝑀 Ramsey 𝐹) ∈ 𝑇 ↔ 𝑇 ≠ ∅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ne0i 4300 | . 2 ⊢ ((𝑀 Ramsey 𝐹) ∈ 𝑇 → 𝑇 ≠ ∅) | |
2 | ramval.c | . . . . . 6 ⊢ 𝐶 = (𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖}) | |
3 | ramval.t | . . . . . 6 ⊢ 𝑇 = {𝑛 ∈ ℕ0 ∣ ∀𝑠(𝑛 ≤ (♯‘𝑠) → ∀𝑓 ∈ (𝑅 ↑m (𝑠𝐶𝑀))∃𝑐 ∈ 𝑅 ∃𝑥 ∈ 𝒫 𝑠((𝐹‘𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (◡𝑓 “ {𝑐})))} | |
4 | 2, 3 | ramcl2lem 16345 | . . . . 5 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ0) → (𝑀 Ramsey 𝐹) = if(𝑇 = ∅, +∞, inf(𝑇, ℝ, < ))) |
5 | ifnefalse 4479 | . . . . 5 ⊢ (𝑇 ≠ ∅ → if(𝑇 = ∅, +∞, inf(𝑇, ℝ, < )) = inf(𝑇, ℝ, < )) | |
6 | 4, 5 | sylan9eq 2876 | . . . 4 ⊢ (((𝑀 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ0) ∧ 𝑇 ≠ ∅) → (𝑀 Ramsey 𝐹) = inf(𝑇, ℝ, < )) |
7 | 3 | ssrab3 4057 | . . . . . . 7 ⊢ 𝑇 ⊆ ℕ0 |
8 | nn0uz 12281 | . . . . . . 7 ⊢ ℕ0 = (ℤ≥‘0) | |
9 | 7, 8 | sseqtri 4003 | . . . . . 6 ⊢ 𝑇 ⊆ (ℤ≥‘0) |
10 | 9 | a1i 11 | . . . . 5 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ0) → 𝑇 ⊆ (ℤ≥‘0)) |
11 | infssuzcl 12333 | . . . . 5 ⊢ ((𝑇 ⊆ (ℤ≥‘0) ∧ 𝑇 ≠ ∅) → inf(𝑇, ℝ, < ) ∈ 𝑇) | |
12 | 10, 11 | sylan 582 | . . . 4 ⊢ (((𝑀 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ0) ∧ 𝑇 ≠ ∅) → inf(𝑇, ℝ, < ) ∈ 𝑇) |
13 | 6, 12 | eqeltrd 2913 | . . 3 ⊢ (((𝑀 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ0) ∧ 𝑇 ≠ ∅) → (𝑀 Ramsey 𝐹) ∈ 𝑇) |
14 | 13 | ex 415 | . 2 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ0) → (𝑇 ≠ ∅ → (𝑀 Ramsey 𝐹) ∈ 𝑇)) |
15 | 1, 14 | impbid2 228 | 1 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ0) → ((𝑀 Ramsey 𝐹) ∈ 𝑇 ↔ 𝑇 ≠ ∅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 ∀wal 1535 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 ∀wral 3138 ∃wrex 3139 {crab 3142 Vcvv 3494 ⊆ wss 3936 ∅c0 4291 ifcif 4467 𝒫 cpw 4539 {csn 4567 class class class wbr 5066 ◡ccnv 5554 “ cima 5558 ⟶wf 6351 ‘cfv 6355 (class class class)co 7156 ∈ cmpo 7158 ↑m cmap 8406 infcinf 8905 ℝcr 10536 0cc0 10537 +∞cpnf 10672 < clt 10675 ≤ cle 10676 ℕ0cn0 11898 ℤ≥cuz 12244 ♯chash 13691 Ramsey cram 16335 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-map 8408 df-en 8510 df-dom 8511 df-sdom 8512 df-sup 8906 df-inf 8907 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-n0 11899 df-z 11983 df-uz 12245 df-ram 16337 |
This theorem is referenced by: ramtcl2 16347 rami 16351 |
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