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Mirrors > Home > MPE Home > Th. List > ramub | 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, 22-Apr-2015.) |
Ref | Expression |
---|---|
rami.c | ⊢ 𝐶 = (𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖}) |
rami.m | ⊢ (𝜑 → 𝑀 ∈ ℕ0) |
rami.r | ⊢ (𝜑 → 𝑅 ∈ 𝑉) |
rami.f | ⊢ (𝜑 → 𝐹:𝑅⟶ℕ0) |
ramub.n | ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
ramub.i | ⊢ ((𝜑 ∧ (𝑁 ≤ (♯‘𝑠) ∧ 𝑓:(𝑠𝐶𝑀)⟶𝑅)) → ∃𝑐 ∈ 𝑅 ∃𝑥 ∈ 𝒫 𝑠((𝐹‘𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (◡𝑓 “ {𝑐}))) |
Ref | Expression |
---|---|
ramub | ⊢ (𝜑 → (𝑀 Ramsey 𝐹) ≤ 𝑁) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rami.m | . 2 ⊢ (𝜑 → 𝑀 ∈ ℕ0) | |
2 | rami.r | . 2 ⊢ (𝜑 → 𝑅 ∈ 𝑉) | |
3 | rami.f | . 2 ⊢ (𝜑 → 𝐹:𝑅⟶ℕ0) | |
4 | breq1 5069 | . . . . 5 ⊢ (𝑛 = 𝑁 → (𝑛 ≤ (♯‘𝑠) ↔ 𝑁 ≤ (♯‘𝑠))) | |
5 | 4 | imbi1d 344 | . . . 4 ⊢ (𝑛 = 𝑁 → ((𝑛 ≤ (♯‘𝑠) → ∀𝑓 ∈ (𝑅 ↑m (𝑠𝐶𝑀))∃𝑐 ∈ 𝑅 ∃𝑥 ∈ 𝒫 𝑠((𝐹‘𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (◡𝑓 “ {𝑐}))) ↔ (𝑁 ≤ (♯‘𝑠) → ∀𝑓 ∈ (𝑅 ↑m (𝑠𝐶𝑀))∃𝑐 ∈ 𝑅 ∃𝑥 ∈ 𝒫 𝑠((𝐹‘𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (◡𝑓 “ {𝑐}))))) |
6 | 5 | albidv 1921 | . . 3 ⊢ (𝑛 = 𝑁 → (∀𝑠(𝑛 ≤ (♯‘𝑠) → ∀𝑓 ∈ (𝑅 ↑m (𝑠𝐶𝑀))∃𝑐 ∈ 𝑅 ∃𝑥 ∈ 𝒫 𝑠((𝐹‘𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (◡𝑓 “ {𝑐}))) ↔ ∀𝑠(𝑁 ≤ (♯‘𝑠) → ∀𝑓 ∈ (𝑅 ↑m (𝑠𝐶𝑀))∃𝑐 ∈ 𝑅 ∃𝑥 ∈ 𝒫 𝑠((𝐹‘𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (◡𝑓 “ {𝑐}))))) |
7 | ramub.n | . . 3 ⊢ (𝜑 → 𝑁 ∈ ℕ0) | |
8 | elmapi 8428 | . . . . . 6 ⊢ (𝑓 ∈ (𝑅 ↑m (𝑠𝐶𝑀)) → 𝑓:(𝑠𝐶𝑀)⟶𝑅) | |
9 | ramub.i | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑁 ≤ (♯‘𝑠) ∧ 𝑓:(𝑠𝐶𝑀)⟶𝑅)) → ∃𝑐 ∈ 𝑅 ∃𝑥 ∈ 𝒫 𝑠((𝐹‘𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (◡𝑓 “ {𝑐}))) | |
10 | 9 | ancom2s 648 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑓:(𝑠𝐶𝑀)⟶𝑅 ∧ 𝑁 ≤ (♯‘𝑠))) → ∃𝑐 ∈ 𝑅 ∃𝑥 ∈ 𝒫 𝑠((𝐹‘𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (◡𝑓 “ {𝑐}))) |
11 | 10 | expr 459 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑓:(𝑠𝐶𝑀)⟶𝑅) → (𝑁 ≤ (♯‘𝑠) → ∃𝑐 ∈ 𝑅 ∃𝑥 ∈ 𝒫 𝑠((𝐹‘𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (◡𝑓 “ {𝑐})))) |
12 | 8, 11 | sylan2 594 | . . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑅 ↑m (𝑠𝐶𝑀))) → (𝑁 ≤ (♯‘𝑠) → ∃𝑐 ∈ 𝑅 ∃𝑥 ∈ 𝒫 𝑠((𝐹‘𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (◡𝑓 “ {𝑐})))) |
13 | 12 | ralrimdva 3189 | . . . 4 ⊢ (𝜑 → (𝑁 ≤ (♯‘𝑠) → ∀𝑓 ∈ (𝑅 ↑m (𝑠𝐶𝑀))∃𝑐 ∈ 𝑅 ∃𝑥 ∈ 𝒫 𝑠((𝐹‘𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (◡𝑓 “ {𝑐})))) |
14 | 13 | alrimiv 1928 | . . 3 ⊢ (𝜑 → ∀𝑠(𝑁 ≤ (♯‘𝑠) → ∀𝑓 ∈ (𝑅 ↑m (𝑠𝐶𝑀))∃𝑐 ∈ 𝑅 ∃𝑥 ∈ 𝒫 𝑠((𝐹‘𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (◡𝑓 “ {𝑐})))) |
15 | 6, 7, 14 | elrabd 3682 | . 2 ⊢ (𝜑 → 𝑁 ∈ {𝑛 ∈ ℕ0 ∣ ∀𝑠(𝑛 ≤ (♯‘𝑠) → ∀𝑓 ∈ (𝑅 ↑m (𝑠𝐶𝑀))∃𝑐 ∈ 𝑅 ∃𝑥 ∈ 𝒫 𝑠((𝐹‘𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (◡𝑓 “ {𝑐})))}) |
16 | rami.c | . . 3 ⊢ 𝐶 = (𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖}) | |
17 | eqid 2821 | . . 3 ⊢ {𝑛 ∈ ℕ0 ∣ ∀𝑠(𝑛 ≤ (♯‘𝑠) → ∀𝑓 ∈ (𝑅 ↑m (𝑠𝐶𝑀))∃𝑐 ∈ 𝑅 ∃𝑥 ∈ 𝒫 𝑠((𝐹‘𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (◡𝑓 “ {𝑐})))} = {𝑛 ∈ ℕ0 ∣ ∀𝑠(𝑛 ≤ (♯‘𝑠) → ∀𝑓 ∈ (𝑅 ↑m (𝑠𝐶𝑀))∃𝑐 ∈ 𝑅 ∃𝑥 ∈ 𝒫 𝑠((𝐹‘𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (◡𝑓 “ {𝑐})))} | |
18 | 16, 17 | ramtub 16348 | . 2 ⊢ (((𝑀 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ0) ∧ 𝑁 ∈ {𝑛 ∈ ℕ0 ∣ ∀𝑠(𝑛 ≤ (♯‘𝑠) → ∀𝑓 ∈ (𝑅 ↑m (𝑠𝐶𝑀))∃𝑐 ∈ 𝑅 ∃𝑥 ∈ 𝒫 𝑠((𝐹‘𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (◡𝑓 “ {𝑐})))}) → (𝑀 Ramsey 𝐹) ≤ 𝑁) |
19 | 1, 2, 3, 15, 18 | syl31anc 1369 | 1 ⊢ (𝜑 → (𝑀 Ramsey 𝐹) ≤ 𝑁) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∀wal 1535 = wceq 1537 ∈ wcel 2114 ∀wral 3138 ∃wrex 3139 {crab 3142 Vcvv 3494 ⊆ wss 3936 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 ≤ cle 10676 ℕ0cn0 11898 ♯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: ramub2 16350 0ram 16356 ram0 16358 ramz2 16360 |
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