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Theorem 0ram 16411
 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 2758 . . 3 (𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖}) = (𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})
2 0nn0 11949 . . . 4 0 ∈ ℕ0
32a1i 11 . . 3 (((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) → 0 ∈ ℕ0)
4 simpl1 1188 . . 3 (((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) → 𝑅𝑉)
5 simpl3 1190 . . 3 (((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) → 𝐹:𝑅⟶ℕ0)
65frnd 6505 . . . 4 (((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) → ran 𝐹 ⊆ ℕ0)
7 nn0ssz 12042 . . . . . 6 0 ⊆ ℤ
86, 7sstrdi 3904 . . . . 5 (((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) → ran 𝐹 ⊆ ℤ)
95fdmd 6508 . . . . . . 7 (((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) → dom 𝐹 = 𝑅)
10 simpl2 1189 . . . . . . 7 (((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) → 𝑅 ≠ ∅)
119, 10eqnetrd 3018 . . . . . 6 (((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) → dom 𝐹 ≠ ∅)
12 dm0rn0 5766 . . . . . . 7 (dom 𝐹 = ∅ ↔ ran 𝐹 = ∅)
1312necon3bii 3003 . . . . . 6 (dom 𝐹 ≠ ∅ ↔ ran 𝐹 ≠ ∅)
1411, 13sylib 221 . . . . 5 (((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) → ran 𝐹 ≠ ∅)
15 simpr 488 . . . . 5 (((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) → ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥)
16 suprzcl2 12378 . . . . 5 ((ran 𝐹 ⊆ ℤ ∧ ran 𝐹 ≠ ∅ ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) → sup(ran 𝐹, ℝ, < ) ∈ ran 𝐹)
178, 14, 15, 16syl3anc 1368 . . . 4 (((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) → sup(ran 𝐹, ℝ, < ) ∈ ran 𝐹)
186, 17sseldd 3893 . . 3 (((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) → sup(ran 𝐹, ℝ, < ) ∈ ℕ0)
191hashbc0 16396 . . . . . . 7 (𝑠 ∈ V → (𝑠(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})0) = {∅})
2019elv 3415 . . . . . 6 (𝑠(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})0) = {∅}
2120feq2i 6490 . . . . 5 (𝑓:(𝑠(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})0)⟶𝑅𝑓:{∅}⟶𝑅)
2221biimpi 219 . . . 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 5177 . . . . . . 7 ∅ ∈ V
2524snid 4558 . . . . . 6 ∅ ∈ {∅}
26 ffvelrn 6840 . . . . . 6 ((𝑓:{∅}⟶𝑅 ∧ ∅ ∈ {∅}) → (𝑓‘∅) ∈ 𝑅)
2723, 25, 26sylancl 589 . . . . 5 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (sup(ran 𝐹, ℝ, < ) ≤ (♯‘𝑠) ∧ 𝑓:{∅}⟶𝑅)) → (𝑓‘∅) ∈ 𝑅)
28 vex 3413 . . . . . . 7 𝑠 ∈ V
2928pwid 4518 . . . . . 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 484 . . . . . . . . 9 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (sup(ran 𝐹, ℝ, < ) ≤ (♯‘𝑠) ∧ 𝑓:{∅}⟶𝑅)) → 𝐹:𝑅⟶ℕ0)
3231, 27ffvelrnd 6843 . . . . . . . 8 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (sup(ran 𝐹, ℝ, < ) ≤ (♯‘𝑠) ∧ 𝑓:{∅}⟶𝑅)) → (𝐹‘(𝑓‘∅)) ∈ ℕ0)
3332nn0red 11995 . . . . . . 7 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (sup(ran 𝐹, ℝ, < ) ≤ (♯‘𝑠) ∧ 𝑓:{∅}⟶𝑅)) → (𝐹‘(𝑓‘∅)) ∈ ℝ)
3433rexrd 10729 . . . . . 6 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (sup(ran 𝐹, ℝ, < ) ≤ (♯‘𝑠) ∧ 𝑓:{∅}⟶𝑅)) → (𝐹‘(𝑓‘∅)) ∈ ℝ*)
3518nn0red 11995 . . . . . . . 8 (((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) → sup(ran 𝐹, ℝ, < ) ∈ ℝ)
3635rexrd 10729 . . . . . . 7 (((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) → sup(ran 𝐹, ℝ, < ) ∈ ℝ*)
3736adantr 484 . . . . . 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 13768 . . . . . . 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 484 . . . . . . 7 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (sup(ran 𝐹, ℝ, < ) ≤ (♯‘𝑠) ∧ 𝑓:{∅}⟶𝑅)) → ran 𝐹 ⊆ ℤ)
4115adantr 484 . . . . . . 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 6499 . . . . . . . 8 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (sup(ran 𝐹, ℝ, < ) ≤ (♯‘𝑠) ∧ 𝑓:{∅}⟶𝑅)) → 𝐹 Fn 𝑅)
43 fnfvelrn 6839 . . . . . . . 8 ((𝐹 Fn 𝑅 ∧ (𝑓‘∅) ∈ 𝑅) → (𝐹‘(𝑓‘∅)) ∈ ran 𝐹)
4442, 27, 43syl2anc 587 . . . . . . 7 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (sup(ran 𝐹, ℝ, < ) ≤ (♯‘𝑠) ∧ 𝑓:{∅}⟶𝑅)) → (𝐹‘(𝑓‘∅)) ∈ ran 𝐹)
45 suprzub 12379 . . . . . . 7 ((ran 𝐹 ⊆ ℤ ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥 ∧ (𝐹‘(𝑓‘∅)) ∈ ran 𝐹) → (𝐹‘(𝑓‘∅)) ≤ sup(ran 𝐹, ℝ, < ))
4640, 41, 44, 45syl3anc 1368 . . . . . 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 12596 . . . . 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 6671 . . . . . . . 8 (𝑓‘∅) ∈ V
5150snid 4558 . . . . . . 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 6498 . . . . . . 7 (𝑓:{∅}⟶𝑅𝑓 Fn {∅})
54 elpreima 6819 . . . . . . 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 6658 . . . . . . . 8 (𝑐 = (𝑓‘∅) → (𝐹𝑐) = (𝐹‘(𝑓‘∅)))
5857breq1d 5042 . . . . . . 7 (𝑐 = (𝑓‘∅) → ((𝐹𝑐) ≤ (♯‘𝑧) ↔ (𝐹‘(𝑓‘∅)) ≤ (♯‘𝑧)))
591hashbc0 16396 . . . . . . . . . . 11 (𝑧 ∈ V → (𝑧(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})0) = {∅})
6059elv 3415 . . . . . . . . . 10 (𝑧(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})0) = {∅}
6160sseq1i 3920 . . . . . . . . 9 ((𝑧(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})0) ⊆ (𝑓 “ {𝑐}) ↔ {∅} ⊆ (𝑓 “ {𝑐}))
6224snss 4676 . . . . . . . . 9 (∅ ∈ (𝑓 “ {𝑐}) ↔ {∅} ⊆ (𝑓 “ {𝑐}))
6361, 62bitr4i 281 . . . . . . . 8 ((𝑧(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})0) ⊆ (𝑓 “ {𝑐}) ↔ ∅ ∈ (𝑓 “ {𝑐}))
64 sneq 4532 . . . . . . . . . 10 (𝑐 = (𝑓‘∅) → {𝑐} = {(𝑓‘∅)})
6564imaeq2d 5901 . . . . . . . . 9 (𝑐 = (𝑓‘∅) → (𝑓 “ {𝑐}) = (𝑓 “ {(𝑓‘∅)}))
6665eleq2d 2837 . . . . . . . 8 (𝑐 = (𝑓‘∅) → (∅ ∈ (𝑓 “ {𝑐}) ↔ ∅ ∈ (𝑓 “ {(𝑓‘∅)})))
6763, 66syl5bb 286 . . . . . . 7 (𝑐 = (𝑓‘∅) → ((𝑧(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})0) ⊆ (𝑓 “ {𝑐}) ↔ ∅ ∈ (𝑓 “ {(𝑓‘∅)})))
6858, 67anbi12d 633 . . . . . 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 6658 . . . . . . . 8 (𝑧 = 𝑠 → (♯‘𝑧) = (♯‘𝑠))
7069breq2d 5044 . . . . . . 7 (𝑧 = 𝑠 → ((𝐹‘(𝑓‘∅)) ≤ (♯‘𝑧) ↔ (𝐹‘(𝑓‘∅)) ≤ (♯‘𝑠)))
7170anbi1d 632 . . . . . 6 (𝑧 = 𝑠 → (((𝐹‘(𝑓‘∅)) ≤ (♯‘𝑧) ∧ ∅ ∈ (𝑓 “ {(𝑓‘∅)})) ↔ ((𝐹‘(𝑓‘∅)) ≤ (♯‘𝑠) ∧ ∅ ∈ (𝑓 “ {(𝑓‘∅)}))))
7268, 71rspc2ev 3553 . . . . 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 1371 . . . 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 16404 . 2 (((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) → (0 Ramsey 𝐹) ≤ sup(ran 𝐹, ℝ, < ))
76 ffn 6498 . . . . 5 (𝐹:𝑅⟶ℕ0𝐹 Fn 𝑅)
77 fvelrnb 6714 . . . . 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 235 . . 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 1209 . . . . . . . . 9 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (𝑐𝑅 ∧ (𝐹𝑐) ∈ ℕ)) → 𝑅𝑉)
82 simpll3 1211 . . . . . . . . 9 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (𝑐𝑅 ∧ (𝐹𝑐) ∈ ℕ)) → 𝐹:𝑅⟶ℕ0)
83 nnm1nn0 11975 . . . . . . . . . 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 3413 . . . . . . . . . . . . 13 𝑐 ∈ V
8624, 85f1osn 6641 . . . . . . . . . . . 12 {⟨∅, 𝑐⟩}:{∅}–1-1-onto→{𝑐}
87 f1of 6602 . . . . . . . . . . . 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 4699 . . . . . . . . . . 11 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (𝑐𝑅 ∧ (𝐹𝑐) ∈ ℕ)) → {𝑐} ⊆ 𝑅)
91 fss 6512 . . . . . . . . . . 11 (({⟨∅, 𝑐⟩}:{∅}⟶{𝑐} ∧ {𝑐} ⊆ 𝑅) → {⟨∅, 𝑐⟩}:{∅}⟶𝑅)
9288, 90, 91sylancr 590 . . . . . . . . . 10 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (𝑐𝑅 ∧ (𝐹𝑐) ∈ ℕ)) → {⟨∅, 𝑐⟩}:{∅}⟶𝑅)
93 ovex 7183 . . . . . . . . . . . 12 (1...((𝐹𝑐) − 1)) ∈ V
941hashbc0 16396 . . . . . . . . . . . 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 6490 . . . . . . . . . 10 ({⟨∅, 𝑐⟩}:((1...((𝐹𝑐) − 1))(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})0)⟶𝑅 ↔ {⟨∅, 𝑐⟩}:{∅}⟶𝑅)
9792, 96sylibr 237 . . . . . . . . 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 3920 . . . . . . . . . . 11 ((𝑧(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})0) ⊆ ({⟨∅, 𝑐⟩} “ {𝑑}) ↔ {∅} ⊆ ({⟨∅, 𝑐⟩} “ {𝑑}))
9924snss 4676 . . . . . . . . . . 11 (∅ ∈ ({⟨∅, 𝑐⟩} “ {𝑑}) ↔ {∅} ⊆ ({⟨∅, 𝑐⟩} “ {𝑑}))
10098, 99bitr4i 281 . . . . . . . . . 10 ((𝑧(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})0) ⊆ ({⟨∅, 𝑐⟩} “ {𝑑}) ↔ ∅ ∈ ({⟨∅, 𝑐⟩} “ {𝑑}))
101 fzfid 13390 . . . . . . . . . . . . . . 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 8573 . . . . . . . . . . . . . . 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 8778 . . . . . . . . . . . . . . 15 (((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (𝑐𝑅 ∧ (𝐹𝑐) ∈ ℕ)) ∧ (𝑑𝑅𝑧 ⊆ (1...((𝐹𝑐) − 1)))) → 𝑧 ∈ Fin)
106 hashdom 13790 . . . . . . . . . . . . . . 15 ((𝑧 ∈ Fin ∧ (1...((𝐹𝑐) − 1)) ∈ Fin) → ((♯‘𝑧) ≤ (♯‘(1...((𝐹𝑐) − 1))) ↔ 𝑧 ≼ (1...((𝐹𝑐) − 1))))
107105, 101, 106syl2anc 587 . . . . . . . . . . . . . 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 260 . . . . . . . . . . . . 13 (((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (𝑐𝑅 ∧ (𝐹𝑐) ∈ ℕ)) ∧ (𝑑𝑅𝑧 ⊆ (1...((𝐹𝑐) − 1)))) → (♯‘𝑧) ≤ (♯‘(1...((𝐹𝑐) − 1))))
10984adantr 484 . . . . . . . . . . . . . 14 (((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (𝑐𝑅 ∧ (𝐹𝑐) ∈ ℕ)) ∧ (𝑑𝑅𝑧 ⊆ (1...((𝐹𝑐) − 1)))) → ((𝐹𝑐) − 1) ∈ ℕ0)
110 hashfz1 13756 . . . . . . . . . . . . . 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 5058 . . . . . . . . . . . 12 (((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (𝑐𝑅 ∧ (𝐹𝑐) ∈ ℕ)) ∧ (𝑑𝑅𝑧 ⊆ (1...((𝐹𝑐) − 1)))) → (♯‘𝑧) ≤ ((𝐹𝑐) − 1))
113 hashcl 13767 . . . . . . . . . . . . . 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 6842 . . . . . . . . . . . . . . 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 484 . . . . . . . . . . . . 13 (((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (𝑐𝑅 ∧ (𝐹𝑐) ∈ ℕ)) ∧ (𝑑𝑅𝑧 ⊆ (1...((𝐹𝑐) − 1)))) → (𝐹𝑐) ∈ ℕ0)
118 nn0ltlem1 12081 . . . . . . . . . . . . 13 (((♯‘𝑧) ∈ ℕ0 ∧ (𝐹𝑐) ∈ ℕ0) → ((♯‘𝑧) < (𝐹𝑐) ↔ (♯‘𝑧) ≤ ((𝐹𝑐) − 1)))
119114, 117, 118syl2anc 587 . . . . . . . . . . . 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 260 . . . . . . . . . . 11 (((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (𝑐𝑅 ∧ (𝐹𝑐) ∈ ℕ)) ∧ (𝑑𝑅𝑧 ⊆ (1...((𝐹𝑐) − 1)))) → (♯‘𝑧) < (𝐹𝑐))
12124, 85fvsn 6934 . . . . . . . . . . . . . . 15 ({⟨∅, 𝑐⟩}‘∅) = 𝑐
122 f1ofn 6603 . . . . . . . . . . . . . . . . 17 ({⟨∅, 𝑐⟩}:{∅}–1-1-onto→{𝑐} → {⟨∅, 𝑐⟩} Fn {∅})
123 elpreima 6819 . . . . . . . . . . . . . . . . 17 ({⟨∅, 𝑐⟩} Fn {∅} → (∅ ∈ ({⟨∅, 𝑐⟩} “ {𝑑}) ↔ (∅ ∈ {∅} ∧ ({⟨∅, 𝑐⟩}‘∅) ∈ {𝑑})))
12486, 122, 123mp2b 10 . . . . . . . . . . . . . . . 16 (∅ ∈ ({⟨∅, 𝑐⟩} “ {𝑑}) ↔ (∅ ∈ {∅} ∧ ({⟨∅, 𝑐⟩}‘∅) ∈ {𝑑}))
125124simprbi 500 . . . . . . . . . . . . . . 15 (∅ ∈ ({⟨∅, 𝑐⟩} “ {𝑑}) → ({⟨∅, 𝑐⟩}‘∅) ∈ {𝑑})
126121, 125eqeltrrid 2857 . . . . . . . . . . . . . 14 (∅ ∈ ({⟨∅, 𝑐⟩} “ {𝑑}) → 𝑐 ∈ {𝑑})
127 elsni 4539 . . . . . . . . . . . . . 14 (𝑐 ∈ {𝑑} → 𝑐 = 𝑑)
128126, 127syl 17 . . . . . . . . . . . . 13 (∅ ∈ ({⟨∅, 𝑐⟩} “ {𝑑}) → 𝑐 = 𝑑)
129128fveq2d 6662 . . . . . . . . . . . 12 (∅ ∈ ({⟨∅, 𝑐⟩} “ {𝑑}) → (𝐹𝑐) = (𝐹𝑑))
130129breq2d 5044 . . . . . . . . . . 11 (∅ ∈ ({⟨∅, 𝑐⟩} “ {𝑑}) → ((♯‘𝑧) < (𝐹𝑐) ↔ (♯‘𝑧) < (𝐹𝑑)))
131120, 130syl5ibcom 248 . . . . . . . . . 10 (((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (𝑐𝑅 ∧ (𝐹𝑐) ∈ ℕ)) ∧ (𝑑𝑅𝑧 ⊆ (1...((𝐹𝑐) − 1)))) → (∅ ∈ ({⟨∅, 𝑐⟩} “ {𝑑}) → (♯‘𝑧) < (𝐹𝑑)))
132100, 131syl5bi 245 . . . . . . . . 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 16410 . . . . . . . 8 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (𝑐𝑅 ∧ (𝐹𝑐) ∈ ℕ)) → ((𝐹𝑐) − 1) < (0 Ramsey 𝐹))
134 ramubcl 16409 . . . . . . . . . . 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 1375 . . . . . . . . . 10 (((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) → (0 Ramsey 𝐹) ∈ ℕ0)
136135adantr 484 . . . . . . . . 9 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (𝑐𝑅 ∧ (𝐹𝑐) ∈ ℕ)) → (0 Ramsey 𝐹) ∈ ℕ0)
137 nn0lem1lt 12086 . . . . . . . . 9 (((𝐹𝑐) ∈ ℕ0 ∧ (0 Ramsey 𝐹) ∈ ℕ0) → ((𝐹𝑐) ≤ (0 Ramsey 𝐹) ↔ ((𝐹𝑐) − 1) < (0 Ramsey 𝐹)))
138116, 136, 137syl2anc 587 . . . . . . . 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 260 . . . . . . 7 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (𝑐𝑅 ∧ (𝐹𝑐) ∈ ℕ)) → (𝐹𝑐) ≤ (0 Ramsey 𝐹))
140139expr 460 . . . . . 6 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ 𝑐𝑅) → ((𝐹𝑐) ∈ ℕ → (𝐹𝑐) ≤ (0 Ramsey 𝐹)))
141135adantr 484 . . . . . . . 8 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ 𝑐𝑅) → (0 Ramsey 𝐹) ∈ ℕ0)
142141nn0ge0d 11997 . . . . . . 7 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ 𝑐𝑅) → 0 ≤ (0 Ramsey 𝐹))
143 breq1 5035 . . . . . . 7 ((𝐹𝑐) = 0 → ((𝐹𝑐) ≤ (0 Ramsey 𝐹) ↔ 0 ≤ (0 Ramsey 𝐹)))
144142, 143syl5ibrcom 250 . . . . . 6 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ 𝑐𝑅) → ((𝐹𝑐) = 0 → (𝐹𝑐) ≤ (0 Ramsey 𝐹)))
145 elnn0 11936 . . . . . . 7 ((𝐹𝑐) ∈ ℕ0 ↔ ((𝐹𝑐) ∈ ℕ ∨ (𝐹𝑐) = 0))
146115, 145sylib 221 . . . . . 6 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ 𝑐𝑅) → ((𝐹𝑐) ∈ ℕ ∨ (𝐹𝑐) = 0))
147140, 144, 146mpjaod 857 . . . . 5 ((((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ 𝑐𝑅) → (𝐹𝑐) ≤ (0 Ramsey 𝐹))
148 breq1 5035 . . . . 5 ((𝐹𝑐) = sup(ran 𝐹, ℝ, < ) → ((𝐹𝑐) ≤ (0 Ramsey 𝐹) ↔ sup(ran 𝐹, ℝ, < ) ≤ (0 Ramsey 𝐹)))
149147, 148syl5ibcom 248 . . . 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 3208 . . 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 11995 . . 3 (((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) → (0 Ramsey 𝐹) ∈ ℝ)
153152, 35letri3d 10820 . 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 209   ∧ wa 399   ∨ wo 844   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   ≠ wne 2951  ∀wral 3070  ∃wrex 3071  {crab 3074  Vcvv 3409   ⊆ wss 3858  ∅c0 4225  𝒫 cpw 4494  {csn 4522  ⟨cop 4528   class class class wbr 5032  ◡ccnv 5523  dom cdm 5524  ran crn 5525   “ cima 5527   Fn wfn 6330  ⟶wf 6331  –1-1-onto→wf1o 6334  ‘cfv 6335  (class class class)co 7150   ∈ cmpo 7152   ≼ cdom 8525  Fincfn 8527  supcsup 8937  ℝcr 10574  0cc0 10575  1c1 10576  ℝ*cxr 10712   < clt 10713   ≤ cle 10714   − cmin 10908  ℕcn 11674  ℕ0cn0 11934  ℤcz 12020  ...cfz 12939  ♯chash 13740   Ramsey cram 16390 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5156  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298  ax-un 7459  ax-cnex 10631  ax-resscn 10632  ax-1cn 10633  ax-icn 10634  ax-addcl 10635  ax-addrcl 10636  ax-mulcl 10637  ax-mulrcl 10638  ax-mulcom 10639  ax-addass 10640  ax-mulass 10641  ax-distr 10642  ax-i2m1 10643  ax-1ne0 10644  ax-1rid 10645  ax-rnegex 10646  ax-rrecex 10647  ax-cnre 10648  ax-pre-lttri 10649  ax-pre-lttrn 10650  ax-pre-ltadd 10651  ax-pre-mulgt0 10652 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-nel 3056  df-ral 3075  df-rex 3076  df-reu 3077  df-rmo 3078  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-pss 3877  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-tp 4527  df-op 4529  df-uni 4799  df-int 4839  df-iun 4885  df-br 5033  df-opab 5095  df-mpt 5113  df-tr 5139  df-id 5430  df-eprel 5435  df-po 5443  df-so 5444  df-fr 5483  df-we 5485  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7580  df-1st 7693  df-2nd 7694  df-wrecs 7957  df-recs 8018  df-rdg 8056  df-1o 8112  df-oadd 8116  df-er 8299  df-map 8418  df-en 8528  df-dom 8529  df-sdom 8530  df-fin 8531  df-sup 8939  df-inf 8940  df-card 9401  df-pnf 10715  df-mnf 10716  df-xr 10717  df-ltxr 10718  df-le 10719  df-sub 10910  df-neg 10911  df-nn 11675  df-n0 11935  df-xnn0 12007  df-z 12021  df-uz 12283  df-fz 12940  df-hash 13741  df-ram 16392 This theorem is referenced by:  0ram2  16412  ramz  16416
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