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Theorem 0ram 16950

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 2733	. . 3 ⊢ (𝑎 ∈ V, 𝑖 ∈ ℕ₀ ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖}) = (𝑎 ∈ V, 𝑖 ∈ ℕ₀ ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})
2		0nn0 12484	. . . 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 6723	. . . 4 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ₀) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) → ran 𝐹 ⊆ ℕ₀)
7		nn0ssz 12578	. . . . . 6 ⊢ ℕ₀ ⊆ ℤ
8	6, 7	sstrdi 3994	. . . . 5 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ₀) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) → ran 𝐹 ⊆ ℤ)
9	5	fdmd 6726	. . . . . . 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 3009	. . . . . 6 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ₀) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) → dom 𝐹 ≠ ∅)
12		dm0rn0 5923	. . . . . . 7 ⊢ (dom 𝐹 = ∅ ↔ ran 𝐹 = ∅)
13	12	necon3bii 2994	. . . . . 6 ⊢ (dom 𝐹 ≠ ∅ ↔ ran 𝐹 ≠ ∅)
14	11, 13	sylib 217	. . . . 5 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ₀) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) → ran 𝐹 ≠ ∅)
15		simpr 486	. . . . 5 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ₀) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) → ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥)
16		suprzcl2 12919	. . . . 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 1372	. . . 4 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ₀) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) → sup(ran 𝐹, ℝ, < ) ∈ ran 𝐹)
18	6, 17	sseldd 3983	. . 3 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ₀) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) → sup(ran 𝐹, ℝ, < ) ∈ ℕ₀)
19	1	hashbc0 16935	. . . . . . 7 ⊢ (𝑠 ∈ V → (𝑠(𝑎 ∈ V, 𝑖 ∈ ℕ₀ ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})0) = {∅})
20	19	elv 3481	. . . . . 6 ⊢ (𝑠(𝑎 ∈ V, 𝑖 ∈ ℕ₀ ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})0) = {∅}
21	20	feq2i 6707	. . . . 5 ⊢ (𝑓:(𝑠(𝑎 ∈ V, 𝑖 ∈ ℕ₀ ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})0)⟶𝑅 ↔ 𝑓:{∅}⟶𝑅)
22	21	biimpi 215	. . . 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 5307	. . . . . . 7 ⊢ ∅ ∈ V
25	24	snid 4664	. . . . . 6 ⊢ ∅ ∈ {∅}
26		ffvelcdm 7081	. . . . . 6 ⊢ ((𝑓:{∅}⟶𝑅 ∧ ∅ ∈ {∅}) → (𝑓‘∅) ∈ 𝑅)
27	23, 25, 26	sylancl 587	. . . . 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 3479	. . . . . . 7 ⊢ 𝑠 ∈ V
29	28	pwid 4624	. . . . . 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 482	. . . . . . . . 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 7085	. . . . . . . 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 12530	. . . . . . 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 11261	. . . . . 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 12530	. . . . . . . 8 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ₀) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) → sup(ran 𝐹, ℝ, < ) ∈ ℝ)
36	35	rexrd 11261	. . . . . . 7 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ₀) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) → sup(ran 𝐹, ℝ, < ) ∈ ℝ^*)
37	36	adantr 482	. . . . . 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 14314	. . . . . . 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 482	. . . . . . 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 482	. . . . . . 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 6716	. . . . . . . 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 7080	. . . . . . . 8 ⊢ ((𝐹 Fn 𝑅 ∧ (𝑓‘∅) ∈ 𝑅) → (𝐹‘(𝑓‘∅)) ∈ ran 𝐹)
44	42, 27, 43	syl2anc 585	. . . . . . 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 12920	. . . . . . 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 1372	. . . . . 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 13138	. . . . 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 6902	. . . . . . . 8 ⊢ (𝑓‘∅) ∈ V
51	50	snid 4664	. . . . . . 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 6715	. . . . . . 7 ⊢ (𝑓:{∅}⟶𝑅 → 𝑓 Fn {∅})
54		elpreima 7057	. . . . . . 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 6889	. . . . . . . 8 ⊢ (𝑐 = (𝑓‘∅) → (𝐹‘𝑐) = (𝐹‘(𝑓‘∅)))
58	57	breq1d 5158	. . . . . . 7 ⊢ (𝑐 = (𝑓‘∅) → ((𝐹‘𝑐) ≤ (♯‘𝑧) ↔ (𝐹‘(𝑓‘∅)) ≤ (♯‘𝑧)))
59	1	hashbc0 16935	. . . . . . . . . . 11 ⊢ (𝑧 ∈ V → (𝑧(𝑎 ∈ V, 𝑖 ∈ ℕ₀ ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})0) = {∅})
60	59	elv 3481	. . . . . . . . . 10 ⊢ (𝑧(𝑎 ∈ V, 𝑖 ∈ ℕ₀ ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})0) = {∅}
61	60	sseq1i 4010	. . . . . . . . 9 ⊢ ((𝑧(𝑎 ∈ V, 𝑖 ∈ ℕ₀ ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})0) ⊆ (^◡𝑓 “ {𝑐}) ↔ {∅} ⊆ (^◡𝑓 “ {𝑐}))
62	24	snss 4789	. . . . . . . . 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 4638	. . . . . . . . . 10 ⊢ (𝑐 = (𝑓‘∅) → {𝑐} = {(𝑓‘∅)})
65	64	imaeq2d 6058	. . . . . . . . 9 ⊢ (𝑐 = (𝑓‘∅) → (^◡𝑓 “ {𝑐}) = (^◡𝑓 “ {(𝑓‘∅)}))
66	65	eleq2d 2820	. . . . . . . 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 6889	. . . . . . . 8 ⊢ (𝑧 = 𝑠 → (♯‘𝑧) = (♯‘𝑠))
70	69	breq2d 5160	. . . . . . 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 3624	. . . . 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 1375	. . . 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 16943	. 2 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ₀) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) → (0 Ramsey 𝐹) ≤ sup(ran 𝐹, ℝ, < ))
76		ffn 6715	. . . . 5 ⊢ (𝐹:𝑅⟶ℕ₀ → 𝐹 Fn 𝑅)
77		fvelrnb 6950	. . . . 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 231	. . 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 12510	. . . . . . . . . 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 3479	. . . . . . . . . . . . 13 ⊢ 𝑐 ∈ V
86	24, 85	f1osn 6871	. . . . . . . . . . . 12 ⊢ {⟨∅, 𝑐⟩}:{∅}–1-1-onto→{𝑐}
87		f1of 6831	. . . . . . . . . . . 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 4812	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ₀) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) ∧ (𝑐 ∈ 𝑅 ∧ (𝐹‘𝑐) ∈ ℕ)) → {𝑐} ⊆ 𝑅)
91		fss 6732	. . . . . . . . . . 11 ⊢ (({⟨∅, 𝑐⟩}:{∅}⟶{𝑐} ∧ {𝑐} ⊆ 𝑅) → {⟨∅, 𝑐⟩}:{∅}⟶𝑅)
92	88, 90, 91	sylancr 588	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ₀) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) ∧ (𝑐 ∈ 𝑅 ∧ (𝐹‘𝑐) ∈ ℕ)) → {⟨∅, 𝑐⟩}:{∅}⟶𝑅)
93		ovex 7439	. . . . . . . . . . . 12 ⊢ (1...((𝐹‘𝑐) − 1)) ∈ V
94	1	hashbc0 16935	. . . . . . . . . . . 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 6707	. . . . . . . . . 10 ⊢ ({⟨∅, 𝑐⟩}:((1...((𝐹‘𝑐) − 1))(𝑎 ∈ V, 𝑖 ∈ ℕ₀ ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})0)⟶𝑅 ↔ {⟨∅, 𝑐⟩}:{∅}⟶𝑅)
97	92, 96	sylibr 233	. . . . . . . . 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 4010	. . . . . . . . . . 11 ⊢ ((𝑧(𝑎 ∈ V, 𝑖 ∈ ℕ₀ ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})0) ⊆ (^◡{⟨∅, 𝑐⟩} “ {𝑑}) ↔ {∅} ⊆ (^◡{⟨∅, 𝑐⟩} “ {𝑑}))
99	24	snss 4789	. . . . . . . . . . 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 13935	. . . . . . . . . . . . . . 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 8993	. . . . . . . . . . . . . . 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 9264	. . . . . . . . . . . . . . 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 14336	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ Fin ∧ (1...((𝐹‘𝑐) − 1)) ∈ Fin) → ((♯‘𝑧) ≤ (♯‘(1...((𝐹‘𝑐) − 1))) ↔ 𝑧 ≼ (1...((𝐹‘𝑐) − 1))))
107	105, 101, 106	syl2anc 585	. . . . . . . . . . . . . 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 482	. . . . . . . . . . . . . 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 14303	. . . . . . . . . . . . . 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 5174	. . . . . . . . . . . 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 14313	. . . . . . . . . . . . . 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 7084	. . . . . . . . . . . . . . 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 482	. . . . . . . . . . . . 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 12619	. . . . . . . . . . . . 13 ⊢ (((♯‘𝑧) ∈ ℕ₀ ∧ (𝐹‘𝑐) ∈ ℕ₀) → ((♯‘𝑧) < (𝐹‘𝑐) ↔ (♯‘𝑧) ≤ ((𝐹‘𝑐) − 1)))
119	114, 117, 118	syl2anc 585	. . . . . . . . . . . 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 7176	. . . . . . . . . . . . . . 15 ⊢ ({⟨∅, 𝑐⟩}‘∅) = 𝑐
122		f1ofn 6832	. . . . . . . . . . . . . . . . 17 ⊢ ({⟨∅, 𝑐⟩}:{∅}–1-1-onto→{𝑐} → {⟨∅, 𝑐⟩} Fn {∅})
123		elpreima 7057	. . . . . . . . . . . . . . . . 17 ⊢ ({⟨∅, 𝑐⟩} Fn {∅} → (∅ ∈ (^◡{⟨∅, 𝑐⟩} “ {𝑑}) ↔ (∅ ∈ {∅} ∧ ({⟨∅, 𝑐⟩}‘∅) ∈ {𝑑})))
124	86, 122, 123	mp2b 10	. . . . . . . . . . . . . . . 16 ⊢ (∅ ∈ (^◡{⟨∅, 𝑐⟩} “ {𝑑}) ↔ (∅ ∈ {∅} ∧ ({⟨∅, 𝑐⟩}‘∅) ∈ {𝑑}))
125	124	simprbi 498	. . . . . . . . . . . . . . 15 ⊢ (∅ ∈ (^◡{⟨∅, 𝑐⟩} “ {𝑑}) → ({⟨∅, 𝑐⟩}‘∅) ∈ {𝑑})
126	121, 125	eqeltrrid 2839	. . . . . . . . . . . . . 14 ⊢ (∅ ∈ (^◡{⟨∅, 𝑐⟩} “ {𝑑}) → 𝑐 ∈ {𝑑})
127		elsni 4645	. . . . . . . . . . . . . 14 ⊢ (𝑐 ∈ {𝑑} → 𝑐 = 𝑑)
128	126, 127	syl 17	. . . . . . . . . . . . 13 ⊢ (∅ ∈ (^◡{⟨∅, 𝑐⟩} “ {𝑑}) → 𝑐 = 𝑑)
129	128	fveq2d 6893	. . . . . . . . . . . 12 ⊢ (∅ ∈ (^◡{⟨∅, 𝑐⟩} “ {𝑑}) → (𝐹‘𝑐) = (𝐹‘𝑑))
130	129	breq2d 5160	. . . . . . . . . . 11 ⊢ (∅ ∈ (^◡{⟨∅, 𝑐⟩} “ {𝑑}) → ((♯‘𝑧) < (𝐹‘𝑐) ↔ (♯‘𝑧) < (𝐹‘𝑑)))
131	120, 130	syl5ibcom 244	. . . . . . . . . 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 241	. . . . . . . . 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 16949	. . . . . . . 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 16948	. . . . . . . . . . 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 1379	. . . . . . . . . 10 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ₀) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) → (0 Ramsey 𝐹) ∈ ℕ₀)
136	135	adantr 482	. . . . . . . . 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 12624	. . . . . . . . 9 ⊢ (((𝐹‘𝑐) ∈ ℕ₀ ∧ (0 Ramsey 𝐹) ∈ ℕ₀) → ((𝐹‘𝑐) ≤ (0 Ramsey 𝐹) ↔ ((𝐹‘𝑐) − 1) < (0 Ramsey 𝐹)))
138	116, 136, 137	syl2anc 585	. . . . . . . 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 458	. . . . . 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 482	. . . . . . . 8 ⊢ ((((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ₀) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) ∧ 𝑐 ∈ 𝑅) → (0 Ramsey 𝐹) ∈ ℕ₀)
142	141	nn0ge0d 12532	. . . . . . 7 ⊢ ((((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ₀) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) ∧ 𝑐 ∈ 𝑅) → 0 ≤ (0 Ramsey 𝐹))
143		breq1 5151	. . . . . . 7 ⊢ ((𝐹‘𝑐) = 0 → ((𝐹‘𝑐) ≤ (0 Ramsey 𝐹) ↔ 0 ≤ (0 Ramsey 𝐹)))
144	142, 143	syl5ibrcom 246	. . . . . 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 12471	. . . . . . 7 ⊢ ((𝐹‘𝑐) ∈ ℕ₀ ↔ ((𝐹‘𝑐) ∈ ℕ ∨ (𝐹‘𝑐) = 0))
146	115, 145	sylib 217	. . . . . 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 859	. . . . 5 ⊢ ((((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ₀) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) ∧ 𝑐 ∈ 𝑅) → (𝐹‘𝑐) ≤ (0 Ramsey 𝐹))
148		breq1 5151	. . . . 5 ⊢ ((𝐹‘𝑐) = sup(ran 𝐹, ℝ, < ) → ((𝐹‘𝑐) ≤ (0 Ramsey 𝐹) ↔ sup(ran 𝐹, ℝ, < ) ≤ (0 Ramsey 𝐹)))
149	147, 148	syl5ibcom 244	. . . 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 3156	. . 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 12530	. . 3 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ₀) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) → (0 Ramsey 𝐹) ∈ ℝ)
153	152, 35	letri3d 11353	. 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, < ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∨ wo 846 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 ∀wral 3062 ∃wrex 3071 {crab 3433 Vcvv 3475 ⊆ wss 3948 ∅c0 4322 𝒫 cpw 4602 {csn 4628 ⟨cop 4634 class class class wbr 5148 ^◡ccnv 5675 dom cdm 5676 ran crn 5677 “ cima 5679 Fn wfn 6536 ⟶wf 6537 –1-1-onto→wf1o 6540 ‘cfv 6541 (class class class)co 7406 ∈ cmpo 7408 ≼ cdom 8934 Fincfn 8936 supcsup 9432 ℝcr 11106 0cc0 11107 1c1 11108 ℝ^*cxr 11244 < clt 11245 ≤ cle 11246 − cmin 11441 ℕcn 12209 ℕ₀cn0 12469 ℤcz 12555 ...cfz 13481 ♯chash 14287 Ramsey cram 16929

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-riota 7362 df-ov 7409 df-oprab 7410 df-mpo 7411 df-om 7853 df-1st 7972 df-2nd 7973 df-frecs 8263 df-wrecs 8294 df-recs 8368 df-rdg 8407 df-1o 8463 df-oadd 8467 df-er 8700 df-map 8819 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-sup 9434 df-inf 9435 df-card 9931 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-nn 12210 df-n0 12470 df-xnn0 12542 df-z 12556 df-uz 12820 df-fz 13482 df-hash 14288 df-ram 16931

This theorem is referenced by: 0ram2 16951 ramz 16955

W3C validator