Theorem 0ram 16346

Description: The Ramsey number when 𝑀 = 0. (Contributed by Mario Carneiro, 22-Apr-2015.)

Assertion

Ref	Expression
0ram	⊢ (((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) → (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.

Step	Hyp	Ref	Expression
1		eqid 2798	. . 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
3	2	a1i 11	. . 3 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) → 0 ∈ ℕ0)
4		simpl1 1188	. . 3 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) → 𝑅 ∈ 𝑉)
5		simpl3 1190	. . 3 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) → 𝐹:𝑅⟶ℕ0)
6	5	frnd 6494	. . . 4 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) → ran 𝐹 ⊆ ℕ0)
7		nn0ssz 11991	. . . . . 6 ⊢ ℕ0 ⊆ ℤ
8	6, 7	sstrdi 3927	. . . . 5 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) → ran 𝐹 ⊆ ℤ)
9	5	fdmd 6497	. . . . . . 7 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) → dom 𝐹 = 𝑅)
10		simpl2 1189	. . . . . . 7 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) → 𝑅 ≠ ∅)
11	9, 10	eqnetrd 3054	. . . . . 6 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) → dom 𝐹 ≠ ∅)
12		dm0rn0 5759	. . . . . . 7 ⊢ (dom 𝐹 = ∅ ↔ ran 𝐹 = ∅)
13	12	necon3bii 3039	. . . . . 6 ⊢ (dom 𝐹 ≠ ∅ ↔ ran 𝐹 ≠ ∅)
14	11, 13	sylib 221	. . . . 5 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) → ran 𝐹 ≠ ∅)
15		simpr 488	. . . . 5 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) → ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥)
16		suprzcl2 12326	. . . . 5 ⊢ ((ran 𝐹 ⊆ ℤ ∧ ran 𝐹 ≠ ∅ ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) → sup(ran 𝐹, ℝ, < ) ∈ ran 𝐹)
17	8, 14, 15, 16	syl3anc 1368	. . . 4 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) → sup(ran 𝐹, ℝ, < ) ∈ ran 𝐹)
18	6, 17	sseldd 3916	. . 3 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) → sup(ran 𝐹, ℝ, < ) ∈ ℕ0)
19	1	hashbc0 16331	. . . . . . 7 ⊢ (𝑠 ∈ V → (𝑠(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})0) = {∅})
20	19	elv 3446	. . . . . 6 ⊢ (𝑠(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})0) = {∅}
21	20	feq2i 6479	. . . . 5 ⊢ (𝑓:(𝑠(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})0)⟶𝑅 ↔ 𝑓:{∅}⟶𝑅)
22	21	biimpi 219	. . . 4 ⊢ (𝑓:(𝑠(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})0)⟶𝑅 → 𝑓:{∅}⟶𝑅)
23		simprr 772	. . . . . 6 ⊢ ((((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) ∧ (sup(ran 𝐹, ℝ, < ) ≤ (♯‘𝑠) ∧ 𝑓:{∅}⟶𝑅)) → 𝑓:{∅}⟶𝑅)
24		0ex 5175	. . . . . . 7 ⊢ ∅ ∈ V
25	24	snid 4561	. . . . . 6 ⊢ ∅ ∈ {∅}
26		ffvelrn 6826	. . . . . 6 ⊢ ((𝑓:{∅}⟶𝑅 ∧ ∅ ∈ {∅}) → (𝑓‘∅) ∈ 𝑅)
27	23, 25, 26	sylancl 589	. . . . 5 ⊢ ((((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) ∧ (sup(ran 𝐹, ℝ, < ) ≤ (♯‘𝑠) ∧ 𝑓:{∅}⟶𝑅)) → (𝑓‘∅) ∈ 𝑅)
28		vex 3444	. . . . . . 7 ⊢ 𝑠 ∈ V
29	28	pwid 4521	. . . . . 6 ⊢ 𝑠 ∈ 𝒫 𝑠
30	29	a1i 11	. . . . 5 ⊢ ((((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) ∧ (sup(ran 𝐹, ℝ, < ) ≤ (♯‘𝑠) ∧ 𝑓:{∅}⟶𝑅)) → 𝑠 ∈ 𝒫 𝑠)
31	5	adantr 484	. . . . . . . . 9 ⊢ ((((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) ∧ (sup(ran 𝐹, ℝ, < ) ≤ (♯‘𝑠) ∧ 𝑓:{∅}⟶𝑅)) → 𝐹:𝑅⟶ℕ0)
32	31, 27	ffvelrnd 6829	. . . . . . . 8 ⊢ ((((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) ∧ (sup(ran 𝐹, ℝ, < ) ≤ (♯‘𝑠) ∧ 𝑓:{∅}⟶𝑅)) → (𝐹‘(𝑓‘∅)) ∈ ℕ0)
33	32	nn0red 11944	. . . . . . 7 ⊢ ((((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) ∧ (sup(ran 𝐹, ℝ, < ) ≤ (♯‘𝑠) ∧ 𝑓:{∅}⟶𝑅)) → (𝐹‘(𝑓‘∅)) ∈ ℝ)
34	33	rexrd 10680	. . . . . 6 ⊢ ((((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) ∧ (sup(ran 𝐹, ℝ, < ) ≤ (♯‘𝑠) ∧ 𝑓:{∅}⟶𝑅)) → (𝐹‘(𝑓‘∅)) ∈ ℝ*)
35	18	nn0red 11944	. . . . . . . 8 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) → sup(ran 𝐹, ℝ, < ) ∈ ℝ)
36	35	rexrd 10680	. . . . . . 7 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) → sup(ran 𝐹, ℝ, < ) ∈ ℝ*)
37	36	adantr 484	. . . . . 6 ⊢ ((((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) ∧ (sup(ran 𝐹, ℝ, < ) ≤ (♯‘𝑠) ∧ 𝑓:{∅}⟶𝑅)) → sup(ran 𝐹, ℝ, < ) ∈ ℝ*)
38		hashxrcl 13714	. . . . . . 7 ⊢ (𝑠 ∈ V → (♯‘𝑠) ∈ ℝ*)
39	28, 38	mp1i 13	. . . . . 6 ⊢ ((((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) ∧ (sup(ran 𝐹, ℝ, < ) ≤ (♯‘𝑠) ∧ 𝑓:{∅}⟶𝑅)) → (♯‘𝑠) ∈ ℝ*)
40	8	adantr 484	. . . . . . 7 ⊢ ((((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) ∧ (sup(ran 𝐹, ℝ, < ) ≤ (♯‘𝑠) ∧ 𝑓:{∅}⟶𝑅)) → ran 𝐹 ⊆ ℤ)
41	15	adantr 484	. . . . . . 7 ⊢ ((((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) ∧ (sup(ran 𝐹, ℝ, < ) ≤ (♯‘𝑠) ∧ 𝑓:{∅}⟶𝑅)) → ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥)
42	31	ffnd 6488	. . . . . . . 8 ⊢ ((((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) ∧ (sup(ran 𝐹, ℝ, < ) ≤ (♯‘𝑠) ∧ 𝑓:{∅}⟶𝑅)) → 𝐹 Fn 𝑅)
43		fnfvelrn 6825	. . . . . . . 8 ⊢ ((𝐹 Fn 𝑅 ∧ (𝑓‘∅) ∈ 𝑅) → (𝐹‘(𝑓‘∅)) ∈ ran 𝐹)
44	42, 27, 43	syl2anc 587	. . . . . . 7 ⊢ ((((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) ∧ (sup(ran 𝐹, ℝ, < ) ≤ (♯‘𝑠) ∧ 𝑓:{∅}⟶𝑅)) → (𝐹‘(𝑓‘∅)) ∈ ran 𝐹)
45		suprzub 12327	. . . . . . 7 ⊢ ((ran 𝐹 ⊆ ℤ ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥 ∧ (𝐹‘(𝑓‘∅)) ∈ ran 𝐹) → (𝐹‘(𝑓‘∅)) ≤ sup(ran 𝐹, ℝ, < ))
46	40, 41, 44, 45	syl3anc 1368	. . . . . 6 ⊢ ((((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) ∧ (sup(ran 𝐹, ℝ, < ) ≤ (♯‘𝑠) ∧ 𝑓:{∅}⟶𝑅)) → (𝐹‘(𝑓‘∅)) ≤ sup(ran 𝐹, ℝ, < ))
47		simprl 770	. . . . . 6 ⊢ ((((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) ∧ (sup(ran 𝐹, ℝ, < ) ≤ (♯‘𝑠) ∧ 𝑓:{∅}⟶𝑅)) → sup(ran 𝐹, ℝ, < ) ≤ (♯‘𝑠))
48	34, 37, 39, 46, 47	xrletrd 12543	. . . . 5 ⊢ ((((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) ∧ (sup(ran 𝐹, ℝ, < ) ≤ (♯‘𝑠) ∧ 𝑓:{∅}⟶𝑅)) → (𝐹‘(𝑓‘∅)) ≤ (♯‘𝑠))
49	25	a1i 11	. . . . . 6 ⊢ ((((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) ∧ (sup(ran 𝐹, ℝ, < ) ≤ (♯‘𝑠) ∧ 𝑓:{∅}⟶𝑅)) → ∅ ∈ {∅})
50		fvex 6658	. . . . . . . 8 ⊢ (𝑓‘∅) ∈ V
51	50	snid 4561	. . . . . . 7 ⊢ (𝑓‘∅) ∈ {(𝑓‘∅)}
52	51	a1i 11	. . . . . 6 ⊢ ((((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) ∧ (sup(ran 𝐹, ℝ, < ) ≤ (♯‘𝑠) ∧ 𝑓:{∅}⟶𝑅)) → (𝑓‘∅) ∈ {(𝑓‘∅)})
53		ffn 6487	. . . . . . 7 ⊢ (𝑓:{∅}⟶𝑅 → 𝑓 Fn {∅})
54		elpreima 6805	. . . . . . 7 ⊢ (𝑓 Fn {∅} → (∅ ∈ (◡𝑓 “ {(𝑓‘∅)}) ↔ (∅ ∈ {∅} ∧ (𝑓‘∅) ∈ {(𝑓‘∅)})))
55	23, 53, 54	3syl 18	. . . . . 6 ⊢ ((((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) ∧ (sup(ran 𝐹, ℝ, < ) ≤ (♯‘𝑠) ∧ 𝑓:{∅}⟶𝑅)) → (∅ ∈ (◡𝑓 “ {(𝑓‘∅)}) ↔ (∅ ∈ {∅} ∧ (𝑓‘∅) ∈ {(𝑓‘∅)})))
56	49, 52, 55	mpbir2and 712	. . . . 5 ⊢ ((((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) ∧ (sup(ran 𝐹, ℝ, < ) ≤ (♯‘𝑠) ∧ 𝑓:{∅}⟶𝑅)) → ∅ ∈ (◡𝑓 “ {(𝑓‘∅)}))
57		fveq2 6645	. . . . . . . 8 ⊢ (𝑐 = (𝑓‘∅) → (𝐹‘𝑐) = (𝐹‘(𝑓‘∅)))
58	57	breq1d 5040	. . . . . . 7 ⊢ (𝑐 = (𝑓‘∅) → ((𝐹‘𝑐) ≤ (♯‘𝑧) ↔ (𝐹‘(𝑓‘∅)) ≤ (♯‘𝑧)))
59	1	hashbc0 16331	. . . . . . . . . . 11 ⊢ (𝑧 ∈ V → (𝑧(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})0) = {∅})
60	59	elv 3446	. . . . . . . . . 10 ⊢ (𝑧(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})0) = {∅}
61	60	sseq1i 3943	. . . . . . . . 9 ⊢ ((𝑧(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})0) ⊆ (◡𝑓 “ {𝑐}) ↔ {∅} ⊆ (◡𝑓 “ {𝑐}))
62	24	snss 4679	. . . . . . . . 9 ⊢ (∅ ∈ (◡𝑓 “ {𝑐}) ↔ {∅} ⊆ (◡𝑓 “ {𝑐}))
63	61, 62	bitr4i 281	. . . . . . . 8 ⊢ ((𝑧(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})0) ⊆ (◡𝑓 “ {𝑐}) ↔ ∅ ∈ (◡𝑓 “ {𝑐}))
64		sneq 4535	. . . . . . . . . 10 ⊢ (𝑐 = (𝑓‘∅) → {𝑐} = {(𝑓‘∅)})
65	64	imaeq2d 5896	. . . . . . . . 9 ⊢ (𝑐 = (𝑓‘∅) → (◡𝑓 “ {𝑐}) = (◡𝑓 “ {(𝑓‘∅)}))
66	65	eleq2d 2875	. . . . . . . 8 ⊢ (𝑐 = (𝑓‘∅) → (∅ ∈ (◡𝑓 “ {𝑐}) ↔ ∅ ∈ (◡𝑓 “ {(𝑓‘∅)})))
67	63, 66	syl5bb 286	. . . . . . 7 ⊢ (𝑐 = (𝑓‘∅) → ((𝑧(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})0) ⊆ (◡𝑓 “ {𝑐}) ↔ ∅ ∈ (◡𝑓 “ {(𝑓‘∅)})))
68	58, 67	anbi12d 633	. . . . . 6 ⊢ (𝑐 = (𝑓‘∅) → (((𝐹‘𝑐) ≤ (♯‘𝑧) ∧ (𝑧(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})0) ⊆ (◡𝑓 “ {𝑐})) ↔ ((𝐹‘(𝑓‘∅)) ≤ (♯‘𝑧) ∧ ∅ ∈ (◡𝑓 “ {(𝑓‘∅)}))))
69		fveq2 6645	. . . . . . . 8 ⊢ (𝑧 = 𝑠 → (♯‘𝑧) = (♯‘𝑠))
70	69	breq2d 5042	. . . . . . 7 ⊢ (𝑧 = 𝑠 → ((𝐹‘(𝑓‘∅)) ≤ (♯‘𝑧) ↔ (𝐹‘(𝑓‘∅)) ≤ (♯‘𝑠)))
71	70	anbi1d 632	. . . . . 6 ⊢ (𝑧 = 𝑠 → (((𝐹‘(𝑓‘∅)) ≤ (♯‘𝑧) ∧ ∅ ∈ (◡𝑓 “ {(𝑓‘∅)})) ↔ ((𝐹‘(𝑓‘∅)) ≤ (♯‘𝑠) ∧ ∅ ∈ (◡𝑓 “ {(𝑓‘∅)}))))
72	68, 71	rspc2ev 3583	. . . . 5 ⊢ (((𝑓‘∅) ∈ 𝑅 ∧ 𝑠 ∈ 𝒫 𝑠 ∧ ((𝐹‘(𝑓‘∅)) ≤ (♯‘𝑠) ∧ ∅ ∈ (◡𝑓 “ {(𝑓‘∅)}))) → ∃𝑐 ∈ 𝑅 ∃𝑧 ∈ 𝒫 𝑠((𝐹‘𝑐) ≤ (♯‘𝑧) ∧ (𝑧(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})0) ⊆ (◡𝑓 “ {𝑐})))
73	27, 30, 48, 56, 72	syl112anc 1371	. . . 4 ⊢ ((((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) ∧ (sup(ran 𝐹, ℝ, < ) ≤ (♯‘𝑠) ∧ 𝑓:{∅}⟶𝑅)) → ∃𝑐 ∈ 𝑅 ∃𝑧 ∈ 𝒫 𝑠((𝐹‘𝑐) ≤ (♯‘𝑧) ∧ (𝑧(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})0) ⊆ (◡𝑓 “ {𝑐})))
74	22, 73	sylanr2 682	. . 3 ⊢ ((((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) ∧ (sup(ran 𝐹, ℝ, < ) ≤ (♯‘𝑠) ∧ 𝑓:(𝑠(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})0)⟶𝑅)) → ∃𝑐 ∈ 𝑅 ∃𝑧 ∈ 𝒫 𝑠((𝐹‘𝑐) ≤ (♯‘𝑧) ∧ (𝑧(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})0) ⊆ (◡𝑓 “ {𝑐})))
75	1, 3, 4, 5, 18, 74	ramub 16339	. 2 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) → (0 Ramsey 𝐹) ≤ sup(ran 𝐹, ℝ, < ))
76		ffn 6487	. . . . 5 ⊢ (𝐹:𝑅⟶ℕ0 → 𝐹 Fn 𝑅)
77		fvelrnb 6701	. . . . 5 ⊢ (𝐹 Fn 𝑅 → (sup(ran 𝐹, ℝ, < ) ∈ ran 𝐹 ↔ ∃𝑐 ∈ 𝑅 (𝐹‘𝑐) = sup(ran 𝐹, ℝ, < )))
78	5, 76, 77	3syl 18	. . . 4 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) → (sup(ran 𝐹, ℝ, < ) ∈ ran 𝐹 ↔ ∃𝑐 ∈ 𝑅 (𝐹‘𝑐) = sup(ran 𝐹, ℝ, < )))
79	17, 78	mpbid 235	. . 3 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) → ∃𝑐 ∈ 𝑅 (𝐹‘𝑐) = sup(ran 𝐹, ℝ, < ))
80	2	a1i 11	. . . . . . . . 9 ⊢ ((((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) ∧ (𝑐 ∈ 𝑅 ∧ (𝐹‘𝑐) ∈ ℕ)) → 0 ∈ ℕ0)
81		simpll1 1209	. . . . . . . . 9 ⊢ ((((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) ∧ (𝑐 ∈ 𝑅 ∧ (𝐹‘𝑐) ∈ ℕ)) → 𝑅 ∈ 𝑉)
82		simpll3 1211	. . . . . . . . 9 ⊢ ((((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) ∧ (𝑐 ∈ 𝑅 ∧ (𝐹‘𝑐) ∈ ℕ)) → 𝐹:𝑅⟶ℕ0)
83		nnm1nn0 11926	. . . . . . . . . 10 ⊢ ((𝐹‘𝑐) ∈ ℕ → ((𝐹‘𝑐) − 1) ∈ ℕ0)
84	83	ad2antll 728	. . . . . . . . 9 ⊢ ((((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) ∧ (𝑐 ∈ 𝑅 ∧ (𝐹‘𝑐) ∈ ℕ)) → ((𝐹‘𝑐) − 1) ∈ ℕ0)
85		vex 3444	. . . . . . . . . . . . 13 ⊢ 𝑐 ∈ V
86	24, 85	f1osn 6629	. . . . . . . . . . . 12 ⊢ {⟨∅, 𝑐⟩}:{∅}–1-1-onto→{𝑐}
87		f1of 6590	. . . . . . . . . . . 12 ⊢ ({⟨∅, 𝑐⟩}:{∅}–1-1-onto→{𝑐} → {⟨∅, 𝑐⟩}:{∅}⟶{𝑐})
88	86, 87	ax-mp 5	. . . . . . . . . . 11 ⊢ {⟨∅, 𝑐⟩}:{∅}⟶{𝑐}
89		simprl 770	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) ∧ (𝑐 ∈ 𝑅 ∧ (𝐹‘𝑐) ∈ ℕ)) → 𝑐 ∈ 𝑅)
90	89	snssd 4702	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) ∧ (𝑐 ∈ 𝑅 ∧ (𝐹‘𝑐) ∈ ℕ)) → {𝑐} ⊆ 𝑅)
91		fss 6501	. . . . . . . . . . 11 ⊢ (({⟨∅, 𝑐⟩}:{∅}⟶{𝑐} ∧ {𝑐} ⊆ 𝑅) → {⟨∅, 𝑐⟩}:{∅}⟶𝑅)
92	88, 90, 91	sylancr 590	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) ∧ (𝑐 ∈ 𝑅 ∧ (𝐹‘𝑐) ∈ ℕ)) → {⟨∅, 𝑐⟩}:{∅}⟶𝑅)
93		ovex 7168	. . . . . . . . . . . 12 ⊢ (1...((𝐹‘𝑐) − 1)) ∈ V
94	1	hashbc0 16331	. . . . . . . . . . . 12 ⊢ ((1...((𝐹‘𝑐) − 1)) ∈ V → ((1...((𝐹‘𝑐) − 1))(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})0) = {∅})
95	93, 94	ax-mp 5	. . . . . . . . . . 11 ⊢ ((1...((𝐹‘𝑐) − 1))(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})0) = {∅}
96	95	feq2i 6479	. . . . . . . . . 10 ⊢ ({⟨∅, 𝑐⟩}:((1...((𝐹‘𝑐) − 1))(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})0)⟶𝑅 ↔ {⟨∅, 𝑐⟩}:{∅}⟶𝑅)
97	92, 96	sylibr 237	. . . . . . . . 9 ⊢ ((((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) ∧ (𝑐 ∈ 𝑅 ∧ (𝐹‘𝑐) ∈ ℕ)) → {⟨∅, 𝑐⟩}:((1...((𝐹‘𝑐) − 1))(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})0)⟶𝑅)
98	60	sseq1i 3943	. . . . . . . . . . 11 ⊢ ((𝑧(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})0) ⊆ (◡{⟨∅, 𝑐⟩} “ {𝑑}) ↔ {∅} ⊆ (◡{⟨∅, 𝑐⟩} “ {𝑑}))
99	24	snss 4679	. . . . . . . . . . 11 ⊢ (∅ ∈ (◡{⟨∅, 𝑐⟩} “ {𝑑}) ↔ {∅} ⊆ (◡{⟨∅, 𝑐⟩} “ {𝑑}))
100	98, 99	bitr4i 281	. . . . . . . . . 10 ⊢ ((𝑧(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})0) ⊆ (◡{⟨∅, 𝑐⟩} “ {𝑑}) ↔ ∅ ∈ (◡{⟨∅, 𝑐⟩} “ {𝑑}))
101		fzfid 13336	. . . . . . . . . . . . . . 15 ⊢ (((((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) ∧ (𝑐 ∈ 𝑅 ∧ (𝐹‘𝑐) ∈ ℕ)) ∧ (𝑑 ∈ 𝑅 ∧ 𝑧 ⊆ (1...((𝐹‘𝑐) − 1)))) → (1...((𝐹‘𝑐) − 1)) ∈ Fin)
102		simprr 772	. . . . . . . . . . . . . . 15 ⊢ (((((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) ∧ (𝑐 ∈ 𝑅 ∧ (𝐹‘𝑐) ∈ ℕ)) ∧ (𝑑 ∈ 𝑅 ∧ 𝑧 ⊆ (1...((𝐹‘𝑐) − 1)))) → 𝑧 ⊆ (1...((𝐹‘𝑐) − 1)))
103		ssdomg 8538	. . . . . . . . . . . . . . 15 ⊢ ((1...((𝐹‘𝑐) − 1)) ∈ Fin → (𝑧 ⊆ (1...((𝐹‘𝑐) − 1)) → 𝑧 ≼ (1...((𝐹‘𝑐) − 1))))
104	101, 102, 103	sylc 65	. . . . . . . . . . . . . 14 ⊢ (((((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) ∧ (𝑐 ∈ 𝑅 ∧ (𝐹‘𝑐) ∈ ℕ)) ∧ (𝑑 ∈ 𝑅 ∧ 𝑧 ⊆ (1...((𝐹‘𝑐) − 1)))) → 𝑧 ≼ (1...((𝐹‘𝑐) − 1)))
105	101, 102	ssfid 8725	. . . . . . . . . . . . . . 15 ⊢ (((((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) ∧ (𝑐 ∈ 𝑅 ∧ (𝐹‘𝑐) ∈ ℕ)) ∧ (𝑑 ∈ 𝑅 ∧ 𝑧 ⊆ (1...((𝐹‘𝑐) − 1)))) → 𝑧 ∈ Fin)
106		hashdom 13736	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ Fin ∧ (1...((𝐹‘𝑐) − 1)) ∈ Fin) → ((♯‘𝑧) ≤ (♯‘(1...((𝐹‘𝑐) − 1))) ↔ 𝑧 ≼ (1...((𝐹‘𝑐) − 1))))
107	105, 101, 106	syl2anc 587	. . . . . . . . . . . . . 14 ⊢ (((((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) ∧ (𝑐 ∈ 𝑅 ∧ (𝐹‘𝑐) ∈ ℕ)) ∧ (𝑑 ∈ 𝑅 ∧ 𝑧 ⊆ (1...((𝐹‘𝑐) − 1)))) → ((♯‘𝑧) ≤ (♯‘(1...((𝐹‘𝑐) − 1))) ↔ 𝑧 ≼ (1...((𝐹‘𝑐) − 1))))
108	104, 107	mpbird 260	. . . . . . . . . . . . 13 ⊢ (((((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) ∧ (𝑐 ∈ 𝑅 ∧ (𝐹‘𝑐) ∈ ℕ)) ∧ (𝑑 ∈ 𝑅 ∧ 𝑧 ⊆ (1...((𝐹‘𝑐) − 1)))) → (♯‘𝑧) ≤ (♯‘(1...((𝐹‘𝑐) − 1))))
109	84	adantr 484	. . . . . . . . . . . . . 14 ⊢ (((((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) ∧ (𝑐 ∈ 𝑅 ∧ (𝐹‘𝑐) ∈ ℕ)) ∧ (𝑑 ∈ 𝑅 ∧ 𝑧 ⊆ (1...((𝐹‘𝑐) − 1)))) → ((𝐹‘𝑐) − 1) ∈ ℕ0)
110		hashfz1 13702	. . . . . . . . . . . . . 14 ⊢ (((𝐹‘𝑐) − 1) ∈ ℕ0 → (♯‘(1...((𝐹‘𝑐) − 1))) = ((𝐹‘𝑐) − 1))
111	109, 110	syl 17	. . . . . . . . . . . . 13 ⊢ (((((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) ∧ (𝑐 ∈ 𝑅 ∧ (𝐹‘𝑐) ∈ ℕ)) ∧ (𝑑 ∈ 𝑅 ∧ 𝑧 ⊆ (1...((𝐹‘𝑐) − 1)))) → (♯‘(1...((𝐹‘𝑐) − 1))) = ((𝐹‘𝑐) − 1))
112	108, 111	breqtrd 5056	. . . . . . . . . . . 12 ⊢ (((((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) ∧ (𝑐 ∈ 𝑅 ∧ (𝐹‘𝑐) ∈ ℕ)) ∧ (𝑑 ∈ 𝑅 ∧ 𝑧 ⊆ (1...((𝐹‘𝑐) − 1)))) → (♯‘𝑧) ≤ ((𝐹‘𝑐) − 1))
113		hashcl 13713	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ Fin → (♯‘𝑧) ∈ ℕ0)
114	105, 113	syl 17	. . . . . . . . . . . . 13 ⊢ (((((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) ∧ (𝑐 ∈ 𝑅 ∧ (𝐹‘𝑐) ∈ ℕ)) ∧ (𝑑 ∈ 𝑅 ∧ 𝑧 ⊆ (1...((𝐹‘𝑐) − 1)))) → (♯‘𝑧) ∈ ℕ0)
115	5	ffvelrnda 6828	. . . . . . . . . . . . . . 15 ⊢ ((((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) ∧ 𝑐 ∈ 𝑅) → (𝐹‘𝑐) ∈ ℕ0)
116	115	adantrr 716	. . . . . . . . . . . . . 14 ⊢ ((((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) ∧ (𝑐 ∈ 𝑅 ∧ (𝐹‘𝑐) ∈ ℕ)) → (𝐹‘𝑐) ∈ ℕ0)
117	116	adantr 484	. . . . . . . . . . . . 13 ⊢ (((((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) ∧ (𝑐 ∈ 𝑅 ∧ (𝐹‘𝑐) ∈ ℕ)) ∧ (𝑑 ∈ 𝑅 ∧ 𝑧 ⊆ (1...((𝐹‘𝑐) − 1)))) → (𝐹‘𝑐) ∈ ℕ0)
118		nn0ltlem1 12030	. . . . . . . . . . . . 13 ⊢ (((♯‘𝑧) ∈ ℕ0 ∧ (𝐹‘𝑐) ∈ ℕ0) → ((♯‘𝑧) < (𝐹‘𝑐) ↔ (♯‘𝑧) ≤ ((𝐹‘𝑐) − 1)))
119	114, 117, 118	syl2anc 587	. . . . . . . . . . . 12 ⊢ (((((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) ∧ (𝑐 ∈ 𝑅 ∧ (𝐹‘𝑐) ∈ ℕ)) ∧ (𝑑 ∈ 𝑅 ∧ 𝑧 ⊆ (1...((𝐹‘𝑐) − 1)))) → ((♯‘𝑧) < (𝐹‘𝑐) ↔ (♯‘𝑧) ≤ ((𝐹‘𝑐) − 1)))
120	112, 119	mpbird 260	. . . . . . . . . . 11 ⊢ (((((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) ∧ (𝑐 ∈ 𝑅 ∧ (𝐹‘𝑐) ∈ ℕ)) ∧ (𝑑 ∈ 𝑅 ∧ 𝑧 ⊆ (1...((𝐹‘𝑐) − 1)))) → (♯‘𝑧) < (𝐹‘𝑐))
121	24, 85	fvsn 6920	. . . . . . . . . . . . . . 15 ⊢ ({⟨∅, 𝑐⟩}‘∅) = 𝑐
122		f1ofn 6591	. . . . . . . . . . . . . . . . 17 ⊢ ({⟨∅, 𝑐⟩}:{∅}–1-1-onto→{𝑐} → {⟨∅, 𝑐⟩} Fn {∅})
123		elpreima 6805	. . . . . . . . . . . . . . . . 17 ⊢ ({⟨∅, 𝑐⟩} Fn {∅} → (∅ ∈ (◡{⟨∅, 𝑐⟩} “ {𝑑}) ↔ (∅ ∈ {∅} ∧ ({⟨∅, 𝑐⟩}‘∅) ∈ {𝑑})))
124	86, 122, 123	mp2b 10	. . . . . . . . . . . . . . . 16 ⊢ (∅ ∈ (◡{⟨∅, 𝑐⟩} “ {𝑑}) ↔ (∅ ∈ {∅} ∧ ({⟨∅, 𝑐⟩}‘∅) ∈ {𝑑}))
125	124	simprbi 500	. . . . . . . . . . . . . . 15 ⊢ (∅ ∈ (◡{⟨∅, 𝑐⟩} “ {𝑑}) → ({⟨∅, 𝑐⟩}‘∅) ∈ {𝑑})
126	121, 125	eqeltrrid 2895	. . . . . . . . . . . . . 14 ⊢ (∅ ∈ (◡{⟨∅, 𝑐⟩} “ {𝑑}) → 𝑐 ∈ {𝑑})
127		elsni 4542	. . . . . . . . . . . . . 14 ⊢ (𝑐 ∈ {𝑑} → 𝑐 = 𝑑)
128	126, 127	syl 17	. . . . . . . . . . . . 13 ⊢ (∅ ∈ (◡{⟨∅, 𝑐⟩} “ {𝑑}) → 𝑐 = 𝑑)
129	128	fveq2d 6649	. . . . . . . . . . . 12 ⊢ (∅ ∈ (◡{⟨∅, 𝑐⟩} “ {𝑑}) → (𝐹‘𝑐) = (𝐹‘𝑑))
130	129	breq2d 5042	. . . . . . . . . . 11 ⊢ (∅ ∈ (◡{⟨∅, 𝑐⟩} “ {𝑑}) → ((♯‘𝑧) < (𝐹‘𝑐) ↔ (♯‘𝑧) < (𝐹‘𝑑)))
131	120, 130	syl5ibcom 248	. . . . . . . . . 10 ⊢ (((((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) ∧ (𝑐 ∈ 𝑅 ∧ (𝐹‘𝑐) ∈ ℕ)) ∧ (𝑑 ∈ 𝑅 ∧ 𝑧 ⊆ (1...((𝐹‘𝑐) − 1)))) → (∅ ∈ (◡{⟨∅, 𝑐⟩} “ {𝑑}) → (♯‘𝑧) < (𝐹‘𝑑)))
132	100, 131	syl5bi 245	. . . . . . . . 9 ⊢ (((((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) ∧ (𝑐 ∈ 𝑅 ∧ (𝐹‘𝑐) ∈ ℕ)) ∧ (𝑑 ∈ 𝑅 ∧ 𝑧 ⊆ (1...((𝐹‘𝑐) − 1)))) → ((𝑧(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})0) ⊆ (◡{⟨∅, 𝑐⟩} “ {𝑑}) → (♯‘𝑧) < (𝐹‘𝑑)))
133	1, 80, 81, 82, 84, 97, 132	ramlb 16345	. . . . . . . 8 ⊢ ((((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) ∧ (𝑐 ∈ 𝑅 ∧ (𝐹‘𝑐) ∈ ℕ)) → ((𝐹‘𝑐) − 1) < (0 Ramsey 𝐹))
134		ramubcl 16344	. . . . . . . . . . 11 ⊢ (((0 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ0) ∧ (sup(ran 𝐹, ℝ, < ) ∈ ℕ0 ∧ (0 Ramsey 𝐹) ≤ sup(ran 𝐹, ℝ, < ))) → (0 Ramsey 𝐹) ∈ ℕ0)
135	3, 4, 5, 18, 75, 134	syl32anc 1375	. . . . . . . . . 10 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) → (0 Ramsey 𝐹) ∈ ℕ0)
136	135	adantr 484	. . . . . . . . 9 ⊢ ((((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) ∧ (𝑐 ∈ 𝑅 ∧ (𝐹‘𝑐) ∈ ℕ)) → (0 Ramsey 𝐹) ∈ ℕ0)
137		nn0lem1lt 12035	. . . . . . . . 9 ⊢ (((𝐹‘𝑐) ∈ ℕ0 ∧ (0 Ramsey 𝐹) ∈ ℕ0) → ((𝐹‘𝑐) ≤ (0 Ramsey 𝐹) ↔ ((𝐹‘𝑐) − 1) < (0 Ramsey 𝐹)))
138	116, 136, 137	syl2anc 587	. . . . . . . 8 ⊢ ((((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) ∧ (𝑐 ∈ 𝑅 ∧ (𝐹‘𝑐) ∈ ℕ)) → ((𝐹‘𝑐) ≤ (0 Ramsey 𝐹) ↔ ((𝐹‘𝑐) − 1) < (0 Ramsey 𝐹)))
139	133, 138	mpbird 260	. . . . . . 7 ⊢ ((((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) ∧ (𝑐 ∈ 𝑅 ∧ (𝐹‘𝑐) ∈ ℕ)) → (𝐹‘𝑐) ≤ (0 Ramsey 𝐹))
140	139	expr 460	. . . . . 6 ⊢ ((((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) ∧ 𝑐 ∈ 𝑅) → ((𝐹‘𝑐) ∈ ℕ → (𝐹‘𝑐) ≤ (0 Ramsey 𝐹)))
141	135	adantr 484	. . . . . . . 8 ⊢ ((((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) ∧ 𝑐 ∈ 𝑅) → (0 Ramsey 𝐹) ∈ ℕ0)
142	141	nn0ge0d 11946	. . . . . . 7 ⊢ ((((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) ∧ 𝑐 ∈ 𝑅) → 0 ≤ (0 Ramsey 𝐹))
143		breq1 5033	. . . . . . 7 ⊢ ((𝐹‘𝑐) = 0 → ((𝐹‘𝑐) ≤ (0 Ramsey 𝐹) ↔ 0 ≤ (0 Ramsey 𝐹)))
144	142, 143	syl5ibrcom 250	. . . . . 6 ⊢ ((((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) ∧ 𝑐 ∈ 𝑅) → ((𝐹‘𝑐) = 0 → (𝐹‘𝑐) ≤ (0 Ramsey 𝐹)))
145		elnn0 11887	. . . . . . 7 ⊢ ((𝐹‘𝑐) ∈ ℕ0 ↔ ((𝐹‘𝑐) ∈ ℕ ∨ (𝐹‘𝑐) = 0))
146	115, 145	sylib 221	. . . . . 6 ⊢ ((((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) ∧ 𝑐 ∈ 𝑅) → ((𝐹‘𝑐) ∈ ℕ ∨ (𝐹‘𝑐) = 0))
147	140, 144, 146	mpjaod 857	. . . . 5 ⊢ ((((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) ∧ 𝑐 ∈ 𝑅) → (𝐹‘𝑐) ≤ (0 Ramsey 𝐹))
148		breq1 5033	. . . . 5 ⊢ ((𝐹‘𝑐) = sup(ran 𝐹, ℝ, < ) → ((𝐹‘𝑐) ≤ (0 Ramsey 𝐹) ↔ sup(ran 𝐹, ℝ, < ) ≤ (0 Ramsey 𝐹)))
149	147, 148	syl5ibcom 248	. . . 4 ⊢ ((((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) ∧ 𝑐 ∈ 𝑅) → ((𝐹‘𝑐) = sup(ran 𝐹, ℝ, < ) → sup(ran 𝐹, ℝ, < ) ≤ (0 Ramsey 𝐹)))
150	149	rexlimdva 3243	. . 3 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) → (∃𝑐 ∈ 𝑅 (𝐹‘𝑐) = sup(ran 𝐹, ℝ, < ) → sup(ran 𝐹, ℝ, < ) ≤ (0 Ramsey 𝐹)))
151	79, 150	mpd 15	. 2 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) → sup(ran 𝐹, ℝ, < ) ≤ (0 Ramsey 𝐹))
152	135	nn0red 11944	. . 3 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) → (0 Ramsey 𝐹) ∈ ℝ)
153	152, 35	letri3d 10771	. 2 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) → ((0 Ramsey 𝐹) = sup(ran 𝐹, ℝ, < ) ↔ ((0 Ramsey 𝐹) ≤ sup(ran 𝐹, ℝ, < ) ∧ sup(ran 𝐹, ℝ, < ) ≤ (0 Ramsey 𝐹))))
154	75, 151, 153	mpbir2and 712	1 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) → (0 Ramsey 𝐹) = sup(ran 𝐹, ℝ, < ))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   ≠ wne 2987  ∀wral 3106  ∃wrex 3107  {crab 3110  Vcvv 3441   ⊆ wss 3881  ∅c0 4243  𝒫 cpw 4497  {csn 4525  ⟨cop 4531   class class class wbr 5030  ^◡ccnv 5518  dom cdm 5519  ran crn 5520   “ cima 5522   Fn wfn 6319  ⟶wf 6320  –1-1-onto→wf1o 6323  ‘cfv 6324  (class class class)co 7135   ∈ cmpo 7137   ≼ cdom 8490  Fincfn 8492  supcsup 8888  ℝcr 10525  0cc0 10526  1c1 10527  ℝ^*cxr 10663   < clt 10664   ≤ cle 10665   − cmin 10859  ℕcn 11625  ℕ₀cn0 11885  ℤcz 11969  ...cfz 12885  ♯chash 13686   Ramsey cram 16325

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 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-1st 7671  df-2nd 7672  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-1o 8085  df-oadd 8089  df-er 8272  df-map 8391  df-en 8493  df-dom 8494  df-sdom 8495  df-fin 8496  df-sup 8890  df-inf 8891  df-card 9352  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11626  df-n0 11886  df-xnn0 11956  df-z 11970  df-uz 12232  df-fz 12886  df-hash 13687  df-ram 16327

This theorem is referenced by:  0ram2  16347  ramz  16351