Theorem 0ram 17045

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 2736	. . 3 ⊢ (𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖}) = (𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})
2		0nn0 12521	. . . 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 1192	. . 3 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) → 𝑅 ∈ 𝑉)
5		simpl3 1194	. . 3 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) → 𝐹:𝑅⟶ℕ0)
6	5	frnd 6719	. . . 4 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) → ran 𝐹 ⊆ ℕ0)
7		nn0ssz 12616	. . . . . 6 ⊢ ℕ0 ⊆ ℤ
8	6, 7	sstrdi 3976	. . . . 5 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) → ran 𝐹 ⊆ ℤ)
9	5	fdmd 6721	. . . . . . 7 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) → dom 𝐹 = 𝑅)
10		simpl2 1193	. . . . . . 7 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) → 𝑅 ≠ ∅)
11	9, 10	eqnetrd 3000	. . . . . 6 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) → dom 𝐹 ≠ ∅)
12		dm0rn0 5909	. . . . . . 7 ⊢ (dom 𝐹 = ∅ ↔ ran 𝐹 = ∅)
13	12	necon3bii 2985	. . . . . 6 ⊢ (dom 𝐹 ≠ ∅ ↔ ran 𝐹 ≠ ∅)
14	11, 13	sylib 218	. . . . 5 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) → ran 𝐹 ≠ ∅)
15		simpr 484	. . . . 5 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) → ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥)
16		suprzcl2 12959	. . . . 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 1373	. . . 4 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) → sup(ran 𝐹, ℝ, < ) ∈ ran 𝐹)
18	6, 17	sseldd 3964	. . 3 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) → sup(ran 𝐹, ℝ, < ) ∈ ℕ0)
19	1	hashbc0 17030	. . . . . . 7 ⊢ (𝑠 ∈ V → (𝑠(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})0) = {∅})
20	19	elv 3469	. . . . . 6 ⊢ (𝑠(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})0) = {∅}
21	20	feq2i 6703	. . . . 5 ⊢ (𝑓:(𝑠(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})0)⟶𝑅 ↔ 𝑓:{∅}⟶𝑅)
22	21	biimpi 216	. . . 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 5282	. . . . . . 7 ⊢ ∅ ∈ V
25	24	snid 4643	. . . . . 6 ⊢ ∅ ∈ {∅}
26		ffvelcdm 7076	. . . . . 6 ⊢ ((𝑓:{∅}⟶𝑅 ∧ ∅ ∈ {∅}) → (𝑓‘∅) ∈ 𝑅)
27	23, 25, 26	sylancl 586	. . . . 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 3468	. . . . . . 7 ⊢ 𝑠 ∈ V
29	28	pwid 4602	. . . . . 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 480	. . . . . . . . 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	ffvelcdmd 7080	. . . . . . . 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 12568	. . . . . . 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 11290	. . . . . 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 12568	. . . . . . . 8 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) → sup(ran 𝐹, ℝ, < ) ∈ ℝ)
36	35	rexrd 11290	. . . . . . 7 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) → sup(ran 𝐹, ℝ, < ) ∈ ℝ*)
37	36	adantr 480	. . . . . 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 14380	. . . . . . 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 480	. . . . . . 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 480	. . . . . . 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 6712	. . . . . . . 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 7075	. . . . . . . 8 ⊢ ((𝐹 Fn 𝑅 ∧ (𝑓‘∅) ∈ 𝑅) → (𝐹‘(𝑓‘∅)) ∈ ran 𝐹)
44	42, 27, 43	syl2anc 584	. . . . . . 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 12960	. . . . . . 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 1373	. . . . . 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 13183	. . . . 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 6894	. . . . . . . 8 ⊢ (𝑓‘∅) ∈ V
51	50	snid 4643	. . . . . . 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 6711	. . . . . . 7 ⊢ (𝑓:{∅}⟶𝑅 → 𝑓 Fn {∅})
54		elpreima 7053	. . . . . . 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 713	. . . . 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 6881	. . . . . . . 8 ⊢ (𝑐 = (𝑓‘∅) → (𝐹‘𝑐) = (𝐹‘(𝑓‘∅)))
58	57	breq1d 5134	. . . . . . 7 ⊢ (𝑐 = (𝑓‘∅) → ((𝐹‘𝑐) ≤ (♯‘𝑧) ↔ (𝐹‘(𝑓‘∅)) ≤ (♯‘𝑧)))
59	1	hashbc0 17030	. . . . . . . . . . 11 ⊢ (𝑧 ∈ V → (𝑧(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})0) = {∅})
60	59	elv 3469	. . . . . . . . . 10 ⊢ (𝑧(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})0) = {∅}
61	60	sseq1i 3992	. . . . . . . . 9 ⊢ ((𝑧(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})0) ⊆ (◡𝑓 “ {𝑐}) ↔ {∅} ⊆ (◡𝑓 “ {𝑐}))
62	24	snss 4766	. . . . . . . . 9 ⊢ (∅ ∈ (◡𝑓 “ {𝑐}) ↔ {∅} ⊆ (◡𝑓 “ {𝑐}))
63	61, 62	bitr4i 278	. . . . . . . 8 ⊢ ((𝑧(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})0) ⊆ (◡𝑓 “ {𝑐}) ↔ ∅ ∈ (◡𝑓 “ {𝑐}))
64		sneq 4616	. . . . . . . . . 10 ⊢ (𝑐 = (𝑓‘∅) → {𝑐} = {(𝑓‘∅)})
65	64	imaeq2d 6052	. . . . . . . . 9 ⊢ (𝑐 = (𝑓‘∅) → (◡𝑓 “ {𝑐}) = (◡𝑓 “ {(𝑓‘∅)}))
66	65	eleq2d 2821	. . . . . . . 8 ⊢ (𝑐 = (𝑓‘∅) → (∅ ∈ (◡𝑓 “ {𝑐}) ↔ ∅ ∈ (◡𝑓 “ {(𝑓‘∅)})))
67	63, 66	bitrid 283	. . . . . . 7 ⊢ (𝑐 = (𝑓‘∅) → ((𝑧(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})0) ⊆ (◡𝑓 “ {𝑐}) ↔ ∅ ∈ (◡𝑓 “ {(𝑓‘∅)})))
68	58, 67	anbi12d 632	. . . . . 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 6881	. . . . . . . 8 ⊢ (𝑧 = 𝑠 → (♯‘𝑧) = (♯‘𝑠))
70	69	breq2d 5136	. . . . . . 7 ⊢ (𝑧 = 𝑠 → ((𝐹‘(𝑓‘∅)) ≤ (♯‘𝑧) ↔ (𝐹‘(𝑓‘∅)) ≤ (♯‘𝑠)))
71	70	anbi1d 631	. . . . . 6 ⊢ (𝑧 = 𝑠 → (((𝐹‘(𝑓‘∅)) ≤ (♯‘𝑧) ∧ ∅ ∈ (◡𝑓 “ {(𝑓‘∅)})) ↔ ((𝐹‘(𝑓‘∅)) ≤ (♯‘𝑠) ∧ ∅ ∈ (◡𝑓 “ {(𝑓‘∅)}))))
72	68, 71	rspc2ev 3619	. . . . 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 1376	. . . 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 683	. . 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 17038	. 2 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) → (0 Ramsey 𝐹) ≤ sup(ran 𝐹, ℝ, < ))
76		ffn 6711	. . . . 5 ⊢ (𝐹:𝑅⟶ℕ0 → 𝐹 Fn 𝑅)
77		fvelrnb 6944	. . . . 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 232	. . 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 1213	. . . . . . . . 9 ⊢ ((((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) ∧ (𝑐 ∈ 𝑅 ∧ (𝐹‘𝑐) ∈ ℕ)) → 𝑅 ∈ 𝑉)
82		simpll3 1215	. . . . . . . . 9 ⊢ ((((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) ∧ (𝑐 ∈ 𝑅 ∧ (𝐹‘𝑐) ∈ ℕ)) → 𝐹:𝑅⟶ℕ0)
83		nnm1nn0 12547	. . . . . . . . . 10 ⊢ ((𝐹‘𝑐) ∈ ℕ → ((𝐹‘𝑐) − 1) ∈ ℕ0)
84	83	ad2antll 729	. . . . . . . . 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 3468	. . . . . . . . . . . . 13 ⊢ 𝑐 ∈ V
86	24, 85	f1osn 6863	. . . . . . . . . . . 12 ⊢ {⟨∅, 𝑐⟩}:{∅}–1-1-onto→{𝑐}
87		f1of 6823	. . . . . . . . . . . 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 4790	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) ∧ (𝑐 ∈ 𝑅 ∧ (𝐹‘𝑐) ∈ ℕ)) → {𝑐} ⊆ 𝑅)
91		fss 6727	. . . . . . . . . . 11 ⊢ (({⟨∅, 𝑐⟩}:{∅}⟶{𝑐} ∧ {𝑐} ⊆ 𝑅) → {⟨∅, 𝑐⟩}:{∅}⟶𝑅)
92	88, 90, 91	sylancr 587	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) ∧ (𝑐 ∈ 𝑅 ∧ (𝐹‘𝑐) ∈ ℕ)) → {⟨∅, 𝑐⟩}:{∅}⟶𝑅)
93		ovex 7443	. . . . . . . . . . . 12 ⊢ (1...((𝐹‘𝑐) − 1)) ∈ V
94	1	hashbc0 17030	. . . . . . . . . . . 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 6703	. . . . . . . . . 10 ⊢ ({⟨∅, 𝑐⟩}:((1...((𝐹‘𝑐) − 1))(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})0)⟶𝑅 ↔ {⟨∅, 𝑐⟩}:{∅}⟶𝑅)
97	92, 96	sylibr 234	. . . . . . . . 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 3992	. . . . . . . . . . 11 ⊢ ((𝑧(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})0) ⊆ (◡{⟨∅, 𝑐⟩} “ {𝑑}) ↔ {∅} ⊆ (◡{⟨∅, 𝑐⟩} “ {𝑑}))
99	24	snss 4766	. . . . . . . . . . 11 ⊢ (∅ ∈ (◡{⟨∅, 𝑐⟩} “ {𝑑}) ↔ {∅} ⊆ (◡{⟨∅, 𝑐⟩} “ {𝑑}))
100	98, 99	bitr4i 278	. . . . . . . . . 10 ⊢ ((𝑧(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})0) ⊆ (◡{⟨∅, 𝑐⟩} “ {𝑑}) ↔ ∅ ∈ (◡{⟨∅, 𝑐⟩} “ {𝑑}))
101		fzfid 13996	. . . . . . . . . . . . . . 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 9019	. . . . . . . . . . . . . . 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 9278	. . . . . . . . . . . . . . 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 14402	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ Fin ∧ (1...((𝐹‘𝑐) − 1)) ∈ Fin) → ((♯‘𝑧) ≤ (♯‘(1...((𝐹‘𝑐) − 1))) ↔ 𝑧 ≼ (1...((𝐹‘𝑐) − 1))))
107	105, 101, 106	syl2anc 584	. . . . . . . . . . . . . 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 257	. . . . . . . . . . . . 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 480	. . . . . . . . . . . . . 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 14369	. . . . . . . . . . . . . 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 5150	. . . . . . . . . . . 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 14379	. . . . . . . . . . . . . 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	ffvelcdmda 7079	. . . . . . . . . . . . . . 15 ⊢ ((((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) ∧ 𝑐 ∈ 𝑅) → (𝐹‘𝑐) ∈ ℕ0)
116	115	adantrr 717	. . . . . . . . . . . . . 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 480	. . . . . . . . . . . . 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 12658	. . . . . . . . . . . . 13 ⊢ (((♯‘𝑧) ∈ ℕ0 ∧ (𝐹‘𝑐) ∈ ℕ0) → ((♯‘𝑧) < (𝐹‘𝑐) ↔ (♯‘𝑧) ≤ ((𝐹‘𝑐) − 1)))
119	114, 117, 118	syl2anc 584	. . . . . . . . . . . 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 257	. . . . . . . . . . 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 7178	. . . . . . . . . . . . . . 15 ⊢ ({⟨∅, 𝑐⟩}‘∅) = 𝑐
122		f1ofn 6824	. . . . . . . . . . . . . . . . 17 ⊢ ({⟨∅, 𝑐⟩}:{∅}–1-1-onto→{𝑐} → {⟨∅, 𝑐⟩} Fn {∅})
123		elpreima 7053	. . . . . . . . . . . . . . . . 17 ⊢ ({⟨∅, 𝑐⟩} Fn {∅} → (∅ ∈ (◡{⟨∅, 𝑐⟩} “ {𝑑}) ↔ (∅ ∈ {∅} ∧ ({⟨∅, 𝑐⟩}‘∅) ∈ {𝑑})))
124	86, 122, 123	mp2b 10	. . . . . . . . . . . . . . . 16 ⊢ (∅ ∈ (◡{⟨∅, 𝑐⟩} “ {𝑑}) ↔ (∅ ∈ {∅} ∧ ({⟨∅, 𝑐⟩}‘∅) ∈ {𝑑}))
125	124	simprbi 496	. . . . . . . . . . . . . . 15 ⊢ (∅ ∈ (◡{⟨∅, 𝑐⟩} “ {𝑑}) → ({⟨∅, 𝑐⟩}‘∅) ∈ {𝑑})
126	121, 125	eqeltrrid 2840	. . . . . . . . . . . . . 14 ⊢ (∅ ∈ (◡{⟨∅, 𝑐⟩} “ {𝑑}) → 𝑐 ∈ {𝑑})
127		elsni 4623	. . . . . . . . . . . . . 14 ⊢ (𝑐 ∈ {𝑑} → 𝑐 = 𝑑)
128	126, 127	syl 17	. . . . . . . . . . . . 13 ⊢ (∅ ∈ (◡{⟨∅, 𝑐⟩} “ {𝑑}) → 𝑐 = 𝑑)
129	128	fveq2d 6885	. . . . . . . . . . . 12 ⊢ (∅ ∈ (◡{⟨∅, 𝑐⟩} “ {𝑑}) → (𝐹‘𝑐) = (𝐹‘𝑑))
130	129	breq2d 5136	. . . . . . . . . . 11 ⊢ (∅ ∈ (◡{⟨∅, 𝑐⟩} “ {𝑑}) → ((♯‘𝑧) < (𝐹‘𝑐) ↔ (♯‘𝑧) < (𝐹‘𝑑)))
131	120, 130	syl5ibcom 245	. . . . . . . . . 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	biimtrid 242	. . . . . . . . 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 17044	. . . . . . . 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 17043	. . . . . . . . . . 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 1380	. . . . . . . . . 10 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) → (0 Ramsey 𝐹) ∈ ℕ0)
136	135	adantr 480	. . . . . . . . 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 12663	. . . . . . . . 9 ⊢ (((𝐹‘𝑐) ∈ ℕ0 ∧ (0 Ramsey 𝐹) ∈ ℕ0) → ((𝐹‘𝑐) ≤ (0 Ramsey 𝐹) ↔ ((𝐹‘𝑐) − 1) < (0 Ramsey 𝐹)))
138	116, 136, 137	syl2anc 584	. . . . . . . 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 257	. . . . . . 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 456	. . . . . 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 480	. . . . . . . 8 ⊢ ((((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) ∧ 𝑐 ∈ 𝑅) → (0 Ramsey 𝐹) ∈ ℕ0)
142	141	nn0ge0d 12570	. . . . . . 7 ⊢ ((((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) ∧ 𝑐 ∈ 𝑅) → 0 ≤ (0 Ramsey 𝐹))
143		breq1 5127	. . . . . . 7 ⊢ ((𝐹‘𝑐) = 0 → ((𝐹‘𝑐) ≤ (0 Ramsey 𝐹) ↔ 0 ≤ (0 Ramsey 𝐹)))
144	142, 143	syl5ibrcom 247	. . . . . 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 12508	. . . . . . 7 ⊢ ((𝐹‘𝑐) ∈ ℕ0 ↔ ((𝐹‘𝑐) ∈ ℕ ∨ (𝐹‘𝑐) = 0))
146	115, 145	sylib 218	. . . . . 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 860	. . . . 5 ⊢ ((((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) ∧ 𝑐 ∈ 𝑅) → (𝐹‘𝑐) ≤ (0 Ramsey 𝐹))
148		breq1 5127	. . . . 5 ⊢ ((𝐹‘𝑐) = sup(ran 𝐹, ℝ, < ) → ((𝐹‘𝑐) ≤ (0 Ramsey 𝐹) ↔ sup(ran 𝐹, ℝ, < ) ≤ (0 Ramsey 𝐹)))
149	147, 148	syl5ibcom 245	. . . 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 3142	. . 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 12568	. . 3 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) → (0 Ramsey 𝐹) ∈ ℝ)
153	152, 35	letri3d 11382	. 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 713	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 206   ∧ wa 395   ∨ wo 847   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2933  ∀wral 3052  ∃wrex 3061  {crab 3420  Vcvv 3464   ⊆ wss 3931  ∅c0 4313  𝒫 cpw 4580  {csn 4606  ⟨cop 4612   class class class wbr 5124  ^◡ccnv 5658  dom cdm 5659  ran crn 5660   “ cima 5662   Fn wfn 6531  ⟶wf 6532  –1-1-onto→wf1o 6535  ‘cfv 6536  (class class class)co 7410   ∈ cmpo 7412   ≼ cdom 8962  Fincfn 8964  supcsup 9457  ℝcr 11133  0cc0 11134  1c1 11135  ℝ^*cxr 11273   < clt 11274   ≤ cle 11275   − cmin 11471  ℕcn 12245  ℕ₀cn0 12506  ℤcz 12593  ...cfz 13529  ♯chash 14353   Ramsey cram 17024

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734  ax-cnex 11190  ax-resscn 11191  ax-1cn 11192  ax-icn 11193  ax-addcl 11194  ax-addrcl 11195  ax-mulcl 11196  ax-mulrcl 11197  ax-mulcom 11198  ax-addass 11199  ax-mulass 11200  ax-distr 11201  ax-i2m1 11202  ax-1ne0 11203  ax-1rid 11204  ax-rnegex 11205  ax-rrecex 11206  ax-cnre 11207  ax-pre-lttri 11208  ax-pre-lttrn 11209  ax-pre-ltadd 11210  ax-pre-mulgt0 11211

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3062  df-rmo 3364  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-int 4928  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6295  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7867  df-1st 7993  df-2nd 7994  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-1o 8485  df-oadd 8489  df-er 8724  df-map 8847  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-sup 9459  df-inf 9460  df-card 9958  df-pnf 11276  df-mnf 11277  df-xr 11278  df-ltxr 11279  df-le 11280  df-sub 11473  df-neg 11474  df-nn 12246  df-n0 12507  df-xnn0 12580  df-z 12594  df-uz 12858  df-fz 13530  df-hash 14354  df-ram 17026

This theorem is referenced by:  0ram2  17046  ramz  17050