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Theorem 0ram 15648
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 2621 . . 3 (𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (#‘𝑏) = 𝑖}) = (𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (#‘𝑏) = 𝑖})
2 0nn0 11251 . . . 4 0 ∈ ℕ0
32a1i 11 . . 3 (((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) → 0 ∈ ℕ0)
4 simpl1 1062 . . 3 (((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) → 𝑅𝑉)
5 simpl3 1064 . . 3 (((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) → 𝐹:𝑅⟶ℕ0)
6 frn 6010 . . . . 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 11342 . . . . . 6 0 ⊆ ℤ
97, 8syl6ss 3595 . . . . 5 (((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) → ran 𝐹 ⊆ ℤ)
10 fdm 6008 . . . . . . . 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 1063 . . . . . . 7 (((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) → 𝑅 ≠ ∅)
1311, 12eqnetrd 2857 . . . . . 6 (((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) → dom 𝐹 ≠ ∅)
14 dm0rn0 5302 . . . . . . 7 (dom 𝐹 = ∅ ↔ ran 𝐹 = ∅)
1514necon3bii 2842 . . . . . 6 (dom 𝐹 ≠ ∅ ↔ ran 𝐹 ≠ ∅)
1613, 15sylib 208 . . . . 5 (((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) → ran 𝐹 ≠ ∅)
17 simpr 477 . . . . 5 (((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) → ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥)
18 suprzcl2 11722 . . . . 5 ((ran 𝐹 ⊆ ℤ ∧ ran 𝐹 ≠ ∅ ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) → sup(ran 𝐹, ℝ, < ) ∈ ran 𝐹)
199, 16, 17, 18syl3anc 1323 . . . 4 (((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) → sup(ran 𝐹, ℝ, < ) ∈ ran 𝐹)
207, 19sseldd 3584 . . 3 (((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) → sup(ran 𝐹, ℝ, < ) ∈ ℕ0)
21 vex 3189 . . . . . . 7 𝑠 ∈ V
221hashbc0 15633 . . . . . . 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 5994 . . . . 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 795 . . . . . 6 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (sup(ran 𝐹, ℝ, < ) ≤ (#‘𝑠) ∧ 𝑓:{∅}⟶𝑅)) → 𝑓:{∅}⟶𝑅)
27 0ex 4750 . . . . . . 7 ∅ ∈ V
2827snid 4179 . . . . . 6 ∅ ∈ {∅}
29 ffvelrn 6313 . . . . . 6 ((𝑓:{∅}⟶𝑅 ∧ ∅ ∈ {∅}) → (𝑓‘∅) ∈ 𝑅)
3026, 28, 29sylancl 693 . . . . 5 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (sup(ran 𝐹, ℝ, < ) ≤ (#‘𝑠) ∧ 𝑓:{∅}⟶𝑅)) → (𝑓‘∅) ∈ 𝑅)
3121pwid 4145 . . . . . 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 481 . . . . . . . . 9 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (sup(ran 𝐹, ℝ, < ) ≤ (#‘𝑠) ∧ 𝑓:{∅}⟶𝑅)) → 𝐹:𝑅⟶ℕ0)
3433, 30ffvelrnd 6316 . . . . . . . 8 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (sup(ran 𝐹, ℝ, < ) ≤ (#‘𝑠) ∧ 𝑓:{∅}⟶𝑅)) → (𝐹‘(𝑓‘∅)) ∈ ℕ0)
3534nn0red 11296 . . . . . . 7 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (sup(ran 𝐹, ℝ, < ) ≤ (#‘𝑠) ∧ 𝑓:{∅}⟶𝑅)) → (𝐹‘(𝑓‘∅)) ∈ ℝ)
3635rexrd 10033 . . . . . 6 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (sup(ran 𝐹, ℝ, < ) ≤ (#‘𝑠) ∧ 𝑓:{∅}⟶𝑅)) → (𝐹‘(𝑓‘∅)) ∈ ℝ*)
3720nn0red 11296 . . . . . . . 8 (((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) → sup(ran 𝐹, ℝ, < ) ∈ ℝ)
3837rexrd 10033 . . . . . . 7 (((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) → sup(ran 𝐹, ℝ, < ) ∈ ℝ*)
3938adantr 481 . . . . . 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 13088 . . . . . . 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 481 . . . . . . 7 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (sup(ran 𝐹, ℝ, < ) ≤ (#‘𝑠) ∧ 𝑓:{∅}⟶𝑅)) → ran 𝐹 ⊆ ℤ)
4317adantr 481 . . . . . . 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 6002 . . . . . . . . 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 6312 . . . . . . . 8 ((𝐹 Fn 𝑅 ∧ (𝑓‘∅) ∈ 𝑅) → (𝐹‘(𝑓‘∅)) ∈ ran 𝐹)
4745, 30, 46syl2anc 692 . . . . . . 7 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (sup(ran 𝐹, ℝ, < ) ≤ (#‘𝑠) ∧ 𝑓:{∅}⟶𝑅)) → (𝐹‘(𝑓‘∅)) ∈ ran 𝐹)
48 suprzub 11723 . . . . . . 7 ((ran 𝐹 ⊆ ℤ ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥 ∧ (𝐹‘(𝑓‘∅)) ∈ ran 𝐹) → (𝐹‘(𝑓‘∅)) ≤ sup(ran 𝐹, ℝ, < ))
4942, 43, 47, 48syl3anc 1323 . . . . . 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 793 . . . . . 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 11937 . . . . 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 6158 . . . . . . . 8 (𝑓‘∅) ∈ V
5453snid 4179 . . . . . . 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 6002 . . . . . . 7 (𝑓:{∅}⟶𝑅𝑓 Fn {∅})
57 elpreima 6293 . . . . . . 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 956 . . . . 5 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (sup(ran 𝐹, ℝ, < ) ≤ (#‘𝑠) ∧ 𝑓:{∅}⟶𝑅)) → ∅ ∈ (𝑓 “ {(𝑓‘∅)}))
60 fveq2 6148 . . . . . . . 8 (𝑐 = (𝑓‘∅) → (𝐹𝑐) = (𝐹‘(𝑓‘∅)))
6160breq1d 4623 . . . . . . 7 (𝑐 = (𝑓‘∅) → ((𝐹𝑐) ≤ (#‘𝑧) ↔ (𝐹‘(𝑓‘∅)) ≤ (#‘𝑧)))
62 vex 3189 . . . . . . . . . . 11 𝑧 ∈ V
631hashbc0 15633 . . . . . . . . . . 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 3608 . . . . . . . . 9 ((𝑧(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (#‘𝑏) = 𝑖})0) ⊆ (𝑓 “ {𝑐}) ↔ {∅} ⊆ (𝑓 “ {𝑐}))
6627snss 4286 . . . . . . . . 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 4158 . . . . . . . . . 10 (𝑐 = (𝑓‘∅) → {𝑐} = {(𝑓‘∅)})
6968imaeq2d 5425 . . . . . . . . 9 (𝑐 = (𝑓‘∅) → (𝑓 “ {𝑐}) = (𝑓 “ {(𝑓‘∅)}))
7069eleq2d 2684 . . . . . . . 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 746 . . . . . 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 6148 . . . . . . . 8 (𝑧 = 𝑠 → (#‘𝑧) = (#‘𝑠))
7473breq2d 4625 . . . . . . 7 (𝑧 = 𝑠 → ((𝐹‘(𝑓‘∅)) ≤ (#‘𝑧) ↔ (𝐹‘(𝑓‘∅)) ≤ (#‘𝑠)))
7574anbi1d 740 . . . . . 6 (𝑧 = 𝑠 → (((𝐹‘(𝑓‘∅)) ≤ (#‘𝑧) ∧ ∅ ∈ (𝑓 “ {(𝑓‘∅)})) ↔ ((𝐹‘(𝑓‘∅)) ≤ (#‘𝑠) ∧ ∅ ∈ (𝑓 “ {(𝑓‘∅)}))))
7672, 75rspc2ev 3308 . . . . 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 1327 . . . 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 684 . . 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 15641 . 2 (((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) → (0 Ramsey 𝐹) ≤ sup(ran 𝐹, ℝ, < ))
80 fvelrnb 6200 . . . . 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 1098 . . . . . . . . 9 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (𝑐𝑅 ∧ (𝐹𝑐) ∈ ℕ)) → 𝑅𝑉)
85 simpll3 1100 . . . . . . . . 9 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (𝑐𝑅 ∧ (𝐹𝑐) ∈ ℕ)) → 𝐹:𝑅⟶ℕ0)
86 nnm1nn0 11278 . . . . . . . . . 10 ((𝐹𝑐) ∈ ℕ → ((𝐹𝑐) − 1) ∈ ℕ0)
8786ad2antll 764 . . . . . . . . 9 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (𝑐𝑅 ∧ (𝐹𝑐) ∈ ℕ)) → ((𝐹𝑐) − 1) ∈ ℕ0)
88 vex 3189 . . . . . . . . . . . . 13 𝑐 ∈ V
8927, 88f1osn 6133 . . . . . . . . . . . 12 {⟨∅, 𝑐⟩}:{∅}–1-1-onto→{𝑐}
90 f1of 6094 . . . . . . . . . . . 12 ({⟨∅, 𝑐⟩}:{∅}–1-1-onto→{𝑐} → {⟨∅, 𝑐⟩}:{∅}⟶{𝑐})
9189, 90ax-mp 5 . . . . . . . . . . 11 {⟨∅, 𝑐⟩}:{∅}⟶{𝑐}
92 simprl 793 . . . . . . . . . . . 12 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (𝑐𝑅 ∧ (𝐹𝑐) ∈ ℕ)) → 𝑐𝑅)
9392snssd 4309 . . . . . . . . . . 11 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (𝑐𝑅 ∧ (𝐹𝑐) ∈ ℕ)) → {𝑐} ⊆ 𝑅)
94 fss 6013 . . . . . . . . . . 11 (({⟨∅, 𝑐⟩}:{∅}⟶{𝑐} ∧ {𝑐} ⊆ 𝑅) → {⟨∅, 𝑐⟩}:{∅}⟶𝑅)
9591, 93, 94sylancr 694 . . . . . . . . . 10 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (𝑐𝑅 ∧ (𝐹𝑐) ∈ ℕ)) → {⟨∅, 𝑐⟩}:{∅}⟶𝑅)
96 ovex 6632 . . . . . . . . . . . 12 (1...((𝐹𝑐) − 1)) ∈ V
971hashbc0 15633 . . . . . . . . . . . 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 5994 . . . . . . . . . 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 3608 . . . . . . . . . . 11 ((𝑧(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (#‘𝑏) = 𝑖})0) ⊆ ({⟨∅, 𝑐⟩} “ {𝑑}) ↔ {∅} ⊆ ({⟨∅, 𝑐⟩} “ {𝑑}))
10227snss 4286 . . . . . . . . . . 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 12712 . . . . . . . . . . . . . . 15 (((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (𝑐𝑅 ∧ (𝐹𝑐) ∈ ℕ)) ∧ (𝑑𝑅𝑧 ⊆ (1...((𝐹𝑐) − 1)))) → (1...((𝐹𝑐) − 1)) ∈ Fin)
105 simprr 795 . . . . . . . . . . . . . . 15 (((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (𝑐𝑅 ∧ (𝐹𝑐) ∈ ℕ)) ∧ (𝑑𝑅𝑧 ⊆ (1...((𝐹𝑐) − 1)))) → 𝑧 ⊆ (1...((𝐹𝑐) − 1)))
106 ssdomg 7945 . . . . . . . . . . . . . . 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 8124 . . . . . . . . . . . . . . . 16 (((1...((𝐹𝑐) − 1)) ∈ Fin ∧ 𝑧 ⊆ (1...((𝐹𝑐) − 1))) → 𝑧 ∈ Fin)
109104, 105, 108syl2anc 692 . . . . . . . . . . . . . . 15 (((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (𝑐𝑅 ∧ (𝐹𝑐) ∈ ℕ)) ∧ (𝑑𝑅𝑧 ⊆ (1...((𝐹𝑐) − 1)))) → 𝑧 ∈ Fin)
110 hashdom 13108 . . . . . . . . . . . . . . 15 ((𝑧 ∈ Fin ∧ (1...((𝐹𝑐) − 1)) ∈ Fin) → ((#‘𝑧) ≤ (#‘(1...((𝐹𝑐) − 1))) ↔ 𝑧 ≼ (1...((𝐹𝑐) − 1))))
111109, 104, 110syl2anc 692 . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . 14 (((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (𝑐𝑅 ∧ (𝐹𝑐) ∈ ℕ)) ∧ (𝑑𝑅𝑧 ⊆ (1...((𝐹𝑐) − 1)))) → ((𝐹𝑐) − 1) ∈ ℕ0)
114 hashfz1 13074 . . . . . . . . . . . . . 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 4639 . . . . . . . . . . . 12 (((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (𝑐𝑅 ∧ (𝐹𝑐) ∈ ℕ)) ∧ (𝑑𝑅𝑧 ⊆ (1...((𝐹𝑐) − 1)))) → (#‘𝑧) ≤ ((𝐹𝑐) − 1))
117 hashcl 13087 . . . . . . . . . . . . . 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 6315 . . . . . . . . . . . . . . 15 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ 𝑐𝑅) → (𝐹𝑐) ∈ ℕ0)
120119adantrr 752 . . . . . . . . . . . . . 14 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (𝑐𝑅 ∧ (𝐹𝑐) ∈ ℕ)) → (𝐹𝑐) ∈ ℕ0)
121120adantr 481 . . . . . . . . . . . . 13 (((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (𝑐𝑅 ∧ (𝐹𝑐) ∈ ℕ)) ∧ (𝑑𝑅𝑧 ⊆ (1...((𝐹𝑐) − 1)))) → (𝐹𝑐) ∈ ℕ0)
122 nn0ltlem1 11381 . . . . . . . . . . . . 13 (((#‘𝑧) ∈ ℕ0 ∧ (𝐹𝑐) ∈ ℕ0) → ((#‘𝑧) < (𝐹𝑐) ↔ (#‘𝑧) ≤ ((𝐹𝑐) − 1)))
123118, 121, 122syl2anc 692 . . . . . . . . . . . 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 6400 . . . . . . . . . . . . . . 15 ({⟨∅, 𝑐⟩}‘∅) = 𝑐
126 f1ofn 6095 . . . . . . . . . . . . . . . . 17 ({⟨∅, 𝑐⟩}:{∅}–1-1-onto→{𝑐} → {⟨∅, 𝑐⟩} Fn {∅})
127 elpreima 6293 . . . . . . . . . . . . . . . . 17 ({⟨∅, 𝑐⟩} Fn {∅} → (∅ ∈ ({⟨∅, 𝑐⟩} “ {𝑑}) ↔ (∅ ∈ {∅} ∧ ({⟨∅, 𝑐⟩}‘∅) ∈ {𝑑})))
12889, 126, 127mp2b 10 . . . . . . . . . . . . . . . 16 (∅ ∈ ({⟨∅, 𝑐⟩} “ {𝑑}) ↔ (∅ ∈ {∅} ∧ ({⟨∅, 𝑐⟩}‘∅) ∈ {𝑑}))
129128simprbi 480 . . . . . . . . . . . . . . 15 (∅ ∈ ({⟨∅, 𝑐⟩} “ {𝑑}) → ({⟨∅, 𝑐⟩}‘∅) ∈ {𝑑})
130125, 129syl5eqelr 2703 . . . . . . . . . . . . . 14 (∅ ∈ ({⟨∅, 𝑐⟩} “ {𝑑}) → 𝑐 ∈ {𝑑})
131 elsni 4165 . . . . . . . . . . . . . 14 (𝑐 ∈ {𝑑} → 𝑐 = 𝑑)
132130, 131syl 17 . . . . . . . . . . . . 13 (∅ ∈ ({⟨∅, 𝑐⟩} “ {𝑑}) → 𝑐 = 𝑑)
133132fveq2d 6152 . . . . . . . . . . . 12 (∅ ∈ ({⟨∅, 𝑐⟩} “ {𝑑}) → (𝐹𝑐) = (𝐹𝑑))
134133breq2d 4625 . . . . . . . . . . 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 15647 . . . . . . . 8 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (𝑐𝑅 ∧ (𝐹𝑐) ∈ ℕ)) → ((𝐹𝑐) − 1) < (0 Ramsey 𝐹))
138 ramubcl 15646 . . . . . . . . . . 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 1331 . . . . . . . . . 10 (((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) → (0 Ramsey 𝐹) ∈ ℕ0)
140139adantr 481 . . . . . . . . 9 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (𝑐𝑅 ∧ (𝐹𝑐) ∈ ℕ)) → (0 Ramsey 𝐹) ∈ ℕ0)
141 nn0lem1lt 11386 . . . . . . . . 9 (((𝐹𝑐) ∈ ℕ0 ∧ (0 Ramsey 𝐹) ∈ ℕ0) → ((𝐹𝑐) ≤ (0 Ramsey 𝐹) ↔ ((𝐹𝑐) − 1) < (0 Ramsey 𝐹)))
142120, 140, 141syl2anc 692 . . . . . . . 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 642 . . . . . 6 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ 𝑐𝑅) → ((𝐹𝑐) ∈ ℕ → (𝐹𝑐) ≤ (0 Ramsey 𝐹)))
145139adantr 481 . . . . . . . 8 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ 𝑐𝑅) → (0 Ramsey 𝐹) ∈ ℕ0)
146145nn0ge0d 11298 . . . . . . 7 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ 𝑐𝑅) → 0 ≤ (0 Ramsey 𝐹))
147 breq1 4616 . . . . . . 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 11238 . . . . . . 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 396 . . . . 5 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ 𝑐𝑅) → (𝐹𝑐) ≤ (0 Ramsey 𝐹))
152 breq1 4616 . . . . 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 3024 . . 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 11296 . . 3 (((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) → (0 Ramsey 𝐹) ∈ ℝ)
157156, 37letri3d 10123 . 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 956 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 383  wa 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790  wral 2907  wrex 2908  {crab 2911  Vcvv 3186  wss 3555  c0 3891  𝒫 cpw 4130  {csn 4148  cop 4154   class class class wbr 4613  ccnv 5073  dom cdm 5074  ran crn 5075  cima 5077   Fn wfn 5842  wf 5843  1-1-ontowf1o 5846  cfv 5847  (class class class)co 6604  cmpt2 6606  cdom 7897  Fincfn 7899  supcsup 8290  cr 9879  0cc0 9880  1c1 9881  *cxr 10017   < clt 10018  cle 10019  cmin 10210  cn 10964  0cn0 11236  cz 11321  ...cfz 12268  #chash 13057   Ramsey cram 15627
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-oadd 7509  df-er 7687  df-map 7804  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-sup 8292  df-inf 8293  df-card 8709  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-nn 10965  df-n0 11237  df-xnn0 11308  df-z 11322  df-uz 11632  df-fz 12269  df-hash 13058  df-ram 15629
This theorem is referenced by:  0ram2  15649  ramz  15653
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