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Mirrors > Home > MPE Home > Th. List > ramubcl | Structured version Visualization version GIF version |
Description: If the Ramsey number is upper bounded, then it is an integer. (Contributed by Mario Carneiro, 20-Apr-2015.) |
Ref | Expression |
---|---|
ramubcl | ⊢ (((𝑀 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ0) ∧ (𝐴 ∈ ℕ0 ∧ (𝑀 Ramsey 𝐹) ≤ 𝐴)) → (𝑀 Ramsey 𝐹) ∈ ℕ0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nn0re 12225 | . . . . . 6 ⊢ (𝐴 ∈ ℕ0 → 𝐴 ∈ ℝ) | |
2 | ltpnf 12838 | . . . . . . 7 ⊢ (𝐴 ∈ ℝ → 𝐴 < +∞) | |
3 | rexr 11005 | . . . . . . . 8 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ*) | |
4 | pnfxr 11013 | . . . . . . . 8 ⊢ +∞ ∈ ℝ* | |
5 | xrltnle 11026 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℝ* ∧ +∞ ∈ ℝ*) → (𝐴 < +∞ ↔ ¬ +∞ ≤ 𝐴)) | |
6 | 3, 4, 5 | sylancl 585 | . . . . . . 7 ⊢ (𝐴 ∈ ℝ → (𝐴 < +∞ ↔ ¬ +∞ ≤ 𝐴)) |
7 | 2, 6 | mpbid 231 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → ¬ +∞ ≤ 𝐴) |
8 | 1, 7 | syl 17 | . . . . 5 ⊢ (𝐴 ∈ ℕ0 → ¬ +∞ ≤ 𝐴) |
9 | 8 | ad2antrl 724 | . . . 4 ⊢ (((𝑀 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ0) ∧ (𝐴 ∈ ℕ0 ∧ (𝑀 Ramsey 𝐹) ≤ 𝐴)) → ¬ +∞ ≤ 𝐴) |
10 | simprr 769 | . . . . 5 ⊢ (((𝑀 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ0) ∧ (𝐴 ∈ ℕ0 ∧ (𝑀 Ramsey 𝐹) ≤ 𝐴)) → (𝑀 Ramsey 𝐹) ≤ 𝐴) | |
11 | breq1 5081 | . . . . 5 ⊢ ((𝑀 Ramsey 𝐹) = +∞ → ((𝑀 Ramsey 𝐹) ≤ 𝐴 ↔ +∞ ≤ 𝐴)) | |
12 | 10, 11 | syl5ibcom 244 | . . . 4 ⊢ (((𝑀 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ0) ∧ (𝐴 ∈ ℕ0 ∧ (𝑀 Ramsey 𝐹) ≤ 𝐴)) → ((𝑀 Ramsey 𝐹) = +∞ → +∞ ≤ 𝐴)) |
13 | 9, 12 | mtod 197 | . . 3 ⊢ (((𝑀 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ0) ∧ (𝐴 ∈ ℕ0 ∧ (𝑀 Ramsey 𝐹) ≤ 𝐴)) → ¬ (𝑀 Ramsey 𝐹) = +∞) |
14 | elsni 4583 | . . 3 ⊢ ((𝑀 Ramsey 𝐹) ∈ {+∞} → (𝑀 Ramsey 𝐹) = +∞) | |
15 | 13, 14 | nsyl 140 | . 2 ⊢ (((𝑀 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ0) ∧ (𝐴 ∈ ℕ0 ∧ (𝑀 Ramsey 𝐹) ≤ 𝐴)) → ¬ (𝑀 Ramsey 𝐹) ∈ {+∞}) |
16 | ramcl2 16698 | . . . . 5 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ0) → (𝑀 Ramsey 𝐹) ∈ (ℕ0 ∪ {+∞})) | |
17 | 16 | adantr 480 | . . . 4 ⊢ (((𝑀 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ0) ∧ (𝐴 ∈ ℕ0 ∧ (𝑀 Ramsey 𝐹) ≤ 𝐴)) → (𝑀 Ramsey 𝐹) ∈ (ℕ0 ∪ {+∞})) |
18 | elun 4087 | . . . 4 ⊢ ((𝑀 Ramsey 𝐹) ∈ (ℕ0 ∪ {+∞}) ↔ ((𝑀 Ramsey 𝐹) ∈ ℕ0 ∨ (𝑀 Ramsey 𝐹) ∈ {+∞})) | |
19 | 17, 18 | sylib 217 | . . 3 ⊢ (((𝑀 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ0) ∧ (𝐴 ∈ ℕ0 ∧ (𝑀 Ramsey 𝐹) ≤ 𝐴)) → ((𝑀 Ramsey 𝐹) ∈ ℕ0 ∨ (𝑀 Ramsey 𝐹) ∈ {+∞})) |
20 | 19 | ord 860 | . 2 ⊢ (((𝑀 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ0) ∧ (𝐴 ∈ ℕ0 ∧ (𝑀 Ramsey 𝐹) ≤ 𝐴)) → (¬ (𝑀 Ramsey 𝐹) ∈ ℕ0 → (𝑀 Ramsey 𝐹) ∈ {+∞})) |
21 | 15, 20 | mt3d 148 | 1 ⊢ (((𝑀 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ0) ∧ (𝐴 ∈ ℕ0 ∧ (𝑀 Ramsey 𝐹) ≤ 𝐴)) → (𝑀 Ramsey 𝐹) ∈ ℕ0) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 ∨ wo 843 ∧ w3a 1085 = wceq 1541 ∈ wcel 2109 ∪ cun 3889 {csn 4566 class class class wbr 5078 ⟶wf 6426 (class class class)co 7268 ℝcr 10854 +∞cpnf 10990 ℝ*cxr 10992 < clt 10993 ≤ cle 10994 ℕ0cn0 12216 Ramsey cram 16681 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 ax-5 1916 ax-6 1974 ax-7 2014 ax-8 2111 ax-9 2119 ax-10 2140 ax-11 2157 ax-12 2174 ax-ext 2710 ax-rep 5213 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7579 ax-cnex 10911 ax-resscn 10912 ax-1cn 10913 ax-icn 10914 ax-addcl 10915 ax-addrcl 10916 ax-mulcl 10917 ax-mulrcl 10918 ax-mulcom 10919 ax-addass 10920 ax-mulass 10921 ax-distr 10922 ax-i2m1 10923 ax-1ne0 10924 ax-1rid 10925 ax-rnegex 10926 ax-rrecex 10927 ax-cnre 10928 ax-pre-lttri 10929 ax-pre-lttrn 10930 ax-pre-ltadd 10931 ax-pre-mulgt0 10932 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1544 df-fal 1554 df-ex 1786 df-nf 1790 df-sb 2071 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3070 df-rex 3071 df-reu 3072 df-rmo 3073 df-rab 3074 df-v 3432 df-sbc 3720 df-csb 3837 df-dif 3894 df-un 3896 df-in 3898 df-ss 3908 df-pss 3910 df-nul 4262 df-if 4465 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4845 df-iun 4931 df-br 5079 df-opab 5141 df-mpt 5162 df-tr 5196 df-id 5488 df-eprel 5494 df-po 5502 df-so 5503 df-fr 5543 df-we 5545 df-xp 5594 df-rel 5595 df-cnv 5596 df-co 5597 df-dm 5598 df-rn 5599 df-res 5600 df-ima 5601 df-pred 6199 df-ord 6266 df-on 6267 df-lim 6268 df-suc 6269 df-iota 6388 df-fun 6432 df-fn 6433 df-f 6434 df-f1 6435 df-fo 6436 df-f1o 6437 df-fv 6438 df-riota 7225 df-ov 7271 df-oprab 7272 df-mpo 7273 df-om 7701 df-1st 7817 df-2nd 7818 df-frecs 8081 df-wrecs 8112 df-recs 8186 df-rdg 8225 df-er 8472 df-map 8591 df-en 8708 df-dom 8709 df-sdom 8710 df-sup 9162 df-inf 9163 df-pnf 10995 df-mnf 10996 df-xr 10997 df-ltxr 10998 df-le 10999 df-sub 11190 df-neg 11191 df-nn 11957 df-n0 12217 df-z 12303 df-uz 12565 df-ram 16683 |
This theorem is referenced by: ramlb 16701 0ram 16702 ram0 16704 ramz2 16706 ramcl 16711 |
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