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Theorem 0ram 16344
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 2818 . . 3 (𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖}) = (𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})
2 0nn0 11900 . . . 4 0 ∈ ℕ0
32a1i 11 . . 3 (((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) → 0 ∈ ℕ0)
4 simpl1 1183 . . 3 (((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) → 𝑅𝑉)
5 simpl3 1185 . . 3 (((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) → 𝐹:𝑅⟶ℕ0)
65frnd 6514 . . . 4 (((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) → ran 𝐹 ⊆ ℕ0)
7 nn0ssz 11991 . . . . . 6 0 ⊆ ℤ
86, 7sstrdi 3976 . . . . 5 (((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) → ran 𝐹 ⊆ ℤ)
95fdmd 6516 . . . . . . 7 (((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) → dom 𝐹 = 𝑅)
10 simpl2 1184 . . . . . . 7 (((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) → 𝑅 ≠ ∅)
119, 10eqnetrd 3080 . . . . . 6 (((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) → dom 𝐹 ≠ ∅)
12 dm0rn0 5788 . . . . . . 7 (dom 𝐹 = ∅ ↔ ran 𝐹 = ∅)
1312necon3bii 3065 . . . . . 6 (dom 𝐹 ≠ ∅ ↔ ran 𝐹 ≠ ∅)
1411, 13sylib 219 . . . . 5 (((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) → ran 𝐹 ≠ ∅)
15 simpr 485 . . . . 5 (((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) → ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥)
16 suprzcl2 12326 . . . . 5 ((ran 𝐹 ⊆ ℤ ∧ ran 𝐹 ≠ ∅ ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) → sup(ran 𝐹, ℝ, < ) ∈ ran 𝐹)
178, 14, 15, 16syl3anc 1363 . . . 4 (((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) → sup(ran 𝐹, ℝ, < ) ∈ ran 𝐹)
186, 17sseldd 3965 . . 3 (((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) → sup(ran 𝐹, ℝ, < ) ∈ ℕ0)
191hashbc0 16329 . . . . . . 7 (𝑠 ∈ V → (𝑠(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})0) = {∅})
2019elv 3497 . . . . . 6 (𝑠(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})0) = {∅}
2120feq2i 6499 . . . . 5 (𝑓:(𝑠(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})0)⟶𝑅𝑓:{∅}⟶𝑅)
2221biimpi 217 . . . 4 (𝑓:(𝑠(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})0)⟶𝑅𝑓:{∅}⟶𝑅)
23 simprr 769 . . . . . 6 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (sup(ran 𝐹, ℝ, < ) ≤ (♯‘𝑠) ∧ 𝑓:{∅}⟶𝑅)) → 𝑓:{∅}⟶𝑅)
24 0ex 5202 . . . . . . 7 ∅ ∈ V
2524snid 4591 . . . . . 6 ∅ ∈ {∅}
26 ffvelrn 6841 . . . . . 6 ((𝑓:{∅}⟶𝑅 ∧ ∅ ∈ {∅}) → (𝑓‘∅) ∈ 𝑅)
2723, 25, 26sylancl 586 . . . . 5 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (sup(ran 𝐹, ℝ, < ) ≤ (♯‘𝑠) ∧ 𝑓:{∅}⟶𝑅)) → (𝑓‘∅) ∈ 𝑅)
28 vex 3495 . . . . . . 7 𝑠 ∈ V
2928pwid 4554 . . . . . 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 481 . . . . . . . . 9 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (sup(ran 𝐹, ℝ, < ) ≤ (♯‘𝑠) ∧ 𝑓:{∅}⟶𝑅)) → 𝐹:𝑅⟶ℕ0)
3231, 27ffvelrnd 6844 . . . . . . . 8 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (sup(ran 𝐹, ℝ, < ) ≤ (♯‘𝑠) ∧ 𝑓:{∅}⟶𝑅)) → (𝐹‘(𝑓‘∅)) ∈ ℕ0)
3332nn0red 11944 . . . . . . 7 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (sup(ran 𝐹, ℝ, < ) ≤ (♯‘𝑠) ∧ 𝑓:{∅}⟶𝑅)) → (𝐹‘(𝑓‘∅)) ∈ ℝ)
3433rexrd 10679 . . . . . 6 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (sup(ran 𝐹, ℝ, < ) ≤ (♯‘𝑠) ∧ 𝑓:{∅}⟶𝑅)) → (𝐹‘(𝑓‘∅)) ∈ ℝ*)
3518nn0red 11944 . . . . . . . 8 (((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) → sup(ran 𝐹, ℝ, < ) ∈ ℝ)
3635rexrd 10679 . . . . . . 7 (((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) → sup(ran 𝐹, ℝ, < ) ∈ ℝ*)
3736adantr 481 . . . . . 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 13706 . . . . . . 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 481 . . . . . . 7 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (sup(ran 𝐹, ℝ, < ) ≤ (♯‘𝑠) ∧ 𝑓:{∅}⟶𝑅)) → ran 𝐹 ⊆ ℤ)
4115adantr 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)
4231ffnd 6508 . . . . . . . 8 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (sup(ran 𝐹, ℝ, < ) ≤ (♯‘𝑠) ∧ 𝑓:{∅}⟶𝑅)) → 𝐹 Fn 𝑅)
43 fnfvelrn 6840 . . . . . . . 8 ((𝐹 Fn 𝑅 ∧ (𝑓‘∅) ∈ 𝑅) → (𝐹‘(𝑓‘∅)) ∈ ran 𝐹)
4442, 27, 43syl2anc 584 . . . . . . 7 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (sup(ran 𝐹, ℝ, < ) ≤ (♯‘𝑠) ∧ 𝑓:{∅}⟶𝑅)) → (𝐹‘(𝑓‘∅)) ∈ ran 𝐹)
45 suprzub 12327 . . . . . . 7 ((ran 𝐹 ⊆ ℤ ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥 ∧ (𝐹‘(𝑓‘∅)) ∈ ran 𝐹) → (𝐹‘(𝑓‘∅)) ≤ sup(ran 𝐹, ℝ, < ))
4640, 41, 44, 45syl3anc 1363 . . . . . 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 767 . . . . . 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 12543 . . . . 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 6676 . . . . . . . 8 (𝑓‘∅) ∈ V
5150snid 4591 . . . . . . 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 6507 . . . . . . 7 (𝑓:{∅}⟶𝑅𝑓 Fn {∅})
54 elpreima 6820 . . . . . . 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 709 . . . . 5 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (sup(ran 𝐹, ℝ, < ) ≤ (♯‘𝑠) ∧ 𝑓:{∅}⟶𝑅)) → ∅ ∈ (𝑓 “ {(𝑓‘∅)}))
57 fveq2 6663 . . . . . . . 8 (𝑐 = (𝑓‘∅) → (𝐹𝑐) = (𝐹‘(𝑓‘∅)))
5857breq1d 5067 . . . . . . 7 (𝑐 = (𝑓‘∅) → ((𝐹𝑐) ≤ (♯‘𝑧) ↔ (𝐹‘(𝑓‘∅)) ≤ (♯‘𝑧)))
591hashbc0 16329 . . . . . . . . . . 11 (𝑧 ∈ V → (𝑧(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})0) = {∅})
6059elv 3497 . . . . . . . . . 10 (𝑧(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})0) = {∅}
6160sseq1i 3992 . . . . . . . . 9 ((𝑧(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})0) ⊆ (𝑓 “ {𝑐}) ↔ {∅} ⊆ (𝑓 “ {𝑐}))
6224snss 4710 . . . . . . . . 9 (∅ ∈ (𝑓 “ {𝑐}) ↔ {∅} ⊆ (𝑓 “ {𝑐}))
6361, 62bitr4i 279 . . . . . . . 8 ((𝑧(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})0) ⊆ (𝑓 “ {𝑐}) ↔ ∅ ∈ (𝑓 “ {𝑐}))
64 sneq 4567 . . . . . . . . . 10 (𝑐 = (𝑓‘∅) → {𝑐} = {(𝑓‘∅)})
6564imaeq2d 5922 . . . . . . . . 9 (𝑐 = (𝑓‘∅) → (𝑓 “ {𝑐}) = (𝑓 “ {(𝑓‘∅)}))
6665eleq2d 2895 . . . . . . . 8 (𝑐 = (𝑓‘∅) → (∅ ∈ (𝑓 “ {𝑐}) ↔ ∅ ∈ (𝑓 “ {(𝑓‘∅)})))
6763, 66syl5bb 284 . . . . . . 7 (𝑐 = (𝑓‘∅) → ((𝑧(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})0) ⊆ (𝑓 “ {𝑐}) ↔ ∅ ∈ (𝑓 “ {(𝑓‘∅)})))
6858, 67anbi12d 630 . . . . . 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 6663 . . . . . . . 8 (𝑧 = 𝑠 → (♯‘𝑧) = (♯‘𝑠))
7069breq2d 5069 . . . . . . 7 (𝑧 = 𝑠 → ((𝐹‘(𝑓‘∅)) ≤ (♯‘𝑧) ↔ (𝐹‘(𝑓‘∅)) ≤ (♯‘𝑠)))
7170anbi1d 629 . . . . . 6 (𝑧 = 𝑠 → (((𝐹‘(𝑓‘∅)) ≤ (♯‘𝑧) ∧ ∅ ∈ (𝑓 “ {(𝑓‘∅)})) ↔ ((𝐹‘(𝑓‘∅)) ≤ (♯‘𝑠) ∧ ∅ ∈ (𝑓 “ {(𝑓‘∅)}))))
7268, 71rspc2ev 3632 . . . . 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 1366 . . . 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 679 . . 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 16337 . 2 (((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) → (0 Ramsey 𝐹) ≤ sup(ran 𝐹, ℝ, < ))
76 ffn 6507 . . . . 5 (𝐹:𝑅⟶ℕ0𝐹 Fn 𝑅)
77 fvelrnb 6719 . . . . 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 233 . . 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 1204 . . . . . . . . 9 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (𝑐𝑅 ∧ (𝐹𝑐) ∈ ℕ)) → 𝑅𝑉)
82 simpll3 1206 . . . . . . . . 9 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (𝑐𝑅 ∧ (𝐹𝑐) ∈ ℕ)) → 𝐹:𝑅⟶ℕ0)
83 nnm1nn0 11926 . . . . . . . . . 10 ((𝐹𝑐) ∈ ℕ → ((𝐹𝑐) − 1) ∈ ℕ0)
8483ad2antll 725 . . . . . . . . 9 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (𝑐𝑅 ∧ (𝐹𝑐) ∈ ℕ)) → ((𝐹𝑐) − 1) ∈ ℕ0)
85 vex 3495 . . . . . . . . . . . . 13 𝑐 ∈ V
8624, 85f1osn 6647 . . . . . . . . . . . 12 {⟨∅, 𝑐⟩}:{∅}–1-1-onto→{𝑐}
87 f1of 6608 . . . . . . . . . . . 12 ({⟨∅, 𝑐⟩}:{∅}–1-1-onto→{𝑐} → {⟨∅, 𝑐⟩}:{∅}⟶{𝑐})
8886, 87ax-mp 5 . . . . . . . . . . 11 {⟨∅, 𝑐⟩}:{∅}⟶{𝑐}
89 simprl 767 . . . . . . . . . . . 12 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (𝑐𝑅 ∧ (𝐹𝑐) ∈ ℕ)) → 𝑐𝑅)
9089snssd 4734 . . . . . . . . . . 11 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (𝑐𝑅 ∧ (𝐹𝑐) ∈ ℕ)) → {𝑐} ⊆ 𝑅)
91 fss 6520 . . . . . . . . . . 11 (({⟨∅, 𝑐⟩}:{∅}⟶{𝑐} ∧ {𝑐} ⊆ 𝑅) → {⟨∅, 𝑐⟩}:{∅}⟶𝑅)
9288, 90, 91sylancr 587 . . . . . . . . . 10 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (𝑐𝑅 ∧ (𝐹𝑐) ∈ ℕ)) → {⟨∅, 𝑐⟩}:{∅}⟶𝑅)
93 ovex 7178 . . . . . . . . . . . 12 (1...((𝐹𝑐) − 1)) ∈ V
941hashbc0 16329 . . . . . . . . . . . 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 6499 . . . . . . . . . 10 ({⟨∅, 𝑐⟩}:((1...((𝐹𝑐) − 1))(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})0)⟶𝑅 ↔ {⟨∅, 𝑐⟩}:{∅}⟶𝑅)
9792, 96sylibr 235 . . . . . . . . 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 3992 . . . . . . . . . . 11 ((𝑧(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})0) ⊆ ({⟨∅, 𝑐⟩} “ {𝑑}) ↔ {∅} ⊆ ({⟨∅, 𝑐⟩} “ {𝑑}))
9924snss 4710 . . . . . . . . . . 11 (∅ ∈ ({⟨∅, 𝑐⟩} “ {𝑑}) ↔ {∅} ⊆ ({⟨∅, 𝑐⟩} “ {𝑑}))
10098, 99bitr4i 279 . . . . . . . . . 10 ((𝑧(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})0) ⊆ ({⟨∅, 𝑐⟩} “ {𝑑}) ↔ ∅ ∈ ({⟨∅, 𝑐⟩} “ {𝑑}))
101 fzfid 13329 . . . . . . . . . . . . . . 15 (((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (𝑐𝑅 ∧ (𝐹𝑐) ∈ ℕ)) ∧ (𝑑𝑅𝑧 ⊆ (1...((𝐹𝑐) − 1)))) → (1...((𝐹𝑐) − 1)) ∈ Fin)
102 simprr 769 . . . . . . . . . . . . . . 15 (((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (𝑐𝑅 ∧ (𝐹𝑐) ∈ ℕ)) ∧ (𝑑𝑅𝑧 ⊆ (1...((𝐹𝑐) − 1)))) → 𝑧 ⊆ (1...((𝐹𝑐) − 1)))
103 ssdomg 8543 . . . . . . . . . . . . . . 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 8729 . . . . . . . . . . . . . . 15 (((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (𝑐𝑅 ∧ (𝐹𝑐) ∈ ℕ)) ∧ (𝑑𝑅𝑧 ⊆ (1...((𝐹𝑐) − 1)))) → 𝑧 ∈ Fin)
106 hashdom 13728 . . . . . . . . . . . . . . 15 ((𝑧 ∈ Fin ∧ (1...((𝐹𝑐) − 1)) ∈ Fin) → ((♯‘𝑧) ≤ (♯‘(1...((𝐹𝑐) − 1))) ↔ 𝑧 ≼ (1...((𝐹𝑐) − 1))))
107105, 101, 106syl2anc 584 . . . . . . . . . . . . . 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 258 . . . . . . . . . . . . 13 (((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (𝑐𝑅 ∧ (𝐹𝑐) ∈ ℕ)) ∧ (𝑑𝑅𝑧 ⊆ (1...((𝐹𝑐) − 1)))) → (♯‘𝑧) ≤ (♯‘(1...((𝐹𝑐) − 1))))
10984adantr 481 . . . . . . . . . . . . . 14 (((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (𝑐𝑅 ∧ (𝐹𝑐) ∈ ℕ)) ∧ (𝑑𝑅𝑧 ⊆ (1...((𝐹𝑐) − 1)))) → ((𝐹𝑐) − 1) ∈ ℕ0)
110 hashfz1 13694 . . . . . . . . . . . . . 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 5083 . . . . . . . . . . . 12 (((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (𝑐𝑅 ∧ (𝐹𝑐) ∈ ℕ)) ∧ (𝑑𝑅𝑧 ⊆ (1...((𝐹𝑐) − 1)))) → (♯‘𝑧) ≤ ((𝐹𝑐) − 1))
113 hashcl 13705 . . . . . . . . . . . . . 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)
1155ffvelrnda 6843 . . . . . . . . . . . . . . 15 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ 𝑐𝑅) → (𝐹𝑐) ∈ ℕ0)
116115adantrr 713 . . . . . . . . . . . . . 14 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (𝑐𝑅 ∧ (𝐹𝑐) ∈ ℕ)) → (𝐹𝑐) ∈ ℕ0)
117116adantr 481 . . . . . . . . . . . . 13 (((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (𝑐𝑅 ∧ (𝐹𝑐) ∈ ℕ)) ∧ (𝑑𝑅𝑧 ⊆ (1...((𝐹𝑐) − 1)))) → (𝐹𝑐) ∈ ℕ0)
118 nn0ltlem1 12030 . . . . . . . . . . . . 13 (((♯‘𝑧) ∈ ℕ0 ∧ (𝐹𝑐) ∈ ℕ0) → ((♯‘𝑧) < (𝐹𝑐) ↔ (♯‘𝑧) ≤ ((𝐹𝑐) − 1)))
119114, 117, 118syl2anc 584 . . . . . . . . . . . 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 258 . . . . . . . . . . 11 (((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (𝑐𝑅 ∧ (𝐹𝑐) ∈ ℕ)) ∧ (𝑑𝑅𝑧 ⊆ (1...((𝐹𝑐) − 1)))) → (♯‘𝑧) < (𝐹𝑐))
12124, 85fvsn 6935 . . . . . . . . . . . . . . 15 ({⟨∅, 𝑐⟩}‘∅) = 𝑐
122 f1ofn 6609 . . . . . . . . . . . . . . . . 17 ({⟨∅, 𝑐⟩}:{∅}–1-1-onto→{𝑐} → {⟨∅, 𝑐⟩} Fn {∅})
123 elpreima 6820 . . . . . . . . . . . . . . . . 17 ({⟨∅, 𝑐⟩} Fn {∅} → (∅ ∈ ({⟨∅, 𝑐⟩} “ {𝑑}) ↔ (∅ ∈ {∅} ∧ ({⟨∅, 𝑐⟩}‘∅) ∈ {𝑑})))
12486, 122, 123mp2b 10 . . . . . . . . . . . . . . . 16 (∅ ∈ ({⟨∅, 𝑐⟩} “ {𝑑}) ↔ (∅ ∈ {∅} ∧ ({⟨∅, 𝑐⟩}‘∅) ∈ {𝑑}))
125124simprbi 497 . . . . . . . . . . . . . . 15 (∅ ∈ ({⟨∅, 𝑐⟩} “ {𝑑}) → ({⟨∅, 𝑐⟩}‘∅) ∈ {𝑑})
126121, 125eqeltrrid 2915 . . . . . . . . . . . . . 14 (∅ ∈ ({⟨∅, 𝑐⟩} “ {𝑑}) → 𝑐 ∈ {𝑑})
127 elsni 4574 . . . . . . . . . . . . . 14 (𝑐 ∈ {𝑑} → 𝑐 = 𝑑)
128126, 127syl 17 . . . . . . . . . . . . 13 (∅ ∈ ({⟨∅, 𝑐⟩} “ {𝑑}) → 𝑐 = 𝑑)
129128fveq2d 6667 . . . . . . . . . . . 12 (∅ ∈ ({⟨∅, 𝑐⟩} “ {𝑑}) → (𝐹𝑐) = (𝐹𝑑))
130129breq2d 5069 . . . . . . . . . . 11 (∅ ∈ ({⟨∅, 𝑐⟩} “ {𝑑}) → ((♯‘𝑧) < (𝐹𝑐) ↔ (♯‘𝑧) < (𝐹𝑑)))
131120, 130syl5ibcom 246 . . . . . . . . . 10 (((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (𝑐𝑅 ∧ (𝐹𝑐) ∈ ℕ)) ∧ (𝑑𝑅𝑧 ⊆ (1...((𝐹𝑐) − 1)))) → (∅ ∈ ({⟨∅, 𝑐⟩} “ {𝑑}) → (♯‘𝑧) < (𝐹𝑑)))
132100, 131syl5bi 243 . . . . . . . . 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 16343 . . . . . . . 8 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (𝑐𝑅 ∧ (𝐹𝑐) ∈ ℕ)) → ((𝐹𝑐) − 1) < (0 Ramsey 𝐹))
134 ramubcl 16342 . . . . . . . . . . 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 1370 . . . . . . . . . 10 (((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) → (0 Ramsey 𝐹) ∈ ℕ0)
136135adantr 481 . . . . . . . . 9 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (𝑐𝑅 ∧ (𝐹𝑐) ∈ ℕ)) → (0 Ramsey 𝐹) ∈ ℕ0)
137 nn0lem1lt 12035 . . . . . . . . 9 (((𝐹𝑐) ∈ ℕ0 ∧ (0 Ramsey 𝐹) ∈ ℕ0) → ((𝐹𝑐) ≤ (0 Ramsey 𝐹) ↔ ((𝐹𝑐) − 1) < (0 Ramsey 𝐹)))
138116, 136, 137syl2anc 584 . . . . . . . 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 258 . . . . . . 7 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (𝑐𝑅 ∧ (𝐹𝑐) ∈ ℕ)) → (𝐹𝑐) ≤ (0 Ramsey 𝐹))
140139expr 457 . . . . . 6 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ 𝑐𝑅) → ((𝐹𝑐) ∈ ℕ → (𝐹𝑐) ≤ (0 Ramsey 𝐹)))
141135adantr 481 . . . . . . . 8 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ 𝑐𝑅) → (0 Ramsey 𝐹) ∈ ℕ0)
142141nn0ge0d 11946 . . . . . . 7 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ 𝑐𝑅) → 0 ≤ (0 Ramsey 𝐹))
143 breq1 5060 . . . . . . 7 ((𝐹𝑐) = 0 → ((𝐹𝑐) ≤ (0 Ramsey 𝐹) ↔ 0 ≤ (0 Ramsey 𝐹)))
144142, 143syl5ibrcom 248 . . . . . 6 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ 𝑐𝑅) → ((𝐹𝑐) = 0 → (𝐹𝑐) ≤ (0 Ramsey 𝐹)))
145 elnn0 11887 . . . . . . 7 ((𝐹𝑐) ∈ ℕ0 ↔ ((𝐹𝑐) ∈ ℕ ∨ (𝐹𝑐) = 0))
146115, 145sylib 219 . . . . . 6 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ 𝑐𝑅) → ((𝐹𝑐) ∈ ℕ ∨ (𝐹𝑐) = 0))
147140, 144, 146mpjaod 854 . . . . 5 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ 𝑐𝑅) → (𝐹𝑐) ≤ (0 Ramsey 𝐹))
148 breq1 5060 . . . . 5 ((𝐹𝑐) = sup(ran 𝐹, ℝ, < ) → ((𝐹𝑐) ≤ (0 Ramsey 𝐹) ↔ sup(ran 𝐹, ℝ, < ) ≤ (0 Ramsey 𝐹)))
149147, 148syl5ibcom 246 . . . 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 3281 . . 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 11944 . . 3 (((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) → (0 Ramsey 𝐹) ∈ ℝ)
153152, 35letri3d 10770 . 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 709 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 207  wa 396  wo 841  w3a 1079   = wceq 1528  wcel 2105  wne 3013  wral 3135  wrex 3136  {crab 3139  Vcvv 3492  wss 3933  c0 4288  𝒫 cpw 4535  {csn 4557  cop 4563   class class class wbr 5057  ccnv 5547  dom cdm 5548  ran crn 5549  cima 5551   Fn wfn 6343  wf 6344  1-1-ontowf1o 6347  cfv 6348  (class class class)co 7145  cmpo 7147  cdom 8495  Fincfn 8497  supcsup 8892  cr 10524  0cc0 10525  1c1 10526  *cxr 10662   < clt 10663  cle 10664  cmin 10858  cn 11626  0cn0 11885  cz 11969  ...cfz 12880  chash 13678   Ramsey cram 16323
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-oadd 8095  df-er 8278  df-map 8397  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-sup 8894  df-inf 8895  df-card 9356  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-nn 11627  df-n0 11886  df-xnn0 11956  df-z 11970  df-uz 12232  df-fz 12881  df-hash 13679  df-ram 16325
This theorem is referenced by:  0ram2  16345  ramz  16349
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