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Theorem ramcl2 15940
Description: The Ramsey number is either a nonnegative integer or plus infinity. (Contributed by Mario Carneiro, 20-Apr-2015.) (Revised by AV, 14-Sep-2020.)
Assertion
Ref Expression
ramcl2 ((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) → (𝑀 Ramsey 𝐹) ∈ (ℕ0 ∪ {+∞}))

Proof of Theorem ramcl2
Dummy variables 𝑓 𝑐 𝑛 𝑠 𝑥 𝑎 𝑏 𝑖 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2813 . . . . 5 (𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖}) = (𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})
2 eqid 2813 . . . . 5 {𝑛 ∈ ℕ0 ∣ ∀𝑠(𝑛 ≤ (♯‘𝑠) → ∀𝑓 ∈ (𝑅𝑚 (𝑠(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})𝑀))∃𝑐𝑅𝑥 ∈ 𝒫 𝑠((𝐹𝑐) ≤ (♯‘𝑥) ∧ (𝑥(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})𝑀) ⊆ (𝑓 “ {𝑐})))} = {𝑛 ∈ ℕ0 ∣ ∀𝑠(𝑛 ≤ (♯‘𝑠) → ∀𝑓 ∈ (𝑅𝑚 (𝑠(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})𝑀))∃𝑐𝑅𝑥 ∈ 𝒫 𝑠((𝐹𝑐) ≤ (♯‘𝑥) ∧ (𝑥(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})𝑀) ⊆ (𝑓 “ {𝑐})))}
31, 2ramcl2lem 15933 . . . 4 ((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) → (𝑀 Ramsey 𝐹) = if({𝑛 ∈ ℕ0 ∣ ∀𝑠(𝑛 ≤ (♯‘𝑠) → ∀𝑓 ∈ (𝑅𝑚 (𝑠(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})𝑀))∃𝑐𝑅𝑥 ∈ 𝒫 𝑠((𝐹𝑐) ≤ (♯‘𝑥) ∧ (𝑥(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})𝑀) ⊆ (𝑓 “ {𝑐})))} = ∅, +∞, inf({𝑛 ∈ ℕ0 ∣ ∀𝑠(𝑛 ≤ (♯‘𝑠) → ∀𝑓 ∈ (𝑅𝑚 (𝑠(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})𝑀))∃𝑐𝑅𝑥 ∈ 𝒫 𝑠((𝐹𝑐) ≤ (♯‘𝑥) ∧ (𝑥(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})𝑀) ⊆ (𝑓 “ {𝑐})))}, ℝ, < )))
4 iftrue 4292 . . . 4 ({𝑛 ∈ ℕ0 ∣ ∀𝑠(𝑛 ≤ (♯‘𝑠) → ∀𝑓 ∈ (𝑅𝑚 (𝑠(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})𝑀))∃𝑐𝑅𝑥 ∈ 𝒫 𝑠((𝐹𝑐) ≤ (♯‘𝑥) ∧ (𝑥(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})𝑀) ⊆ (𝑓 “ {𝑐})))} = ∅ → if({𝑛 ∈ ℕ0 ∣ ∀𝑠(𝑛 ≤ (♯‘𝑠) → ∀𝑓 ∈ (𝑅𝑚 (𝑠(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})𝑀))∃𝑐𝑅𝑥 ∈ 𝒫 𝑠((𝐹𝑐) ≤ (♯‘𝑥) ∧ (𝑥(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})𝑀) ⊆ (𝑓 “ {𝑐})))} = ∅, +∞, inf({𝑛 ∈ ℕ0 ∣ ∀𝑠(𝑛 ≤ (♯‘𝑠) → ∀𝑓 ∈ (𝑅𝑚 (𝑠(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})𝑀))∃𝑐𝑅𝑥 ∈ 𝒫 𝑠((𝐹𝑐) ≤ (♯‘𝑥) ∧ (𝑥(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})𝑀) ⊆ (𝑓 “ {𝑐})))}, ℝ, < )) = +∞)
53, 4sylan9eq 2867 . . 3 (((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ {𝑛 ∈ ℕ0 ∣ ∀𝑠(𝑛 ≤ (♯‘𝑠) → ∀𝑓 ∈ (𝑅𝑚 (𝑠(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})𝑀))∃𝑐𝑅𝑥 ∈ 𝒫 𝑠((𝐹𝑐) ≤ (♯‘𝑥) ∧ (𝑥(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})𝑀) ⊆ (𝑓 “ {𝑐})))} = ∅) → (𝑀 Ramsey 𝐹) = +∞)
6 ssun2 3983 . . . 4 {+∞} ⊆ (ℕ0 ∪ {+∞})
7 pnfex 10381 . . . . 5 +∞ ∈ V
87snss 4513 . . . 4 (+∞ ∈ (ℕ0 ∪ {+∞}) ↔ {+∞} ⊆ (ℕ0 ∪ {+∞}))
96, 8mpbir 222 . . 3 +∞ ∈ (ℕ0 ∪ {+∞})
105, 9syl6eqel 2900 . 2 (((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ {𝑛 ∈ ℕ0 ∣ ∀𝑠(𝑛 ≤ (♯‘𝑠) → ∀𝑓 ∈ (𝑅𝑚 (𝑠(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})𝑀))∃𝑐𝑅𝑥 ∈ 𝒫 𝑠((𝐹𝑐) ≤ (♯‘𝑥) ∧ (𝑥(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})𝑀) ⊆ (𝑓 “ {𝑐})))} = ∅) → (𝑀 Ramsey 𝐹) ∈ (ℕ0 ∪ {+∞}))
11 ssun1 3982 . . 3 0 ⊆ (ℕ0 ∪ {+∞})
121, 2ramtcl2 15935 . . . 4 ((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) → ((𝑀 Ramsey 𝐹) ∈ ℕ0 ↔ {𝑛 ∈ ℕ0 ∣ ∀𝑠(𝑛 ≤ (♯‘𝑠) → ∀𝑓 ∈ (𝑅𝑚 (𝑠(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})𝑀))∃𝑐𝑅𝑥 ∈ 𝒫 𝑠((𝐹𝑐) ≤ (♯‘𝑥) ∧ (𝑥(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})𝑀) ⊆ (𝑓 “ {𝑐})))} ≠ ∅))
1312biimpar 465 . . 3 (((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ {𝑛 ∈ ℕ0 ∣ ∀𝑠(𝑛 ≤ (♯‘𝑠) → ∀𝑓 ∈ (𝑅𝑚 (𝑠(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})𝑀))∃𝑐𝑅𝑥 ∈ 𝒫 𝑠((𝐹𝑐) ≤ (♯‘𝑥) ∧ (𝑥(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})𝑀) ⊆ (𝑓 “ {𝑐})))} ≠ ∅) → (𝑀 Ramsey 𝐹) ∈ ℕ0)
1411, 13sseldi 3803 . 2 (((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ {𝑛 ∈ ℕ0 ∣ ∀𝑠(𝑛 ≤ (♯‘𝑠) → ∀𝑓 ∈ (𝑅𝑚 (𝑠(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})𝑀))∃𝑐𝑅𝑥 ∈ 𝒫 𝑠((𝐹𝑐) ≤ (♯‘𝑥) ∧ (𝑥(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})𝑀) ⊆ (𝑓 “ {𝑐})))} ≠ ∅) → (𝑀 Ramsey 𝐹) ∈ (ℕ0 ∪ {+∞}))
1510, 14pm2.61dane 3072 1 ((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) → (𝑀 Ramsey 𝐹) ∈ (ℕ0 ∪ {+∞}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1100  wal 1635   = wceq 1637  wcel 2157  wne 2985  wral 3103  wrex 3104  {crab 3107  Vcvv 3398  cun 3774  wss 3776  c0 4123  ifcif 4286  𝒫 cpw 4358  {csn 4377   class class class wbr 4851  ccnv 5317  cima 5321  wf 6100  cfv 6104  (class class class)co 6877  cmpt2 6879  𝑚 cmap 8095  infcinf 8589  cr 10223  +∞cpnf 10359   < clt 10362  cle 10363  0cn0 11562  chash 13340   Ramsey cram 15923
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-8 2159  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2791  ax-rep 4971  ax-sep 4982  ax-nul 4990  ax-pow 5042  ax-pr 5103  ax-un 7182  ax-cnex 10280  ax-resscn 10281  ax-1cn 10282  ax-icn 10283  ax-addcl 10284  ax-addrcl 10285  ax-mulcl 10286  ax-mulrcl 10287  ax-mulcom 10288  ax-addass 10289  ax-mulass 10290  ax-distr 10291  ax-i2m1 10292  ax-1ne0 10293  ax-1rid 10294  ax-rnegex 10295  ax-rrecex 10296  ax-cnre 10297  ax-pre-lttri 10298  ax-pre-lttrn 10299  ax-pre-ltadd 10300  ax-pre-mulgt0 10301
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2638  df-clab 2800  df-cleq 2806  df-clel 2809  df-nfc 2944  df-ne 2986  df-nel 3089  df-ral 3108  df-rex 3109  df-reu 3110  df-rmo 3111  df-rab 3112  df-v 3400  df-sbc 3641  df-csb 3736  df-dif 3779  df-un 3781  df-in 3783  df-ss 3790  df-pss 3792  df-nul 4124  df-if 4287  df-pw 4360  df-sn 4378  df-pr 4380  df-tp 4382  df-op 4384  df-uni 4638  df-iun 4721  df-br 4852  df-opab 4914  df-mpt 4931  df-tr 4954  df-id 5226  df-eprel 5231  df-po 5239  df-so 5240  df-fr 5277  df-we 5279  df-xp 5324  df-rel 5325  df-cnv 5326  df-co 5327  df-dm 5328  df-rn 5329  df-res 5330  df-ima 5331  df-pred 5900  df-ord 5946  df-on 5947  df-lim 5948  df-suc 5949  df-iota 6067  df-fun 6106  df-fn 6107  df-f 6108  df-f1 6109  df-fo 6110  df-f1o 6111  df-fv 6112  df-riota 6838  df-ov 6880  df-oprab 6881  df-mpt2 6882  df-om 7299  df-1st 7401  df-2nd 7402  df-wrecs 7645  df-recs 7707  df-rdg 7745  df-er 7982  df-map 8097  df-en 8196  df-dom 8197  df-sdom 8198  df-sup 8590  df-inf 8591  df-pnf 10364  df-mnf 10365  df-xr 10366  df-ltxr 10367  df-le 10368  df-sub 10556  df-neg 10557  df-nn 11309  df-n0 11563  df-z 11647  df-uz 11908  df-ram 15925
This theorem is referenced by:  ramxrcl  15941  ramubcl  15942
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