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Mirrors > Home > MPE Home > Th. List > ramxrcl | Structured version Visualization version GIF version |
Description: The Ramsey number is an extended real number. (This theorem does not imply Ramsey's theorem, unlike ramcl 16758.) (Contributed by Mario Carneiro, 20-Apr-2015.) |
Ref | Expression |
---|---|
ramxrcl | ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ0) → (𝑀 Ramsey 𝐹) ∈ ℝ*) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nn0ssre 12265 | . . . 4 ⊢ ℕ0 ⊆ ℝ | |
2 | ressxr 11047 | . . . 4 ⊢ ℝ ⊆ ℝ* | |
3 | 1, 2 | sstri 3932 | . . 3 ⊢ ℕ0 ⊆ ℝ* |
4 | pnfxr 11057 | . . . 4 ⊢ +∞ ∈ ℝ* | |
5 | snssi 4744 | . . . 4 ⊢ (+∞ ∈ ℝ* → {+∞} ⊆ ℝ*) | |
6 | 4, 5 | ax-mp 5 | . . 3 ⊢ {+∞} ⊆ ℝ* |
7 | 3, 6 | unssi 4122 | . 2 ⊢ (ℕ0 ∪ {+∞}) ⊆ ℝ* |
8 | ramcl2 16745 | . 2 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ0) → (𝑀 Ramsey 𝐹) ∈ (ℕ0 ∪ {+∞})) | |
9 | 7, 8 | sselid 3921 | 1 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ0) → (𝑀 Ramsey 𝐹) ∈ ℝ*) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1085 ∈ wcel 2101 ∪ cun 3887 ⊆ wss 3889 {csn 4564 ⟶wf 6443 (class class class)co 7295 ℝcr 10898 +∞cpnf 11034 ℝ*cxr 11036 ℕ0cn0 12261 Ramsey cram 16728 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2103 ax-9 2111 ax-10 2132 ax-11 2149 ax-12 2166 ax-ext 2704 ax-rep 5212 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7608 ax-cnex 10955 ax-resscn 10956 ax-1cn 10957 ax-icn 10958 ax-addcl 10959 ax-addrcl 10960 ax-mulcl 10961 ax-mulrcl 10962 ax-mulcom 10963 ax-addass 10964 ax-mulass 10965 ax-distr 10966 ax-i2m1 10967 ax-1ne0 10968 ax-1rid 10969 ax-rnegex 10970 ax-rrecex 10971 ax-cnre 10972 ax-pre-lttri 10973 ax-pre-lttrn 10974 ax-pre-ltadd 10975 ax-pre-mulgt0 10976 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2063 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2884 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3222 df-reu 3223 df-rab 3224 df-v 3436 df-sbc 3719 df-csb 3835 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3908 df-nul 4260 df-if 4463 df-pw 4538 df-sn 4565 df-pr 4567 df-op 4571 df-uni 4842 df-iun 4929 df-br 5078 df-opab 5140 df-mpt 5161 df-tr 5195 df-id 5491 df-eprel 5497 df-po 5505 df-so 5506 df-fr 5546 df-we 5548 df-xp 5597 df-rel 5598 df-cnv 5599 df-co 5600 df-dm 5601 df-rn 5602 df-res 5603 df-ima 5604 df-pred 6206 df-ord 6273 df-on 6274 df-lim 6275 df-suc 6276 df-iota 6399 df-fun 6449 df-fn 6450 df-f 6451 df-f1 6452 df-fo 6453 df-f1o 6454 df-fv 6455 df-riota 7252 df-ov 7298 df-oprab 7299 df-mpo 7300 df-om 7733 df-1st 7851 df-2nd 7852 df-frecs 8117 df-wrecs 8148 df-recs 8222 df-rdg 8261 df-er 8518 df-map 8637 df-en 8754 df-dom 8755 df-sdom 8756 df-sup 9229 df-inf 9230 df-pnf 11039 df-mnf 11040 df-xr 11041 df-ltxr 11042 df-le 11043 df-sub 11235 df-neg 11236 df-nn 12002 df-n0 12262 df-z 12348 df-uz 12611 df-ram 16730 |
This theorem is referenced by: ramlb 16748 |
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