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Theorem ramz2 16962

Description: The Ramsey number when 𝐹 has value zero for some color 𝐶. (Contributed by Mario Carneiro, 22-Apr-2015.)

**Assertion**
Ref	Expression
ramz2	⊢ (((𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ₀) ∧ (𝐶 ∈ 𝑅 ∧ (𝐹‘𝐶) = 0)) → (𝑀 Ramsey 𝐹) = 0)

Proof of Theorem ramz2 Dummy variables 𝑏 𝑓 𝑐 𝑠 𝑥 𝑎 𝑖 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2724	. . 3 ⊢ (𝑎 ∈ V, 𝑖 ∈ ℕ₀ ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖}) = (𝑎 ∈ V, 𝑖 ∈ ℕ₀ ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})
2		simpl1 1188	. . . 4 ⊢ (((𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ₀) ∧ (𝐶 ∈ 𝑅 ∧ (𝐹‘𝐶) = 0)) → 𝑀 ∈ ℕ)
3	2	nnnn0d 12531	. . 3 ⊢ (((𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ₀) ∧ (𝐶 ∈ 𝑅 ∧ (𝐹‘𝐶) = 0)) → 𝑀 ∈ ℕ₀)
4		simpl2 1189	. . 3 ⊢ (((𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ₀) ∧ (𝐶 ∈ 𝑅 ∧ (𝐹‘𝐶) = 0)) → 𝑅 ∈ 𝑉)
5		simpl3 1190	. . 3 ⊢ (((𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ₀) ∧ (𝐶 ∈ 𝑅 ∧ (𝐹‘𝐶) = 0)) → 𝐹:𝑅⟶ℕ₀)
6		0nn0 12486	. . . 4 ⊢ 0 ∈ ℕ₀
7	6	a1i 11	. . 3 ⊢ (((𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ₀) ∧ (𝐶 ∈ 𝑅 ∧ (𝐹‘𝐶) = 0)) → 0 ∈ ℕ₀)
8		simplrl 774	. . . 4 ⊢ ((((𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ₀) ∧ (𝐶 ∈ 𝑅 ∧ (𝐹‘𝐶) = 0)) ∧ (0 ≤ (♯‘𝑠) ∧ 𝑓:(𝑠(𝑎 ∈ V, 𝑖 ∈ ℕ₀ ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})𝑀)⟶𝑅)) → 𝐶 ∈ 𝑅)
9		0elpw 5345	. . . . 5 ⊢ ∅ ∈ 𝒫 𝑠
10	9	a1i 11	. . . 4 ⊢ ((((𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ₀) ∧ (𝐶 ∈ 𝑅 ∧ (𝐹‘𝐶) = 0)) ∧ (0 ≤ (♯‘𝑠) ∧ 𝑓:(𝑠(𝑎 ∈ V, 𝑖 ∈ ℕ₀ ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})𝑀)⟶𝑅)) → ∅ ∈ 𝒫 𝑠)
11		simplrr 775	. . . . 5 ⊢ ((((𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ₀) ∧ (𝐶 ∈ 𝑅 ∧ (𝐹‘𝐶) = 0)) ∧ (0 ≤ (♯‘𝑠) ∧ 𝑓:(𝑠(𝑎 ∈ V, 𝑖 ∈ ℕ₀ ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})𝑀)⟶𝑅)) → (𝐹‘𝐶) = 0)
12		0le0 12312	. . . . 5 ⊢ 0 ≤ 0
13	11, 12	eqbrtrdi 5178	. . . 4 ⊢ ((((𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ₀) ∧ (𝐶 ∈ 𝑅 ∧ (𝐹‘𝐶) = 0)) ∧ (0 ≤ (♯‘𝑠) ∧ 𝑓:(𝑠(𝑎 ∈ V, 𝑖 ∈ ℕ₀ ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})𝑀)⟶𝑅)) → (𝐹‘𝐶) ≤ 0)
14		simpll1 1209	. . . . . 6 ⊢ ((((𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ₀) ∧ (𝐶 ∈ 𝑅 ∧ (𝐹‘𝐶) = 0)) ∧ (0 ≤ (♯‘𝑠) ∧ 𝑓:(𝑠(𝑎 ∈ V, 𝑖 ∈ ℕ₀ ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})𝑀)⟶𝑅)) → 𝑀 ∈ ℕ)
15	1	0hashbc 16945	. . . . . 6 ⊢ (𝑀 ∈ ℕ → (∅(𝑎 ∈ V, 𝑖 ∈ ℕ₀ ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})𝑀) = ∅)
16	14, 15	syl 17	. . . . 5 ⊢ ((((𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ₀) ∧ (𝐶 ∈ 𝑅 ∧ (𝐹‘𝐶) = 0)) ∧ (0 ≤ (♯‘𝑠) ∧ 𝑓:(𝑠(𝑎 ∈ V, 𝑖 ∈ ℕ₀ ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})𝑀)⟶𝑅)) → (∅(𝑎 ∈ V, 𝑖 ∈ ℕ₀ ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})𝑀) = ∅)
17		0ss 4389	. . . . 5 ⊢ ∅ ⊆ (^◡𝑓 “ {𝐶})
18	16, 17	eqsstrdi 4029	. . . 4 ⊢ ((((𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ₀) ∧ (𝐶 ∈ 𝑅 ∧ (𝐹‘𝐶) = 0)) ∧ (0 ≤ (♯‘𝑠) ∧ 𝑓:(𝑠(𝑎 ∈ V, 𝑖 ∈ ℕ₀ ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})𝑀)⟶𝑅)) → (∅(𝑎 ∈ V, 𝑖 ∈ ℕ₀ ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})𝑀) ⊆ (^◡𝑓 “ {𝐶}))
19		fveq2 6882	. . . . . . 7 ⊢ (𝑐 = 𝐶 → (𝐹‘𝑐) = (𝐹‘𝐶))
20	19	breq1d 5149	. . . . . 6 ⊢ (𝑐 = 𝐶 → ((𝐹‘𝑐) ≤ (♯‘𝑥) ↔ (𝐹‘𝐶) ≤ (♯‘𝑥)))
21		sneq 4631	. . . . . . . 8 ⊢ (𝑐 = 𝐶 → {𝑐} = {𝐶})
22	21	imaeq2d 6050	. . . . . . 7 ⊢ (𝑐 = 𝐶 → (^◡𝑓 “ {𝑐}) = (^◡𝑓 “ {𝐶}))
23	22	sseq2d 4007	. . . . . 6 ⊢ (𝑐 = 𝐶 → ((𝑥(𝑎 ∈ V, 𝑖 ∈ ℕ₀ ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})𝑀) ⊆ (^◡𝑓 “ {𝑐}) ↔ (𝑥(𝑎 ∈ V, 𝑖 ∈ ℕ₀ ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})𝑀) ⊆ (^◡𝑓 “ {𝐶})))
24	20, 23	anbi12d 630	. . . . 5 ⊢ (𝑐 = 𝐶 → (((𝐹‘𝑐) ≤ (♯‘𝑥) ∧ (𝑥(𝑎 ∈ V, 𝑖 ∈ ℕ₀ ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})𝑀) ⊆ (^◡𝑓 “ {𝑐})) ↔ ((𝐹‘𝐶) ≤ (♯‘𝑥) ∧ (𝑥(𝑎 ∈ V, 𝑖 ∈ ℕ₀ ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})𝑀) ⊆ (^◡𝑓 “ {𝐶}))))
25		fveq2 6882	. . . . . . . 8 ⊢ (𝑥 = ∅ → (♯‘𝑥) = (♯‘∅))
26		hash0 14328	. . . . . . . 8 ⊢ (♯‘∅) = 0
27	25, 26	eqtrdi 2780	. . . . . . 7 ⊢ (𝑥 = ∅ → (♯‘𝑥) = 0)
28	27	breq2d 5151	. . . . . 6 ⊢ (𝑥 = ∅ → ((𝐹‘𝐶) ≤ (♯‘𝑥) ↔ (𝐹‘𝐶) ≤ 0))
29		oveq1 7409	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝑥(𝑎 ∈ V, 𝑖 ∈ ℕ₀ ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})𝑀) = (∅(𝑎 ∈ V, 𝑖 ∈ ℕ₀ ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})𝑀))
30	29	sseq1d 4006	. . . . . 6 ⊢ (𝑥 = ∅ → ((𝑥(𝑎 ∈ V, 𝑖 ∈ ℕ₀ ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})𝑀) ⊆ (^◡𝑓 “ {𝐶}) ↔ (∅(𝑎 ∈ V, 𝑖 ∈ ℕ₀ ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})𝑀) ⊆ (^◡𝑓 “ {𝐶})))
31	28, 30	anbi12d 630	. . . . 5 ⊢ (𝑥 = ∅ → (((𝐹‘𝐶) ≤ (♯‘𝑥) ∧ (𝑥(𝑎 ∈ V, 𝑖 ∈ ℕ₀ ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})𝑀) ⊆ (^◡𝑓 “ {𝐶})) ↔ ((𝐹‘𝐶) ≤ 0 ∧ (∅(𝑎 ∈ V, 𝑖 ∈ ℕ₀ ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})𝑀) ⊆ (^◡𝑓 “ {𝐶}))))
32	24, 31	rspc2ev 3617	. . . 4 ⊢ ((𝐶 ∈ 𝑅 ∧ ∅ ∈ 𝒫 𝑠 ∧ ((𝐹‘𝐶) ≤ 0 ∧ (∅(𝑎 ∈ V, 𝑖 ∈ ℕ₀ ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})𝑀) ⊆ (^◡𝑓 “ {𝐶}))) → ∃𝑐 ∈ 𝑅 ∃𝑥 ∈ 𝒫 𝑠((𝐹‘𝑐) ≤ (♯‘𝑥) ∧ (𝑥(𝑎 ∈ V, 𝑖 ∈ ℕ₀ ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})𝑀) ⊆ (^◡𝑓 “ {𝑐})))
33	8, 10, 13, 18, 32	syl112anc 1371	. . 3 ⊢ ((((𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ₀) ∧ (𝐶 ∈ 𝑅 ∧ (𝐹‘𝐶) = 0)) ∧ (0 ≤ (♯‘𝑠) ∧ 𝑓:(𝑠(𝑎 ∈ V, 𝑖 ∈ ℕ₀ ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})𝑀)⟶𝑅)) → ∃𝑐 ∈ 𝑅 ∃𝑥 ∈ 𝒫 𝑠((𝐹‘𝑐) ≤ (♯‘𝑥) ∧ (𝑥(𝑎 ∈ V, 𝑖 ∈ ℕ₀ ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})𝑀) ⊆ (^◡𝑓 “ {𝑐})))
34	1, 3, 4, 5, 7, 33	ramub 16951	. 2 ⊢ (((𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ₀) ∧ (𝐶 ∈ 𝑅 ∧ (𝐹‘𝐶) = 0)) → (𝑀 Ramsey 𝐹) ≤ 0)
35		ramubcl 16956	. . . 4 ⊢ (((𝑀 ∈ ℕ₀ ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ₀) ∧ (0 ∈ ℕ₀ ∧ (𝑀 Ramsey 𝐹) ≤ 0)) → (𝑀 Ramsey 𝐹) ∈ ℕ₀)
36	3, 4, 5, 7, 34, 35	syl32anc 1375	. . 3 ⊢ (((𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ₀) ∧ (𝐶 ∈ 𝑅 ∧ (𝐹‘𝐶) = 0)) → (𝑀 Ramsey 𝐹) ∈ ℕ₀)
37		nn0le0eq0 12499	. . 3 ⊢ ((𝑀 Ramsey 𝐹) ∈ ℕ₀ → ((𝑀 Ramsey 𝐹) ≤ 0 ↔ (𝑀 Ramsey 𝐹) = 0))
38	36, 37	syl 17	. 2 ⊢ (((𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ₀) ∧ (𝐶 ∈ 𝑅 ∧ (𝐹‘𝐶) = 0)) → ((𝑀 Ramsey 𝐹) ≤ 0 ↔ (𝑀 Ramsey 𝐹) = 0))
39	34, 38	mpbid 231	1 ⊢ (((𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ₀) ∧ (𝐶 ∈ 𝑅 ∧ (𝐹‘𝐶) = 0)) → (𝑀 Ramsey 𝐹) = 0)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ∃wrex 3062 {crab 3424 Vcvv 3466 ⊆ wss 3941 ∅c0 4315 𝒫 cpw 4595 {csn 4621 class class class wbr 5139 ^◡ccnv 5666 “ cima 5670 ⟶wf 6530 ‘cfv 6534 (class class class)co 7402 ∈ cmpo 7404 0cc0 11107 ≤ cle 11248 ℕcn 12211 ℕ₀cn0 12471 ♯chash 14291 Ramsey cram 16937

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 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 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-int 4942 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-1st 7969 df-2nd 7970 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-oadd 8466 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-dju 9893 df-card 9931 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-div 11871 df-nn 12212 df-n0 12472 df-z 12558 df-uz 12822 df-rp 12976 df-fz 13486 df-seq 13968 df-fac 14235 df-bc 14264 df-hash 14292 df-ram 16939

This theorem is referenced by: ramz 16963 ramcl 16967

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