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Theorem 0ram 16949
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 2733 . . 3 (𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖}) = (𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})
2 0nn0 12483 . . . 4 0 ∈ ℕ0
32a1i 11 . . 3 (((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) → 0 ∈ ℕ0)
4 simpl1 1192 . . 3 (((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) → 𝑅𝑉)
5 simpl3 1194 . . 3 (((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) → 𝐹:𝑅⟶ℕ0)
65frnd 6722 . . . 4 (((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) → ran 𝐹 ⊆ ℕ0)
7 nn0ssz 12577 . . . . . 6 0 ⊆ ℤ
86, 7sstrdi 3993 . . . . 5 (((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) → ran 𝐹 ⊆ ℤ)
95fdmd 6725 . . . . . . 7 (((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) → dom 𝐹 = 𝑅)
10 simpl2 1193 . . . . . . 7 (((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) → 𝑅 ≠ ∅)
119, 10eqnetrd 3009 . . . . . 6 (((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) → dom 𝐹 ≠ ∅)
12 dm0rn0 5922 . . . . . . 7 (dom 𝐹 = ∅ ↔ ran 𝐹 = ∅)
1312necon3bii 2994 . . . . . 6 (dom 𝐹 ≠ ∅ ↔ ran 𝐹 ≠ ∅)
1411, 13sylib 217 . . . . 5 (((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) → ran 𝐹 ≠ ∅)
15 simpr 486 . . . . 5 (((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) → ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥)
16 suprzcl2 12918 . . . . 5 ((ran 𝐹 ⊆ ℤ ∧ ran 𝐹 ≠ ∅ ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) → sup(ran 𝐹, ℝ, < ) ∈ ran 𝐹)
178, 14, 15, 16syl3anc 1372 . . . 4 (((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) → sup(ran 𝐹, ℝ, < ) ∈ ran 𝐹)
186, 17sseldd 3982 . . 3 (((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) → sup(ran 𝐹, ℝ, < ) ∈ ℕ0)
191hashbc0 16934 . . . . . . 7 (𝑠 ∈ V → (𝑠(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})0) = {∅})
2019elv 3481 . . . . . 6 (𝑠(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})0) = {∅}
2120feq2i 6706 . . . . 5 (𝑓:(𝑠(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})0)⟶𝑅𝑓:{∅}⟶𝑅)
2221biimpi 215 . . . 4 (𝑓:(𝑠(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})0)⟶𝑅𝑓:{∅}⟶𝑅)
23 simprr 772 . . . . . 6 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (sup(ran 𝐹, ℝ, < ) ≤ (♯‘𝑠) ∧ 𝑓:{∅}⟶𝑅)) → 𝑓:{∅}⟶𝑅)
24 0ex 5306 . . . . . . 7 ∅ ∈ V
2524snid 4663 . . . . . 6 ∅ ∈ {∅}
26 ffvelcdm 7079 . . . . . 6 ((𝑓:{∅}⟶𝑅 ∧ ∅ ∈ {∅}) → (𝑓‘∅) ∈ 𝑅)
2723, 25, 26sylancl 587 . . . . 5 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (sup(ran 𝐹, ℝ, < ) ≤ (♯‘𝑠) ∧ 𝑓:{∅}⟶𝑅)) → (𝑓‘∅) ∈ 𝑅)
28 vex 3479 . . . . . . 7 𝑠 ∈ V
2928pwid 4623 . . . . . 6 𝑠 ∈ 𝒫 𝑠
3029a1i 11 . . . . 5 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (sup(ran 𝐹, ℝ, < ) ≤ (♯‘𝑠) ∧ 𝑓:{∅}⟶𝑅)) → 𝑠 ∈ 𝒫 𝑠)
315adantr 482 . . . . . . . . 9 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (sup(ran 𝐹, ℝ, < ) ≤ (♯‘𝑠) ∧ 𝑓:{∅}⟶𝑅)) → 𝐹:𝑅⟶ℕ0)
3231, 27ffvelcdmd 7083 . . . . . . . 8 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (sup(ran 𝐹, ℝ, < ) ≤ (♯‘𝑠) ∧ 𝑓:{∅}⟶𝑅)) → (𝐹‘(𝑓‘∅)) ∈ ℕ0)
3332nn0red 12529 . . . . . . 7 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (sup(ran 𝐹, ℝ, < ) ≤ (♯‘𝑠) ∧ 𝑓:{∅}⟶𝑅)) → (𝐹‘(𝑓‘∅)) ∈ ℝ)
3433rexrd 11260 . . . . . 6 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (sup(ran 𝐹, ℝ, < ) ≤ (♯‘𝑠) ∧ 𝑓:{∅}⟶𝑅)) → (𝐹‘(𝑓‘∅)) ∈ ℝ*)
3518nn0red 12529 . . . . . . . 8 (((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) → sup(ran 𝐹, ℝ, < ) ∈ ℝ)
3635rexrd 11260 . . . . . . 7 (((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) → sup(ran 𝐹, ℝ, < ) ∈ ℝ*)
3736adantr 482 . . . . . 6 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (sup(ran 𝐹, ℝ, < ) ≤ (♯‘𝑠) ∧ 𝑓:{∅}⟶𝑅)) → sup(ran 𝐹, ℝ, < ) ∈ ℝ*)
38 hashxrcl 14313 . . . . . . 7 (𝑠 ∈ V → (♯‘𝑠) ∈ ℝ*)
3928, 38mp1i 13 . . . . . 6 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (sup(ran 𝐹, ℝ, < ) ≤ (♯‘𝑠) ∧ 𝑓:{∅}⟶𝑅)) → (♯‘𝑠) ∈ ℝ*)
408adantr 482 . . . . . . 7 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (sup(ran 𝐹, ℝ, < ) ≤ (♯‘𝑠) ∧ 𝑓:{∅}⟶𝑅)) → ran 𝐹 ⊆ ℤ)
4115adantr 482 . . . . . . 7 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (sup(ran 𝐹, ℝ, < ) ≤ (♯‘𝑠) ∧ 𝑓:{∅}⟶𝑅)) → ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥)
4231ffnd 6715 . . . . . . . 8 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (sup(ran 𝐹, ℝ, < ) ≤ (♯‘𝑠) ∧ 𝑓:{∅}⟶𝑅)) → 𝐹 Fn 𝑅)
43 fnfvelrn 7078 . . . . . . . 8 ((𝐹 Fn 𝑅 ∧ (𝑓‘∅) ∈ 𝑅) → (𝐹‘(𝑓‘∅)) ∈ ran 𝐹)
4442, 27, 43syl2anc 585 . . . . . . 7 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (sup(ran 𝐹, ℝ, < ) ≤ (♯‘𝑠) ∧ 𝑓:{∅}⟶𝑅)) → (𝐹‘(𝑓‘∅)) ∈ ran 𝐹)
45 suprzub 12919 . . . . . . 7 ((ran 𝐹 ⊆ ℤ ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥 ∧ (𝐹‘(𝑓‘∅)) ∈ ran 𝐹) → (𝐹‘(𝑓‘∅)) ≤ sup(ran 𝐹, ℝ, < ))
4640, 41, 44, 45syl3anc 1372 . . . . . 6 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (sup(ran 𝐹, ℝ, < ) ≤ (♯‘𝑠) ∧ 𝑓:{∅}⟶𝑅)) → (𝐹‘(𝑓‘∅)) ≤ sup(ran 𝐹, ℝ, < ))
47 simprl 770 . . . . . 6 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (sup(ran 𝐹, ℝ, < ) ≤ (♯‘𝑠) ∧ 𝑓:{∅}⟶𝑅)) → sup(ran 𝐹, ℝ, < ) ≤ (♯‘𝑠))
4834, 37, 39, 46, 47xrletrd 13137 . . . . 5 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (sup(ran 𝐹, ℝ, < ) ≤ (♯‘𝑠) ∧ 𝑓:{∅}⟶𝑅)) → (𝐹‘(𝑓‘∅)) ≤ (♯‘𝑠))
4925a1i 11 . . . . . 6 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (sup(ran 𝐹, ℝ, < ) ≤ (♯‘𝑠) ∧ 𝑓:{∅}⟶𝑅)) → ∅ ∈ {∅})
50 fvex 6901 . . . . . . . 8 (𝑓‘∅) ∈ V
5150snid 4663 . . . . . . 7 (𝑓‘∅) ∈ {(𝑓‘∅)}
5251a1i 11 . . . . . 6 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (sup(ran 𝐹, ℝ, < ) ≤ (♯‘𝑠) ∧ 𝑓:{∅}⟶𝑅)) → (𝑓‘∅) ∈ {(𝑓‘∅)})
53 ffn 6714 . . . . . . 7 (𝑓:{∅}⟶𝑅𝑓 Fn {∅})
54 elpreima 7055 . . . . . . 7 (𝑓 Fn {∅} → (∅ ∈ (𝑓 “ {(𝑓‘∅)}) ↔ (∅ ∈ {∅} ∧ (𝑓‘∅) ∈ {(𝑓‘∅)})))
5523, 53, 543syl 18 . . . . . 6 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (sup(ran 𝐹, ℝ, < ) ≤ (♯‘𝑠) ∧ 𝑓:{∅}⟶𝑅)) → (∅ ∈ (𝑓 “ {(𝑓‘∅)}) ↔ (∅ ∈ {∅} ∧ (𝑓‘∅) ∈ {(𝑓‘∅)})))
5649, 52, 55mpbir2and 712 . . . . 5 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (sup(ran 𝐹, ℝ, < ) ≤ (♯‘𝑠) ∧ 𝑓:{∅}⟶𝑅)) → ∅ ∈ (𝑓 “ {(𝑓‘∅)}))
57 fveq2 6888 . . . . . . . 8 (𝑐 = (𝑓‘∅) → (𝐹𝑐) = (𝐹‘(𝑓‘∅)))
5857breq1d 5157 . . . . . . 7 (𝑐 = (𝑓‘∅) → ((𝐹𝑐) ≤ (♯‘𝑧) ↔ (𝐹‘(𝑓‘∅)) ≤ (♯‘𝑧)))
591hashbc0 16934 . . . . . . . . . . 11 (𝑧 ∈ V → (𝑧(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})0) = {∅})
6059elv 3481 . . . . . . . . . 10 (𝑧(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})0) = {∅}
6160sseq1i 4009 . . . . . . . . 9 ((𝑧(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})0) ⊆ (𝑓 “ {𝑐}) ↔ {∅} ⊆ (𝑓 “ {𝑐}))
6224snss 4788 . . . . . . . . 9 (∅ ∈ (𝑓 “ {𝑐}) ↔ {∅} ⊆ (𝑓 “ {𝑐}))
6361, 62bitr4i 278 . . . . . . . 8 ((𝑧(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})0) ⊆ (𝑓 “ {𝑐}) ↔ ∅ ∈ (𝑓 “ {𝑐}))
64 sneq 4637 . . . . . . . . . 10 (𝑐 = (𝑓‘∅) → {𝑐} = {(𝑓‘∅)})
6564imaeq2d 6057 . . . . . . . . 9 (𝑐 = (𝑓‘∅) → (𝑓 “ {𝑐}) = (𝑓 “ {(𝑓‘∅)}))
6665eleq2d 2820 . . . . . . . 8 (𝑐 = (𝑓‘∅) → (∅ ∈ (𝑓 “ {𝑐}) ↔ ∅ ∈ (𝑓 “ {(𝑓‘∅)})))
6763, 66bitrid 283 . . . . . . 7 (𝑐 = (𝑓‘∅) → ((𝑧(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})0) ⊆ (𝑓 “ {𝑐}) ↔ ∅ ∈ (𝑓 “ {(𝑓‘∅)})))
6858, 67anbi12d 632 . . . . . 6 (𝑐 = (𝑓‘∅) → (((𝐹𝑐) ≤ (♯‘𝑧) ∧ (𝑧(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})0) ⊆ (𝑓 “ {𝑐})) ↔ ((𝐹‘(𝑓‘∅)) ≤ (♯‘𝑧) ∧ ∅ ∈ (𝑓 “ {(𝑓‘∅)}))))
69 fveq2 6888 . . . . . . . 8 (𝑧 = 𝑠 → (♯‘𝑧) = (♯‘𝑠))
7069breq2d 5159 . . . . . . 7 (𝑧 = 𝑠 → ((𝐹‘(𝑓‘∅)) ≤ (♯‘𝑧) ↔ (𝐹‘(𝑓‘∅)) ≤ (♯‘𝑠)))
7170anbi1d 631 . . . . . 6 (𝑧 = 𝑠 → (((𝐹‘(𝑓‘∅)) ≤ (♯‘𝑧) ∧ ∅ ∈ (𝑓 “ {(𝑓‘∅)})) ↔ ((𝐹‘(𝑓‘∅)) ≤ (♯‘𝑠) ∧ ∅ ∈ (𝑓 “ {(𝑓‘∅)}))))
7268, 71rspc2ev 3623 . . . . 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})))
7327, 30, 48, 56, 72syl112anc 1375 . . . 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})))
7422, 73sylanr2 682 . . 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})))
751, 3, 4, 5, 18, 74ramub 16942 . 2 (((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) → (0 Ramsey 𝐹) ≤ sup(ran 𝐹, ℝ, < ))
76 ffn 6714 . . . . 5 (𝐹:𝑅⟶ℕ0𝐹 Fn 𝑅)
77 fvelrnb 6949 . . . . 5 (𝐹 Fn 𝑅 → (sup(ran 𝐹, ℝ, < ) ∈ ran 𝐹 ↔ ∃𝑐𝑅 (𝐹𝑐) = sup(ran 𝐹, ℝ, < )))
785, 76, 773syl 18 . . . 4 (((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) → (sup(ran 𝐹, ℝ, < ) ∈ ran 𝐹 ↔ ∃𝑐𝑅 (𝐹𝑐) = sup(ran 𝐹, ℝ, < )))
7917, 78mpbid 231 . . 3 (((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) → ∃𝑐𝑅 (𝐹𝑐) = sup(ran 𝐹, ℝ, < ))
802a1i 11 . . . . . . . . 9 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (𝑐𝑅 ∧ (𝐹𝑐) ∈ ℕ)) → 0 ∈ ℕ0)
81 simpll1 1213 . . . . . . . . 9 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (𝑐𝑅 ∧ (𝐹𝑐) ∈ ℕ)) → 𝑅𝑉)
82 simpll3 1215 . . . . . . . . 9 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (𝑐𝑅 ∧ (𝐹𝑐) ∈ ℕ)) → 𝐹:𝑅⟶ℕ0)
83 nnm1nn0 12509 . . . . . . . . . 10 ((𝐹𝑐) ∈ ℕ → ((𝐹𝑐) − 1) ∈ ℕ0)
8483ad2antll 728 . . . . . . . . 9 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (𝑐𝑅 ∧ (𝐹𝑐) ∈ ℕ)) → ((𝐹𝑐) − 1) ∈ ℕ0)
85 vex 3479 . . . . . . . . . . . . 13 𝑐 ∈ V
8624, 85f1osn 6870 . . . . . . . . . . . 12 {⟨∅, 𝑐⟩}:{∅}–1-1-onto→{𝑐}
87 f1of 6830 . . . . . . . . . . . 12 ({⟨∅, 𝑐⟩}:{∅}–1-1-onto→{𝑐} → {⟨∅, 𝑐⟩}:{∅}⟶{𝑐})
8886, 87ax-mp 5 . . . . . . . . . . 11 {⟨∅, 𝑐⟩}:{∅}⟶{𝑐}
89 simprl 770 . . . . . . . . . . . 12 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (𝑐𝑅 ∧ (𝐹𝑐) ∈ ℕ)) → 𝑐𝑅)
9089snssd 4811 . . . . . . . . . . 11 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (𝑐𝑅 ∧ (𝐹𝑐) ∈ ℕ)) → {𝑐} ⊆ 𝑅)
91 fss 6731 . . . . . . . . . . 11 (({⟨∅, 𝑐⟩}:{∅}⟶{𝑐} ∧ {𝑐} ⊆ 𝑅) → {⟨∅, 𝑐⟩}:{∅}⟶𝑅)
9288, 90, 91sylancr 588 . . . . . . . . . 10 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (𝑐𝑅 ∧ (𝐹𝑐) ∈ ℕ)) → {⟨∅, 𝑐⟩}:{∅}⟶𝑅)
93 ovex 7437 . . . . . . . . . . . 12 (1...((𝐹𝑐) − 1)) ∈ V
941hashbc0 16934 . . . . . . . . . . . 12 ((1...((𝐹𝑐) − 1)) ∈ V → ((1...((𝐹𝑐) − 1))(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})0) = {∅})
9593, 94ax-mp 5 . . . . . . . . . . 11 ((1...((𝐹𝑐) − 1))(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})0) = {∅}
9695feq2i 6706 . . . . . . . . . 10 ({⟨∅, 𝑐⟩}:((1...((𝐹𝑐) − 1))(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})0)⟶𝑅 ↔ {⟨∅, 𝑐⟩}:{∅}⟶𝑅)
9792, 96sylibr 233 . . . . . . . . 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)
9860sseq1i 4009 . . . . . . . . . . 11 ((𝑧(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})0) ⊆ ({⟨∅, 𝑐⟩} “ {𝑑}) ↔ {∅} ⊆ ({⟨∅, 𝑐⟩} “ {𝑑}))
9924snss 4788 . . . . . . . . . . 11 (∅ ∈ ({⟨∅, 𝑐⟩} “ {𝑑}) ↔ {∅} ⊆ ({⟨∅, 𝑐⟩} “ {𝑑}))
10098, 99bitr4i 278 . . . . . . . . . 10 ((𝑧(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})0) ⊆ ({⟨∅, 𝑐⟩} “ {𝑑}) ↔ ∅ ∈ ({⟨∅, 𝑐⟩} “ {𝑑}))
101 fzfid 13934 . . . . . . . . . . . . . . 15 (((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (𝑐𝑅 ∧ (𝐹𝑐) ∈ ℕ)) ∧ (𝑑𝑅𝑧 ⊆ (1...((𝐹𝑐) − 1)))) → (1...((𝐹𝑐) − 1)) ∈ Fin)
102 simprr 772 . . . . . . . . . . . . . . 15 (((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (𝑐𝑅 ∧ (𝐹𝑐) ∈ ℕ)) ∧ (𝑑𝑅𝑧 ⊆ (1...((𝐹𝑐) − 1)))) → 𝑧 ⊆ (1...((𝐹𝑐) − 1)))
103 ssdomg 8992 . . . . . . . . . . . . . . 15 ((1...((𝐹𝑐) − 1)) ∈ Fin → (𝑧 ⊆ (1...((𝐹𝑐) − 1)) → 𝑧 ≼ (1...((𝐹𝑐) − 1))))
104101, 102, 103sylc 65 . . . . . . . . . . . . . 14 (((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (𝑐𝑅 ∧ (𝐹𝑐) ∈ ℕ)) ∧ (𝑑𝑅𝑧 ⊆ (1...((𝐹𝑐) − 1)))) → 𝑧 ≼ (1...((𝐹𝑐) − 1)))
105101, 102ssfid 9263 . . . . . . . . . . . . . . 15 (((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (𝑐𝑅 ∧ (𝐹𝑐) ∈ ℕ)) ∧ (𝑑𝑅𝑧 ⊆ (1...((𝐹𝑐) − 1)))) → 𝑧 ∈ Fin)
106 hashdom 14335 . . . . . . . . . . . . . . 15 ((𝑧 ∈ Fin ∧ (1...((𝐹𝑐) − 1)) ∈ Fin) → ((♯‘𝑧) ≤ (♯‘(1...((𝐹𝑐) − 1))) ↔ 𝑧 ≼ (1...((𝐹𝑐) − 1))))
107105, 101, 106syl2anc 585 . . . . . . . . . . . . . 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))))
108104, 107mpbird 257 . . . . . . . . . . . . 13 (((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (𝑐𝑅 ∧ (𝐹𝑐) ∈ ℕ)) ∧ (𝑑𝑅𝑧 ⊆ (1...((𝐹𝑐) − 1)))) → (♯‘𝑧) ≤ (♯‘(1...((𝐹𝑐) − 1))))
10984adantr 482 . . . . . . . . . . . . . 14 (((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (𝑐𝑅 ∧ (𝐹𝑐) ∈ ℕ)) ∧ (𝑑𝑅𝑧 ⊆ (1...((𝐹𝑐) − 1)))) → ((𝐹𝑐) − 1) ∈ ℕ0)
110 hashfz1 14302 . . . . . . . . . . . . . 14 (((𝐹𝑐) − 1) ∈ ℕ0 → (♯‘(1...((𝐹𝑐) − 1))) = ((𝐹𝑐) − 1))
111109, 110syl 17 . . . . . . . . . . . . 13 (((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (𝑐𝑅 ∧ (𝐹𝑐) ∈ ℕ)) ∧ (𝑑𝑅𝑧 ⊆ (1...((𝐹𝑐) − 1)))) → (♯‘(1...((𝐹𝑐) − 1))) = ((𝐹𝑐) − 1))
112108, 111breqtrd 5173 . . . . . . . . . . . 12 (((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (𝑐𝑅 ∧ (𝐹𝑐) ∈ ℕ)) ∧ (𝑑𝑅𝑧 ⊆ (1...((𝐹𝑐) − 1)))) → (♯‘𝑧) ≤ ((𝐹𝑐) − 1))
113 hashcl 14312 . . . . . . . . . . . . . 14 (𝑧 ∈ Fin → (♯‘𝑧) ∈ ℕ0)
114105, 113syl 17 . . . . . . . . . . . . 13 (((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (𝑐𝑅 ∧ (𝐹𝑐) ∈ ℕ)) ∧ (𝑑𝑅𝑧 ⊆ (1...((𝐹𝑐) − 1)))) → (♯‘𝑧) ∈ ℕ0)
1155ffvelcdmda 7082 . . . . . . . . . . . . . . 15 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ 𝑐𝑅) → (𝐹𝑐) ∈ ℕ0)
116115adantrr 716 . . . . . . . . . . . . . 14 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (𝑐𝑅 ∧ (𝐹𝑐) ∈ ℕ)) → (𝐹𝑐) ∈ ℕ0)
117116adantr 482 . . . . . . . . . . . . 13 (((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (𝑐𝑅 ∧ (𝐹𝑐) ∈ ℕ)) ∧ (𝑑𝑅𝑧 ⊆ (1...((𝐹𝑐) − 1)))) → (𝐹𝑐) ∈ ℕ0)
118 nn0ltlem1 12618 . . . . . . . . . . . . 13 (((♯‘𝑧) ∈ ℕ0 ∧ (𝐹𝑐) ∈ ℕ0) → ((♯‘𝑧) < (𝐹𝑐) ↔ (♯‘𝑧) ≤ ((𝐹𝑐) − 1)))
119114, 117, 118syl2anc 585 . . . . . . . . . . . 12 (((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (𝑐𝑅 ∧ (𝐹𝑐) ∈ ℕ)) ∧ (𝑑𝑅𝑧 ⊆ (1...((𝐹𝑐) − 1)))) → ((♯‘𝑧) < (𝐹𝑐) ↔ (♯‘𝑧) ≤ ((𝐹𝑐) − 1)))
120112, 119mpbird 257 . . . . . . . . . . 11 (((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (𝑐𝑅 ∧ (𝐹𝑐) ∈ ℕ)) ∧ (𝑑𝑅𝑧 ⊆ (1...((𝐹𝑐) − 1)))) → (♯‘𝑧) < (𝐹𝑐))
12124, 85fvsn 7174 . . . . . . . . . . . . . . 15 ({⟨∅, 𝑐⟩}‘∅) = 𝑐
122 f1ofn 6831 . . . . . . . . . . . . . . . . 17 ({⟨∅, 𝑐⟩}:{∅}–1-1-onto→{𝑐} → {⟨∅, 𝑐⟩} Fn {∅})
123 elpreima 7055 . . . . . . . . . . . . . . . . 17 ({⟨∅, 𝑐⟩} Fn {∅} → (∅ ∈ ({⟨∅, 𝑐⟩} “ {𝑑}) ↔ (∅ ∈ {∅} ∧ ({⟨∅, 𝑐⟩}‘∅) ∈ {𝑑})))
12486, 122, 123mp2b 10 . . . . . . . . . . . . . . . 16 (∅ ∈ ({⟨∅, 𝑐⟩} “ {𝑑}) ↔ (∅ ∈ {∅} ∧ ({⟨∅, 𝑐⟩}‘∅) ∈ {𝑑}))
125124simprbi 498 . . . . . . . . . . . . . . 15 (∅ ∈ ({⟨∅, 𝑐⟩} “ {𝑑}) → ({⟨∅, 𝑐⟩}‘∅) ∈ {𝑑})
126121, 125eqeltrrid 2839 . . . . . . . . . . . . . 14 (∅ ∈ ({⟨∅, 𝑐⟩} “ {𝑑}) → 𝑐 ∈ {𝑑})
127 elsni 4644 . . . . . . . . . . . . . 14 (𝑐 ∈ {𝑑} → 𝑐 = 𝑑)
128126, 127syl 17 . . . . . . . . . . . . 13 (∅ ∈ ({⟨∅, 𝑐⟩} “ {𝑑}) → 𝑐 = 𝑑)
129128fveq2d 6892 . . . . . . . . . . . 12 (∅ ∈ ({⟨∅, 𝑐⟩} “ {𝑑}) → (𝐹𝑐) = (𝐹𝑑))
130129breq2d 5159 . . . . . . . . . . 11 (∅ ∈ ({⟨∅, 𝑐⟩} “ {𝑑}) → ((♯‘𝑧) < (𝐹𝑐) ↔ (♯‘𝑧) < (𝐹𝑑)))
131120, 130syl5ibcom 244 . . . . . . . . . 10 (((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (𝑐𝑅 ∧ (𝐹𝑐) ∈ ℕ)) ∧ (𝑑𝑅𝑧 ⊆ (1...((𝐹𝑐) − 1)))) → (∅ ∈ ({⟨∅, 𝑐⟩} “ {𝑑}) → (♯‘𝑧) < (𝐹𝑑)))
132100, 131biimtrid 241 . . . . . . . . 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)))
1331, 80, 81, 82, 84, 97, 132ramlb 16948 . . . . . . . 8 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (𝑐𝑅 ∧ (𝐹𝑐) ∈ ℕ)) → ((𝐹𝑐) − 1) < (0 Ramsey 𝐹))
134 ramubcl 16947 . . . . . . . . . . 11 (((0 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (sup(ran 𝐹, ℝ, < ) ∈ ℕ0 ∧ (0 Ramsey 𝐹) ≤ sup(ran 𝐹, ℝ, < ))) → (0 Ramsey 𝐹) ∈ ℕ0)
1353, 4, 5, 18, 75, 134syl32anc 1379 . . . . . . . . . 10 (((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) → (0 Ramsey 𝐹) ∈ ℕ0)
136135adantr 482 . . . . . . . . 9 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (𝑐𝑅 ∧ (𝐹𝑐) ∈ ℕ)) → (0 Ramsey 𝐹) ∈ ℕ0)
137 nn0lem1lt 12623 . . . . . . . . 9 (((𝐹𝑐) ∈ ℕ0 ∧ (0 Ramsey 𝐹) ∈ ℕ0) → ((𝐹𝑐) ≤ (0 Ramsey 𝐹) ↔ ((𝐹𝑐) − 1) < (0 Ramsey 𝐹)))
138116, 136, 137syl2anc 585 . . . . . . . 8 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (𝑐𝑅 ∧ (𝐹𝑐) ∈ ℕ)) → ((𝐹𝑐) ≤ (0 Ramsey 𝐹) ↔ ((𝐹𝑐) − 1) < (0 Ramsey 𝐹)))
139133, 138mpbird 257 . . . . . . 7 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (𝑐𝑅 ∧ (𝐹𝑐) ∈ ℕ)) → (𝐹𝑐) ≤ (0 Ramsey 𝐹))
140139expr 458 . . . . . 6 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ 𝑐𝑅) → ((𝐹𝑐) ∈ ℕ → (𝐹𝑐) ≤ (0 Ramsey 𝐹)))
141135adantr 482 . . . . . . . 8 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ 𝑐𝑅) → (0 Ramsey 𝐹) ∈ ℕ0)
142141nn0ge0d 12531 . . . . . . 7 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ 𝑐𝑅) → 0 ≤ (0 Ramsey 𝐹))
143 breq1 5150 . . . . . . 7 ((𝐹𝑐) = 0 → ((𝐹𝑐) ≤ (0 Ramsey 𝐹) ↔ 0 ≤ (0 Ramsey 𝐹)))
144142, 143syl5ibrcom 246 . . . . . 6 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ 𝑐𝑅) → ((𝐹𝑐) = 0 → (𝐹𝑐) ≤ (0 Ramsey 𝐹)))
145 elnn0 12470 . . . . . . 7 ((𝐹𝑐) ∈ ℕ0 ↔ ((𝐹𝑐) ∈ ℕ ∨ (𝐹𝑐) = 0))
146115, 145sylib 217 . . . . . 6 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ 𝑐𝑅) → ((𝐹𝑐) ∈ ℕ ∨ (𝐹𝑐) = 0))
147140, 144, 146mpjaod 859 . . . . 5 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ 𝑐𝑅) → (𝐹𝑐) ≤ (0 Ramsey 𝐹))
148 breq1 5150 . . . . 5 ((𝐹𝑐) = sup(ran 𝐹, ℝ, < ) → ((𝐹𝑐) ≤ (0 Ramsey 𝐹) ↔ sup(ran 𝐹, ℝ, < ) ≤ (0 Ramsey 𝐹)))
149147, 148syl5ibcom 244 . . . 4 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ 𝑐𝑅) → ((𝐹𝑐) = sup(ran 𝐹, ℝ, < ) → sup(ran 𝐹, ℝ, < ) ≤ (0 Ramsey 𝐹)))
150149rexlimdva 3156 . . 3 (((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) → (∃𝑐𝑅 (𝐹𝑐) = sup(ran 𝐹, ℝ, < ) → sup(ran 𝐹, ℝ, < ) ≤ (0 Ramsey 𝐹)))
15179, 150mpd 15 . 2 (((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) → sup(ran 𝐹, ℝ, < ) ≤ (0 Ramsey 𝐹))
152135nn0red 12529 . . 3 (((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) → (0 Ramsey 𝐹) ∈ ℝ)
153152, 35letri3d 11352 . 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))))
15475, 151, 153mpbir2and 712 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 205  wa 397  wo 846  w3a 1088   = wceq 1542  wcel 2107  wne 2941  wral 3062  wrex 3071  {crab 3433  Vcvv 3475  wss 3947  c0 4321  𝒫 cpw 4601  {csn 4627  cop 4633   class class class wbr 5147  ccnv 5674  dom cdm 5675  ran crn 5676  cima 5678   Fn wfn 6535  wf 6536  1-1-ontowf1o 6539  cfv 6540  (class class class)co 7404  cmpo 7406  cdom 8933  Fincfn 8935  supcsup 9431  cr 11105  0cc0 11106  1c1 11107  *cxr 11243   < clt 11244  cle 11245  cmin 11440  cn 12208  0cn0 12468  cz 12554  ...cfz 13480  chash 14286   Ramsey cram 16928
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7720  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7360  df-ov 7407  df-oprab 7408  df-mpo 7409  df-om 7851  df-1st 7970  df-2nd 7971  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-1o 8461  df-oadd 8465  df-er 8699  df-map 8818  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-sup 9433  df-inf 9434  df-card 9930  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-nn 12209  df-n0 12469  df-xnn0 12541  df-z 12555  df-uz 12819  df-fz 13481  df-hash 14287  df-ram 16930
This theorem is referenced by:  0ram2  16950  ramz  16954
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