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Theorem 0ram 15926
 Description: The Ramsey number when 𝑀 = 0. (Contributed by Mario Carneiro, 22-Apr-2015.)
Assertion
Ref Expression
0ram (((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) → (0 Ramsey 𝐹) = sup(ran 𝐹, ℝ, < ))
Distinct variable groups:   𝑥,𝑦,𝑅   𝑥,𝐹,𝑦   𝑥,𝑉
Allowed substitution hint:   𝑉(𝑦)

Proof of Theorem 0ram
Dummy variables 𝑏 𝑑 𝑧 𝑓 𝑐 𝑠 𝑎 𝑖 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2760 . . 3 (𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖}) = (𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})
2 0nn0 11499 . . . 4 0 ∈ ℕ0
32a1i 11 . . 3 (((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) → 0 ∈ ℕ0)
4 simpl1 1228 . . 3 (((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) → 𝑅𝑉)
5 simpl3 1232 . . 3 (((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) → 𝐹:𝑅⟶ℕ0)
6 frn 6214 . . . . 5 (𝐹:𝑅⟶ℕ0 → ran 𝐹 ⊆ ℕ0)
75, 6syl 17 . . . 4 (((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) → ran 𝐹 ⊆ ℕ0)
8 nn0ssz 11590 . . . . . 6 0 ⊆ ℤ
97, 8syl6ss 3756 . . . . 5 (((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) → ran 𝐹 ⊆ ℤ)
10 fdm 6212 . . . . . . . 8 (𝐹:𝑅⟶ℕ0 → dom 𝐹 = 𝑅)
115, 10syl 17 . . . . . . 7 (((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) → dom 𝐹 = 𝑅)
12 simpl2 1230 . . . . . . 7 (((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) → 𝑅 ≠ ∅)
1311, 12eqnetrd 2999 . . . . . 6 (((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) → dom 𝐹 ≠ ∅)
14 dm0rn0 5497 . . . . . . 7 (dom 𝐹 = ∅ ↔ ran 𝐹 = ∅)
1514necon3bii 2984 . . . . . 6 (dom 𝐹 ≠ ∅ ↔ ran 𝐹 ≠ ∅)
1613, 15sylib 208 . . . . 5 (((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) → ran 𝐹 ≠ ∅)
17 simpr 479 . . . . 5 (((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) → ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥)
18 suprzcl2 11971 . . . . 5 ((ran 𝐹 ⊆ ℤ ∧ ran 𝐹 ≠ ∅ ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) → sup(ran 𝐹, ℝ, < ) ∈ ran 𝐹)
199, 16, 17, 18syl3anc 1477 . . . 4 (((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) → sup(ran 𝐹, ℝ, < ) ∈ ran 𝐹)
207, 19sseldd 3745 . . 3 (((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) → sup(ran 𝐹, ℝ, < ) ∈ ℕ0)
21 vex 3343 . . . . . . 7 𝑠 ∈ V
221hashbc0 15911 . . . . . . 7 (𝑠 ∈ V → (𝑠(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})0) = {∅})
2321, 22ax-mp 5 . . . . . 6 (𝑠(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})0) = {∅}
2423feq2i 6198 . . . . 5 (𝑓:(𝑠(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})0)⟶𝑅𝑓:{∅}⟶𝑅)
2524biimpi 206 . . . 4 (𝑓:(𝑠(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})0)⟶𝑅𝑓:{∅}⟶𝑅)
26 simprr 813 . . . . . 6 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (sup(ran 𝐹, ℝ, < ) ≤ (♯‘𝑠) ∧ 𝑓:{∅}⟶𝑅)) → 𝑓:{∅}⟶𝑅)
27 0ex 4942 . . . . . . 7 ∅ ∈ V
2827snid 4353 . . . . . 6 ∅ ∈ {∅}
29 ffvelrn 6520 . . . . . 6 ((𝑓:{∅}⟶𝑅 ∧ ∅ ∈ {∅}) → (𝑓‘∅) ∈ 𝑅)
3026, 28, 29sylancl 697 . . . . 5 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (sup(ran 𝐹, ℝ, < ) ≤ (♯‘𝑠) ∧ 𝑓:{∅}⟶𝑅)) → (𝑓‘∅) ∈ 𝑅)
3121pwid 4318 . . . . . 6 𝑠 ∈ 𝒫 𝑠
3231a1i 11 . . . . 5 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (sup(ran 𝐹, ℝ, < ) ≤ (♯‘𝑠) ∧ 𝑓:{∅}⟶𝑅)) → 𝑠 ∈ 𝒫 𝑠)
335adantr 472 . . . . . . . . 9 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (sup(ran 𝐹, ℝ, < ) ≤ (♯‘𝑠) ∧ 𝑓:{∅}⟶𝑅)) → 𝐹:𝑅⟶ℕ0)
3433, 30ffvelrnd 6523 . . . . . . . 8 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (sup(ran 𝐹, ℝ, < ) ≤ (♯‘𝑠) ∧ 𝑓:{∅}⟶𝑅)) → (𝐹‘(𝑓‘∅)) ∈ ℕ0)
3534nn0red 11544 . . . . . . 7 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (sup(ran 𝐹, ℝ, < ) ≤ (♯‘𝑠) ∧ 𝑓:{∅}⟶𝑅)) → (𝐹‘(𝑓‘∅)) ∈ ℝ)
3635rexrd 10281 . . . . . 6 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (sup(ran 𝐹, ℝ, < ) ≤ (♯‘𝑠) ∧ 𝑓:{∅}⟶𝑅)) → (𝐹‘(𝑓‘∅)) ∈ ℝ*)
3720nn0red 11544 . . . . . . . 8 (((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) → sup(ran 𝐹, ℝ, < ) ∈ ℝ)
3837rexrd 10281 . . . . . . 7 (((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) → sup(ran 𝐹, ℝ, < ) ∈ ℝ*)
3938adantr 472 . . . . . 6 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (sup(ran 𝐹, ℝ, < ) ≤ (♯‘𝑠) ∧ 𝑓:{∅}⟶𝑅)) → sup(ran 𝐹, ℝ, < ) ∈ ℝ*)
40 hashxrcl 13340 . . . . . . 7 (𝑠 ∈ V → (♯‘𝑠) ∈ ℝ*)
4121, 40mp1i 13 . . . . . 6 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (sup(ran 𝐹, ℝ, < ) ≤ (♯‘𝑠) ∧ 𝑓:{∅}⟶𝑅)) → (♯‘𝑠) ∈ ℝ*)
429adantr 472 . . . . . . 7 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (sup(ran 𝐹, ℝ, < ) ≤ (♯‘𝑠) ∧ 𝑓:{∅}⟶𝑅)) → ran 𝐹 ⊆ ℤ)
4317adantr 472 . . . . . . 7 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (sup(ran 𝐹, ℝ, < ) ≤ (♯‘𝑠) ∧ 𝑓:{∅}⟶𝑅)) → ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥)
44 ffn 6206 . . . . . . . . 9 (𝐹:𝑅⟶ℕ0𝐹 Fn 𝑅)
4533, 44syl 17 . . . . . . . 8 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (sup(ran 𝐹, ℝ, < ) ≤ (♯‘𝑠) ∧ 𝑓:{∅}⟶𝑅)) → 𝐹 Fn 𝑅)
46 fnfvelrn 6519 . . . . . . . 8 ((𝐹 Fn 𝑅 ∧ (𝑓‘∅) ∈ 𝑅) → (𝐹‘(𝑓‘∅)) ∈ ran 𝐹)
4745, 30, 46syl2anc 696 . . . . . . 7 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (sup(ran 𝐹, ℝ, < ) ≤ (♯‘𝑠) ∧ 𝑓:{∅}⟶𝑅)) → (𝐹‘(𝑓‘∅)) ∈ ran 𝐹)
48 suprzub 11972 . . . . . . 7 ((ran 𝐹 ⊆ ℤ ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥 ∧ (𝐹‘(𝑓‘∅)) ∈ ran 𝐹) → (𝐹‘(𝑓‘∅)) ≤ sup(ran 𝐹, ℝ, < ))
4942, 43, 47, 48syl3anc 1477 . . . . . 6 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (sup(ran 𝐹, ℝ, < ) ≤ (♯‘𝑠) ∧ 𝑓:{∅}⟶𝑅)) → (𝐹‘(𝑓‘∅)) ≤ sup(ran 𝐹, ℝ, < ))
50 simprl 811 . . . . . 6 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (sup(ran 𝐹, ℝ, < ) ≤ (♯‘𝑠) ∧ 𝑓:{∅}⟶𝑅)) → sup(ran 𝐹, ℝ, < ) ≤ (♯‘𝑠))
5136, 39, 41, 49, 50xrletrd 12186 . . . . 5 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (sup(ran 𝐹, ℝ, < ) ≤ (♯‘𝑠) ∧ 𝑓:{∅}⟶𝑅)) → (𝐹‘(𝑓‘∅)) ≤ (♯‘𝑠))
5228a1i 11 . . . . . 6 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (sup(ran 𝐹, ℝ, < ) ≤ (♯‘𝑠) ∧ 𝑓:{∅}⟶𝑅)) → ∅ ∈ {∅})
53 fvex 6362 . . . . . . . 8 (𝑓‘∅) ∈ V
5453snid 4353 . . . . . . 7 (𝑓‘∅) ∈ {(𝑓‘∅)}
5554a1i 11 . . . . . 6 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (sup(ran 𝐹, ℝ, < ) ≤ (♯‘𝑠) ∧ 𝑓:{∅}⟶𝑅)) → (𝑓‘∅) ∈ {(𝑓‘∅)})
56 ffn 6206 . . . . . . 7 (𝑓:{∅}⟶𝑅𝑓 Fn {∅})
57 elpreima 6500 . . . . . . 7 (𝑓 Fn {∅} → (∅ ∈ (𝑓 “ {(𝑓‘∅)}) ↔ (∅ ∈ {∅} ∧ (𝑓‘∅) ∈ {(𝑓‘∅)})))
5826, 56, 573syl 18 . . . . . 6 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (sup(ran 𝐹, ℝ, < ) ≤ (♯‘𝑠) ∧ 𝑓:{∅}⟶𝑅)) → (∅ ∈ (𝑓 “ {(𝑓‘∅)}) ↔ (∅ ∈ {∅} ∧ (𝑓‘∅) ∈ {(𝑓‘∅)})))
5952, 55, 58mpbir2and 995 . . . . 5 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (sup(ran 𝐹, ℝ, < ) ≤ (♯‘𝑠) ∧ 𝑓:{∅}⟶𝑅)) → ∅ ∈ (𝑓 “ {(𝑓‘∅)}))
60 fveq2 6352 . . . . . . . 8 (𝑐 = (𝑓‘∅) → (𝐹𝑐) = (𝐹‘(𝑓‘∅)))
6160breq1d 4814 . . . . . . 7 (𝑐 = (𝑓‘∅) → ((𝐹𝑐) ≤ (♯‘𝑧) ↔ (𝐹‘(𝑓‘∅)) ≤ (♯‘𝑧)))
62 vex 3343 . . . . . . . . . . 11 𝑧 ∈ V
631hashbc0 15911 . . . . . . . . . . 11 (𝑧 ∈ V → (𝑧(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})0) = {∅})
6462, 63ax-mp 5 . . . . . . . . . 10 (𝑧(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})0) = {∅}
6564sseq1i 3770 . . . . . . . . 9 ((𝑧(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})0) ⊆ (𝑓 “ {𝑐}) ↔ {∅} ⊆ (𝑓 “ {𝑐}))
6627snss 4460 . . . . . . . . 9 (∅ ∈ (𝑓 “ {𝑐}) ↔ {∅} ⊆ (𝑓 “ {𝑐}))
6765, 66bitr4i 267 . . . . . . . 8 ((𝑧(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})0) ⊆ (𝑓 “ {𝑐}) ↔ ∅ ∈ (𝑓 “ {𝑐}))
68 sneq 4331 . . . . . . . . . 10 (𝑐 = (𝑓‘∅) → {𝑐} = {(𝑓‘∅)})
6968imaeq2d 5624 . . . . . . . . 9 (𝑐 = (𝑓‘∅) → (𝑓 “ {𝑐}) = (𝑓 “ {(𝑓‘∅)}))
7069eleq2d 2825 . . . . . . . 8 (𝑐 = (𝑓‘∅) → (∅ ∈ (𝑓 “ {𝑐}) ↔ ∅ ∈ (𝑓 “ {(𝑓‘∅)})))
7167, 70syl5bb 272 . . . . . . 7 (𝑐 = (𝑓‘∅) → ((𝑧(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})0) ⊆ (𝑓 “ {𝑐}) ↔ ∅ ∈ (𝑓 “ {(𝑓‘∅)})))
7261, 71anbi12d 749 . . . . . 6 (𝑐 = (𝑓‘∅) → (((𝐹𝑐) ≤ (♯‘𝑧) ∧ (𝑧(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})0) ⊆ (𝑓 “ {𝑐})) ↔ ((𝐹‘(𝑓‘∅)) ≤ (♯‘𝑧) ∧ ∅ ∈ (𝑓 “ {(𝑓‘∅)}))))
73 fveq2 6352 . . . . . . . 8 (𝑧 = 𝑠 → (♯‘𝑧) = (♯‘𝑠))
7473breq2d 4816 . . . . . . 7 (𝑧 = 𝑠 → ((𝐹‘(𝑓‘∅)) ≤ (♯‘𝑧) ↔ (𝐹‘(𝑓‘∅)) ≤ (♯‘𝑠)))
7574anbi1d 743 . . . . . 6 (𝑧 = 𝑠 → (((𝐹‘(𝑓‘∅)) ≤ (♯‘𝑧) ∧ ∅ ∈ (𝑓 “ {(𝑓‘∅)})) ↔ ((𝐹‘(𝑓‘∅)) ≤ (♯‘𝑠) ∧ ∅ ∈ (𝑓 “ {(𝑓‘∅)}))))
7672, 75rspc2ev 3463 . . . . 5 (((𝑓‘∅) ∈ 𝑅𝑠 ∈ 𝒫 𝑠 ∧ ((𝐹‘(𝑓‘∅)) ≤ (♯‘𝑠) ∧ ∅ ∈ (𝑓 “ {(𝑓‘∅)}))) → ∃𝑐𝑅𝑧 ∈ 𝒫 𝑠((𝐹𝑐) ≤ (♯‘𝑧) ∧ (𝑧(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})0) ⊆ (𝑓 “ {𝑐})))
7730, 32, 51, 59, 76syl112anc 1481 . . . 4 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (sup(ran 𝐹, ℝ, < ) ≤ (♯‘𝑠) ∧ 𝑓:{∅}⟶𝑅)) → ∃𝑐𝑅𝑧 ∈ 𝒫 𝑠((𝐹𝑐) ≤ (♯‘𝑧) ∧ (𝑧(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})0) ⊆ (𝑓 “ {𝑐})))
7825, 77sylanr2 688 . . 3 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (sup(ran 𝐹, ℝ, < ) ≤ (♯‘𝑠) ∧ 𝑓:(𝑠(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})0)⟶𝑅)) → ∃𝑐𝑅𝑧 ∈ 𝒫 𝑠((𝐹𝑐) ≤ (♯‘𝑧) ∧ (𝑧(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})0) ⊆ (𝑓 “ {𝑐})))
791, 3, 4, 5, 20, 78ramub 15919 . 2 (((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) → (0 Ramsey 𝐹) ≤ sup(ran 𝐹, ℝ, < ))
80 fvelrnb 6405 . . . . 5 (𝐹 Fn 𝑅 → (sup(ran 𝐹, ℝ, < ) ∈ ran 𝐹 ↔ ∃𝑐𝑅 (𝐹𝑐) = sup(ran 𝐹, ℝ, < )))
815, 44, 803syl 18 . . . 4 (((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) → (sup(ran 𝐹, ℝ, < ) ∈ ran 𝐹 ↔ ∃𝑐𝑅 (𝐹𝑐) = sup(ran 𝐹, ℝ, < )))
8219, 81mpbid 222 . . 3 (((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) → ∃𝑐𝑅 (𝐹𝑐) = sup(ran 𝐹, ℝ, < ))
832a1i 11 . . . . . . . . 9 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (𝑐𝑅 ∧ (𝐹𝑐) ∈ ℕ)) → 0 ∈ ℕ0)
84 simpll1 1255 . . . . . . . . 9 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (𝑐𝑅 ∧ (𝐹𝑐) ∈ ℕ)) → 𝑅𝑉)
85 simpll3 1259 . . . . . . . . 9 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (𝑐𝑅 ∧ (𝐹𝑐) ∈ ℕ)) → 𝐹:𝑅⟶ℕ0)
86 nnm1nn0 11526 . . . . . . . . . 10 ((𝐹𝑐) ∈ ℕ → ((𝐹𝑐) − 1) ∈ ℕ0)
8786ad2antll 767 . . . . . . . . 9 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (𝑐𝑅 ∧ (𝐹𝑐) ∈ ℕ)) → ((𝐹𝑐) − 1) ∈ ℕ0)
88 vex 3343 . . . . . . . . . . . . 13 𝑐 ∈ V
8927, 88f1osn 6337 . . . . . . . . . . . 12 {⟨∅, 𝑐⟩}:{∅}–1-1-onto→{𝑐}
90 f1of 6298 . . . . . . . . . . . 12 ({⟨∅, 𝑐⟩}:{∅}–1-1-onto→{𝑐} → {⟨∅, 𝑐⟩}:{∅}⟶{𝑐})
9189, 90ax-mp 5 . . . . . . . . . . 11 {⟨∅, 𝑐⟩}:{∅}⟶{𝑐}
92 simprl 811 . . . . . . . . . . . 12 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (𝑐𝑅 ∧ (𝐹𝑐) ∈ ℕ)) → 𝑐𝑅)
9392snssd 4485 . . . . . . . . . . 11 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (𝑐𝑅 ∧ (𝐹𝑐) ∈ ℕ)) → {𝑐} ⊆ 𝑅)
94 fss 6217 . . . . . . . . . . 11 (({⟨∅, 𝑐⟩}:{∅}⟶{𝑐} ∧ {𝑐} ⊆ 𝑅) → {⟨∅, 𝑐⟩}:{∅}⟶𝑅)
9591, 93, 94sylancr 698 . . . . . . . . . 10 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (𝑐𝑅 ∧ (𝐹𝑐) ∈ ℕ)) → {⟨∅, 𝑐⟩}:{∅}⟶𝑅)
96 ovex 6841 . . . . . . . . . . . 12 (1...((𝐹𝑐) − 1)) ∈ V
971hashbc0 15911 . . . . . . . . . . . 12 ((1...((𝐹𝑐) − 1)) ∈ V → ((1...((𝐹𝑐) − 1))(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})0) = {∅})
9896, 97ax-mp 5 . . . . . . . . . . 11 ((1...((𝐹𝑐) − 1))(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})0) = {∅}
9998feq2i 6198 . . . . . . . . . 10 ({⟨∅, 𝑐⟩}:((1...((𝐹𝑐) − 1))(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})0)⟶𝑅 ↔ {⟨∅, 𝑐⟩}:{∅}⟶𝑅)
10095, 99sylibr 224 . . . . . . . . 9 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (𝑐𝑅 ∧ (𝐹𝑐) ∈ ℕ)) → {⟨∅, 𝑐⟩}:((1...((𝐹𝑐) − 1))(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})0)⟶𝑅)
10164sseq1i 3770 . . . . . . . . . . 11 ((𝑧(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})0) ⊆ ({⟨∅, 𝑐⟩} “ {𝑑}) ↔ {∅} ⊆ ({⟨∅, 𝑐⟩} “ {𝑑}))
10227snss 4460 . . . . . . . . . . 11 (∅ ∈ ({⟨∅, 𝑐⟩} “ {𝑑}) ↔ {∅} ⊆ ({⟨∅, 𝑐⟩} “ {𝑑}))
103101, 102bitr4i 267 . . . . . . . . . 10 ((𝑧(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})0) ⊆ ({⟨∅, 𝑐⟩} “ {𝑑}) ↔ ∅ ∈ ({⟨∅, 𝑐⟩} “ {𝑑}))
104 fzfid 12966 . . . . . . . . . . . . . . 15 (((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (𝑐𝑅 ∧ (𝐹𝑐) ∈ ℕ)) ∧ (𝑑𝑅𝑧 ⊆ (1...((𝐹𝑐) − 1)))) → (1...((𝐹𝑐) − 1)) ∈ Fin)
105 simprr 813 . . . . . . . . . . . . . . 15 (((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (𝑐𝑅 ∧ (𝐹𝑐) ∈ ℕ)) ∧ (𝑑𝑅𝑧 ⊆ (1...((𝐹𝑐) − 1)))) → 𝑧 ⊆ (1...((𝐹𝑐) − 1)))
106 ssdomg 8167 . . . . . . . . . . . . . . 15 ((1...((𝐹𝑐) − 1)) ∈ Fin → (𝑧 ⊆ (1...((𝐹𝑐) − 1)) → 𝑧 ≼ (1...((𝐹𝑐) − 1))))
107104, 105, 106sylc 65 . . . . . . . . . . . . . 14 (((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (𝑐𝑅 ∧ (𝐹𝑐) ∈ ℕ)) ∧ (𝑑𝑅𝑧 ⊆ (1...((𝐹𝑐) − 1)))) → 𝑧 ≼ (1...((𝐹𝑐) − 1)))
108 ssfi 8345 . . . . . . . . . . . . . . . 16 (((1...((𝐹𝑐) − 1)) ∈ Fin ∧ 𝑧 ⊆ (1...((𝐹𝑐) − 1))) → 𝑧 ∈ Fin)
109104, 105, 108syl2anc 696 . . . . . . . . . . . . . . 15 (((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (𝑐𝑅 ∧ (𝐹𝑐) ∈ ℕ)) ∧ (𝑑𝑅𝑧 ⊆ (1...((𝐹𝑐) − 1)))) → 𝑧 ∈ Fin)
110 hashdom 13360 . . . . . . . . . . . . . . 15 ((𝑧 ∈ Fin ∧ (1...((𝐹𝑐) − 1)) ∈ Fin) → ((♯‘𝑧) ≤ (♯‘(1...((𝐹𝑐) − 1))) ↔ 𝑧 ≼ (1...((𝐹𝑐) − 1))))
111109, 104, 110syl2anc 696 . . . . . . . . . . . . . 14 (((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (𝑐𝑅 ∧ (𝐹𝑐) ∈ ℕ)) ∧ (𝑑𝑅𝑧 ⊆ (1...((𝐹𝑐) − 1)))) → ((♯‘𝑧) ≤ (♯‘(1...((𝐹𝑐) − 1))) ↔ 𝑧 ≼ (1...((𝐹𝑐) − 1))))
112107, 111mpbird 247 . . . . . . . . . . . . 13 (((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (𝑐𝑅 ∧ (𝐹𝑐) ∈ ℕ)) ∧ (𝑑𝑅𝑧 ⊆ (1...((𝐹𝑐) − 1)))) → (♯‘𝑧) ≤ (♯‘(1...((𝐹𝑐) − 1))))
11387adantr 472 . . . . . . . . . . . . . 14 (((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (𝑐𝑅 ∧ (𝐹𝑐) ∈ ℕ)) ∧ (𝑑𝑅𝑧 ⊆ (1...((𝐹𝑐) − 1)))) → ((𝐹𝑐) − 1) ∈ ℕ0)
114 hashfz1 13328 . . . . . . . . . . . . . 14 (((𝐹𝑐) − 1) ∈ ℕ0 → (♯‘(1...((𝐹𝑐) − 1))) = ((𝐹𝑐) − 1))
115113, 114syl 17 . . . . . . . . . . . . 13 (((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (𝑐𝑅 ∧ (𝐹𝑐) ∈ ℕ)) ∧ (𝑑𝑅𝑧 ⊆ (1...((𝐹𝑐) − 1)))) → (♯‘(1...((𝐹𝑐) − 1))) = ((𝐹𝑐) − 1))
116112, 115breqtrd 4830 . . . . . . . . . . . 12 (((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (𝑐𝑅 ∧ (𝐹𝑐) ∈ ℕ)) ∧ (𝑑𝑅𝑧 ⊆ (1...((𝐹𝑐) − 1)))) → (♯‘𝑧) ≤ ((𝐹𝑐) − 1))
117 hashcl 13339 . . . . . . . . . . . . . 14 (𝑧 ∈ Fin → (♯‘𝑧) ∈ ℕ0)
118109, 117syl 17 . . . . . . . . . . . . 13 (((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (𝑐𝑅 ∧ (𝐹𝑐) ∈ ℕ)) ∧ (𝑑𝑅𝑧 ⊆ (1...((𝐹𝑐) − 1)))) → (♯‘𝑧) ∈ ℕ0)
1195ffvelrnda 6522 . . . . . . . . . . . . . . 15 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ 𝑐𝑅) → (𝐹𝑐) ∈ ℕ0)
120119adantrr 755 . . . . . . . . . . . . . 14 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (𝑐𝑅 ∧ (𝐹𝑐) ∈ ℕ)) → (𝐹𝑐) ∈ ℕ0)
121120adantr 472 . . . . . . . . . . . . 13 (((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (𝑐𝑅 ∧ (𝐹𝑐) ∈ ℕ)) ∧ (𝑑𝑅𝑧 ⊆ (1...((𝐹𝑐) − 1)))) → (𝐹𝑐) ∈ ℕ0)
122 nn0ltlem1 11629 . . . . . . . . . . . . 13 (((♯‘𝑧) ∈ ℕ0 ∧ (𝐹𝑐) ∈ ℕ0) → ((♯‘𝑧) < (𝐹𝑐) ↔ (♯‘𝑧) ≤ ((𝐹𝑐) − 1)))
123118, 121, 122syl2anc 696 . . . . . . . . . . . 12 (((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (𝑐𝑅 ∧ (𝐹𝑐) ∈ ℕ)) ∧ (𝑑𝑅𝑧 ⊆ (1...((𝐹𝑐) − 1)))) → ((♯‘𝑧) < (𝐹𝑐) ↔ (♯‘𝑧) ≤ ((𝐹𝑐) − 1)))
124116, 123mpbird 247 . . . . . . . . . . 11 (((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (𝑐𝑅 ∧ (𝐹𝑐) ∈ ℕ)) ∧ (𝑑𝑅𝑧 ⊆ (1...((𝐹𝑐) − 1)))) → (♯‘𝑧) < (𝐹𝑐))
12527, 88fvsn 6610 . . . . . . . . . . . . . . 15 ({⟨∅, 𝑐⟩}‘∅) = 𝑐
126 f1ofn 6299 . . . . . . . . . . . . . . . . 17 ({⟨∅, 𝑐⟩}:{∅}–1-1-onto→{𝑐} → {⟨∅, 𝑐⟩} Fn {∅})
127 elpreima 6500 . . . . . . . . . . . . . . . . 17 ({⟨∅, 𝑐⟩} Fn {∅} → (∅ ∈ ({⟨∅, 𝑐⟩} “ {𝑑}) ↔ (∅ ∈ {∅} ∧ ({⟨∅, 𝑐⟩}‘∅) ∈ {𝑑})))
12889, 126, 127mp2b 10 . . . . . . . . . . . . . . . 16 (∅ ∈ ({⟨∅, 𝑐⟩} “ {𝑑}) ↔ (∅ ∈ {∅} ∧ ({⟨∅, 𝑐⟩}‘∅) ∈ {𝑑}))
129128simprbi 483 . . . . . . . . . . . . . . 15 (∅ ∈ ({⟨∅, 𝑐⟩} “ {𝑑}) → ({⟨∅, 𝑐⟩}‘∅) ∈ {𝑑})
130125, 129syl5eqelr 2844 . . . . . . . . . . . . . 14 (∅ ∈ ({⟨∅, 𝑐⟩} “ {𝑑}) → 𝑐 ∈ {𝑑})
131 elsni 4338 . . . . . . . . . . . . . 14 (𝑐 ∈ {𝑑} → 𝑐 = 𝑑)
132130, 131syl 17 . . . . . . . . . . . . 13 (∅ ∈ ({⟨∅, 𝑐⟩} “ {𝑑}) → 𝑐 = 𝑑)
133132fveq2d 6356 . . . . . . . . . . . 12 (∅ ∈ ({⟨∅, 𝑐⟩} “ {𝑑}) → (𝐹𝑐) = (𝐹𝑑))
134133breq2d 4816 . . . . . . . . . . 11 (∅ ∈ ({⟨∅, 𝑐⟩} “ {𝑑}) → ((♯‘𝑧) < (𝐹𝑐) ↔ (♯‘𝑧) < (𝐹𝑑)))
135124, 134syl5ibcom 235 . . . . . . . . . 10 (((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (𝑐𝑅 ∧ (𝐹𝑐) ∈ ℕ)) ∧ (𝑑𝑅𝑧 ⊆ (1...((𝐹𝑐) − 1)))) → (∅ ∈ ({⟨∅, 𝑐⟩} “ {𝑑}) → (♯‘𝑧) < (𝐹𝑑)))
136103, 135syl5bi 232 . . . . . . . . 9 (((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (𝑐𝑅 ∧ (𝐹𝑐) ∈ ℕ)) ∧ (𝑑𝑅𝑧 ⊆ (1...((𝐹𝑐) − 1)))) → ((𝑧(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})0) ⊆ ({⟨∅, 𝑐⟩} “ {𝑑}) → (♯‘𝑧) < (𝐹𝑑)))
1371, 83, 84, 85, 87, 100, 136ramlb 15925 . . . . . . . 8 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (𝑐𝑅 ∧ (𝐹𝑐) ∈ ℕ)) → ((𝐹𝑐) − 1) < (0 Ramsey 𝐹))
138 ramubcl 15924 . . . . . . . . . . 11 (((0 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (sup(ran 𝐹, ℝ, < ) ∈ ℕ0 ∧ (0 Ramsey 𝐹) ≤ sup(ran 𝐹, ℝ, < ))) → (0 Ramsey 𝐹) ∈ ℕ0)
1393, 4, 5, 20, 79, 138syl32anc 1485 . . . . . . . . . 10 (((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) → (0 Ramsey 𝐹) ∈ ℕ0)
140139adantr 472 . . . . . . . . 9 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (𝑐𝑅 ∧ (𝐹𝑐) ∈ ℕ)) → (0 Ramsey 𝐹) ∈ ℕ0)
141 nn0lem1lt 11634 . . . . . . . . 9 (((𝐹𝑐) ∈ ℕ0 ∧ (0 Ramsey 𝐹) ∈ ℕ0) → ((𝐹𝑐) ≤ (0 Ramsey 𝐹) ↔ ((𝐹𝑐) − 1) < (0 Ramsey 𝐹)))
142120, 140, 141syl2anc 696 . . . . . . . 8 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (𝑐𝑅 ∧ (𝐹𝑐) ∈ ℕ)) → ((𝐹𝑐) ≤ (0 Ramsey 𝐹) ↔ ((𝐹𝑐) − 1) < (0 Ramsey 𝐹)))
143137, 142mpbird 247 . . . . . . 7 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (𝑐𝑅 ∧ (𝐹𝑐) ∈ ℕ)) → (𝐹𝑐) ≤ (0 Ramsey 𝐹))
144143expr 644 . . . . . 6 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ 𝑐𝑅) → ((𝐹𝑐) ∈ ℕ → (𝐹𝑐) ≤ (0 Ramsey 𝐹)))
145139adantr 472 . . . . . . . 8 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ 𝑐𝑅) → (0 Ramsey 𝐹) ∈ ℕ0)
146145nn0ge0d 11546 . . . . . . 7 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ 𝑐𝑅) → 0 ≤ (0 Ramsey 𝐹))
147 breq1 4807 . . . . . . 7 ((𝐹𝑐) = 0 → ((𝐹𝑐) ≤ (0 Ramsey 𝐹) ↔ 0 ≤ (0 Ramsey 𝐹)))
148146, 147syl5ibrcom 237 . . . . . 6 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ 𝑐𝑅) → ((𝐹𝑐) = 0 → (𝐹𝑐) ≤ (0 Ramsey 𝐹)))
149 elnn0 11486 . . . . . . 7 ((𝐹𝑐) ∈ ℕ0 ↔ ((𝐹𝑐) ∈ ℕ ∨ (𝐹𝑐) = 0))
150119, 149sylib 208 . . . . . 6 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ 𝑐𝑅) → ((𝐹𝑐) ∈ ℕ ∨ (𝐹𝑐) = 0))
151144, 148, 150mpjaod 395 . . . . 5 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ 𝑐𝑅) → (𝐹𝑐) ≤ (0 Ramsey 𝐹))
152 breq1 4807 . . . . 5 ((𝐹𝑐) = sup(ran 𝐹, ℝ, < ) → ((𝐹𝑐) ≤ (0 Ramsey 𝐹) ↔ sup(ran 𝐹, ℝ, < ) ≤ (0 Ramsey 𝐹)))
153151, 152syl5ibcom 235 . . . 4 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ 𝑐𝑅) → ((𝐹𝑐) = sup(ran 𝐹, ℝ, < ) → sup(ran 𝐹, ℝ, < ) ≤ (0 Ramsey 𝐹)))
154153rexlimdva 3169 . . 3 (((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) → (∃𝑐𝑅 (𝐹𝑐) = sup(ran 𝐹, ℝ, < ) → sup(ran 𝐹, ℝ, < ) ≤ (0 Ramsey 𝐹)))
15582, 154mpd 15 . 2 (((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) → sup(ran 𝐹, ℝ, < ) ≤ (0 Ramsey 𝐹))
156139nn0red 11544 . . 3 (((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) → (0 Ramsey 𝐹) ∈ ℝ)
157156, 37letri3d 10371 . 2 (((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) → ((0 Ramsey 𝐹) = sup(ran 𝐹, ℝ, < ) ↔ ((0 Ramsey 𝐹) ≤ sup(ran 𝐹, ℝ, < ) ∧ sup(ran 𝐹, ℝ, < ) ≤ (0 Ramsey 𝐹))))
15879, 155, 157mpbir2and 995 1 (((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) → (0 Ramsey 𝐹) = sup(ran 𝐹, ℝ, < ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∨ wo 382   ∧ wa 383   ∧ w3a 1072   = wceq 1632   ∈ wcel 2139   ≠ wne 2932  ∀wral 3050  ∃wrex 3051  {crab 3054  Vcvv 3340   ⊆ wss 3715  ∅c0 4058  𝒫 cpw 4302  {csn 4321  ⟨cop 4327   class class class wbr 4804  ◡ccnv 5265  dom cdm 5266  ran crn 5267   “ cima 5269   Fn wfn 6044  ⟶wf 6045  –1-1-onto→wf1o 6048  ‘cfv 6049  (class class class)co 6813   ↦ cmpt2 6815   ≼ cdom 8119  Fincfn 8121  supcsup 8511  ℝcr 10127  0cc0 10128  1c1 10129  ℝ*cxr 10265   < clt 10266   ≤ cle 10267   − cmin 10458  ℕcn 11212  ℕ0cn0 11484  ℤcz 11569  ...cfz 12519  ♯chash 13311   Ramsey cram 15905 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-cnex 10184  ax-resscn 10185  ax-1cn 10186  ax-icn 10187  ax-addcl 10188  ax-addrcl 10189  ax-mulcl 10190  ax-mulrcl 10191  ax-mulcom 10192  ax-addass 10193  ax-mulass 10194  ax-distr 10195  ax-i2m1 10196  ax-1ne0 10197  ax-1rid 10198  ax-rnegex 10199  ax-rrecex 10200  ax-cnre 10201  ax-pre-lttri 10202  ax-pre-lttrn 10203  ax-pre-ltadd 10204  ax-pre-mulgt0 10205 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-om 7231  df-1st 7333  df-2nd 7334  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-1o 7729  df-oadd 7733  df-er 7911  df-map 8025  df-en 8122  df-dom 8123  df-sdom 8124  df-fin 8125  df-sup 8513  df-inf 8514  df-card 8955  df-pnf 10268  df-mnf 10269  df-xr 10270  df-ltxr 10271  df-le 10272  df-sub 10460  df-neg 10461  df-nn 11213  df-n0 11485  df-xnn0 11556  df-z 11570  df-uz 11880  df-fz 12520  df-hash 13312  df-ram 15907 This theorem is referenced by:  0ram2  15927  ramz  15931
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