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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 17024.) (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 12520 | . . . 4 ⊢ ℕ0 ⊆ ℝ | |
2 | ressxr 11297 | . . . 4 ⊢ ℝ ⊆ ℝ* | |
3 | 1, 2 | sstri 3989 | . . 3 ⊢ ℕ0 ⊆ ℝ* |
4 | pnfxr 11307 | . . . 4 ⊢ +∞ ∈ ℝ* | |
5 | snssi 4808 | . . . 4 ⊢ (+∞ ∈ ℝ* → {+∞} ⊆ ℝ*) | |
6 | 4, 5 | ax-mp 5 | . . 3 ⊢ {+∞} ⊆ ℝ* |
7 | 3, 6 | unssi 4184 | . 2 ⊢ (ℕ0 ∪ {+∞}) ⊆ ℝ* |
8 | ramcl2 17011 | . 2 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ0) → (𝑀 Ramsey 𝐹) ∈ (ℕ0 ∪ {+∞})) | |
9 | 7, 8 | sselid 3977 | 1 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ0) → (𝑀 Ramsey 𝐹) ∈ ℝ*) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1084 ∈ wcel 2099 ∪ cun 3945 ⊆ wss 3947 {csn 4624 ⟶wf 6540 (class class class)co 7414 ℝcr 11146 +∞cpnf 11284 ℝ*cxr 11286 ℕ0cn0 12516 Ramsey cram 16994 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7736 ax-cnex 11203 ax-resscn 11204 ax-1cn 11205 ax-icn 11206 ax-addcl 11207 ax-addrcl 11208 ax-mulcl 11209 ax-mulrcl 11210 ax-mulcom 11211 ax-addass 11212 ax-mulass 11213 ax-distr 11214 ax-i2m1 11215 ax-1ne0 11216 ax-1rid 11217 ax-rnegex 11218 ax-rrecex 11219 ax-cnre 11220 ax-pre-lttri 11221 ax-pre-lttrn 11222 ax-pre-ltadd 11223 ax-pre-mulgt0 11224 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3365 df-reu 3366 df-rab 3421 df-v 3465 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4324 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4907 df-iun 4996 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6303 df-ord 6369 df-on 6370 df-lim 6371 df-suc 6372 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7867 df-1st 7993 df-2nd 7994 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-er 8724 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-sup 9476 df-inf 9477 df-pnf 11289 df-mnf 11290 df-xr 11291 df-ltxr 11292 df-le 11293 df-sub 11485 df-neg 11486 df-nn 12257 df-n0 12517 df-z 12603 df-uz 12867 df-ram 16996 |
This theorem is referenced by: ramlb 17014 |
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