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Theorem ramub1lem1 17047

Description: Lemma for ramub1 17049. (Contributed by Mario Carneiro, 23-Apr-2015.)

**Hypotheses**
Ref	Expression
ramub1.m	⊢ (𝜑 → 𝑀 ∈ ℕ)
ramub1.r	⊢ (𝜑 → 𝑅 ∈ Fin)
ramub1.f	⊢ (𝜑 → 𝐹:𝑅⟶ℕ)
ramub1.g	⊢ 𝐺 = (𝑥 ∈ 𝑅 ↦ (𝑀 Ramsey (𝑦 ∈ 𝑅 ↦ if(𝑦 = 𝑥, ((𝐹‘𝑥) − 1), (𝐹‘𝑦)))))
ramub1.1	⊢ (𝜑 → 𝐺:𝑅⟶ℕ₀)
ramub1.2	⊢ (𝜑 → ((𝑀 − 1) Ramsey 𝐺) ∈ ℕ₀)
ramub1.3	⊢ 𝐶 = (𝑎 ∈ V, 𝑖 ∈ ℕ₀ ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})
ramub1.4	⊢ (𝜑 → 𝑆 ∈ Fin)
ramub1.5	⊢ (𝜑 → (♯‘𝑆) = (((𝑀 − 1) Ramsey 𝐺) + 1))
ramub1.6	⊢ (𝜑 → 𝐾:(𝑆𝐶𝑀)⟶𝑅)
ramub1.x	⊢ (𝜑 → 𝑋 ∈ 𝑆)
ramub1.h	⊢ 𝐻 = (𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1)) ↦ (𝐾‘(𝑢 ∪ {𝑋})))
ramub1.d	⊢ (𝜑 → 𝐷 ∈ 𝑅)
ramub1.w	⊢ (𝜑 → 𝑊 ⊆ (𝑆 ∖ {𝑋}))
ramub1.7	⊢ (𝜑 → (𝐺‘𝐷) ≤ (♯‘𝑊))
ramub1.8	⊢ (𝜑 → (𝑊𝐶(𝑀 − 1)) ⊆ (^◡𝐻 “ {𝐷}))
ramub1.e	⊢ (𝜑 → 𝐸 ∈ 𝑅)
ramub1.v	⊢ (𝜑 → 𝑉 ⊆ 𝑊)
ramub1.9	⊢ (𝜑 → if(𝐸 = 𝐷, ((𝐹‘𝐷) − 1), (𝐹‘𝐸)) ≤ (♯‘𝑉))
ramub1.s	⊢ (𝜑 → (𝑉𝐶𝑀) ⊆ (^◡𝐾 “ {𝐸}))

**Assertion**
Ref	Expression
ramub1lem1	⊢ (𝜑 → ∃𝑧 ∈ 𝒫 𝑆((𝐹‘𝐸) ≤ (♯‘𝑧) ∧ (𝑧𝐶𝑀) ⊆ (^◡𝐾 “ {𝐸})))

Distinct variable groups: 𝑥,𝑢,𝐷 𝑦,𝑢,𝑧,𝐹,𝑥 𝑎,𝑏,𝑖,𝑢,𝑥,𝑦,𝑧,𝑀 𝐺,𝑎,𝑖,𝑢,𝑥,𝑦,𝑧 𝑢,𝑅,𝑥,𝑦,𝑧 𝑊,𝑎,𝑖,𝑢 𝜑,𝑢,𝑥,𝑦,𝑧 𝑆,𝑎,𝑖,𝑢,𝑥,𝑦,𝑧 𝑉,𝑎,𝑖,𝑥,𝑧 𝑢,𝐶,𝑥,𝑦,𝑧 𝑢,𝐻,𝑥,𝑦,𝑧 𝑢,𝐾,𝑥,𝑦,𝑧 𝑥,𝐸,𝑧 𝑋,𝑎,𝑖,𝑢,𝑥,𝑦,𝑧 Allowed substitution hints: 𝜑(𝑖,𝑎,𝑏) 𝐶(𝑖,𝑎,𝑏) 𝐷(𝑦,𝑧,𝑖,𝑎,𝑏) 𝑅(𝑖,𝑎,𝑏) 𝑆(𝑏) 𝐸(𝑦,𝑢,𝑖,𝑎,𝑏) 𝐹(𝑖,𝑎,𝑏) 𝐺(𝑏) 𝐻(𝑖,𝑎,𝑏) 𝐾(𝑖,𝑎,𝑏) 𝑉(𝑦,𝑢,𝑏) 𝑊(𝑥,𝑦,𝑧,𝑏) 𝑋(𝑏)

**Proof of Theorem ramub1lem1**
Step	Hyp	Ref	Expression
1		ramub1.4	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ Fin)
2		ramub1.v	. . . . . . . 8 ⊢ (𝜑 → 𝑉 ⊆ 𝑊)
3		ramub1.w	. . . . . . . 8 ⊢ (𝜑 → 𝑊 ⊆ (𝑆 ∖ {𝑋}))
4	2, 3	sstrd 3976	. . . . . . 7 ⊢ (𝜑 → 𝑉 ⊆ (𝑆 ∖ {𝑋}))
5	4	difss2d 4121	. . . . . 6 ⊢ (𝜑 → 𝑉 ⊆ 𝑆)
6		ramub1.x	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝑆)
7	6	snssd 4791	. . . . . 6 ⊢ (𝜑 → {𝑋} ⊆ 𝑆)
8	5, 7	unssd 4174	. . . . 5 ⊢ (𝜑 → (𝑉 ∪ {𝑋}) ⊆ 𝑆)
9	1, 8	sselpwd 5310	. . . 4 ⊢ (𝜑 → (𝑉 ∪ {𝑋}) ∈ 𝒫 𝑆)
10	9	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝐸 = 𝐷) → (𝑉 ∪ {𝑋}) ∈ 𝒫 𝑆)
11		iftrue 4513	. . . . . . 7 ⊢ (𝐸 = 𝐷 → if(𝐸 = 𝐷, ((𝐹‘𝐷) − 1), (𝐹‘𝐸)) = ((𝐹‘𝐷) − 1))
12	11	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝐸 = 𝐷) → if(𝐸 = 𝐷, ((𝐹‘𝐷) − 1), (𝐹‘𝐸)) = ((𝐹‘𝐷) − 1))
13		ramub1.9	. . . . . . 7 ⊢ (𝜑 → if(𝐸 = 𝐷, ((𝐹‘𝐷) − 1), (𝐹‘𝐸)) ≤ (♯‘𝑉))
14	13	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝐸 = 𝐷) → if(𝐸 = 𝐷, ((𝐹‘𝐷) − 1), (𝐹‘𝐸)) ≤ (♯‘𝑉))
15	12, 14	eqbrtrrd 5149	. . . . 5 ⊢ ((𝜑 ∧ 𝐸 = 𝐷) → ((𝐹‘𝐷) − 1) ≤ (♯‘𝑉))
16		ramub1.f	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:𝑅⟶ℕ)
17		ramub1.d	. . . . . . . . 9 ⊢ (𝜑 → 𝐷 ∈ 𝑅)
18	16, 17	ffvelcdmd 7086	. . . . . . . 8 ⊢ (𝜑 → (𝐹‘𝐷) ∈ ℕ)
19	18	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐸 = 𝐷) → (𝐹‘𝐷) ∈ ℕ)
20	19	nnred 12264	. . . . . 6 ⊢ ((𝜑 ∧ 𝐸 = 𝐷) → (𝐹‘𝐷) ∈ ℝ)
21		1red 11245	. . . . . 6 ⊢ ((𝜑 ∧ 𝐸 = 𝐷) → 1 ∈ ℝ)
22	1, 5	ssfid 9284	. . . . . . . 8 ⊢ (𝜑 → 𝑉 ∈ Fin)
23		hashcl 14378	. . . . . . . 8 ⊢ (𝑉 ∈ Fin → (♯‘𝑉) ∈ ℕ₀)
24		nn0re 12519	. . . . . . . 8 ⊢ ((♯‘𝑉) ∈ ℕ₀ → (♯‘𝑉) ∈ ℝ)
25	22, 23, 24	3syl 18	. . . . . . 7 ⊢ (𝜑 → (♯‘𝑉) ∈ ℝ)
26	25	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝐸 = 𝐷) → (♯‘𝑉) ∈ ℝ)
27	20, 21, 26	lesubaddd 11843	. . . . 5 ⊢ ((𝜑 ∧ 𝐸 = 𝐷) → (((𝐹‘𝐷) − 1) ≤ (♯‘𝑉) ↔ (𝐹‘𝐷) ≤ ((♯‘𝑉) + 1)))
28	15, 27	mpbid 232	. . . 4 ⊢ ((𝜑 ∧ 𝐸 = 𝐷) → (𝐹‘𝐷) ≤ ((♯‘𝑉) + 1))
29		fveq2 6887	. . . . 5 ⊢ (𝐸 = 𝐷 → (𝐹‘𝐸) = (𝐹‘𝐷))
30		snidg 4642	. . . . . . . 8 ⊢ (𝑋 ∈ 𝑆 → 𝑋 ∈ {𝑋})
31	6, 30	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ {𝑋})
32	4	sseld 3964	. . . . . . . 8 ⊢ (𝜑 → (𝑋 ∈ 𝑉 → 𝑋 ∈ (𝑆 ∖ {𝑋})))
33		eldifn 4114	. . . . . . . 8 ⊢ (𝑋 ∈ (𝑆 ∖ {𝑋}) → ¬ 𝑋 ∈ {𝑋})
34	32, 33	syl6 35	. . . . . . 7 ⊢ (𝜑 → (𝑋 ∈ 𝑉 → ¬ 𝑋 ∈ {𝑋}))
35	31, 34	mt2d 136	. . . . . 6 ⊢ (𝜑 → ¬ 𝑋 ∈ 𝑉)
36		hashunsng 14414	. . . . . . 7 ⊢ (𝑋 ∈ 𝑆 → ((𝑉 ∈ Fin ∧ ¬ 𝑋 ∈ 𝑉) → (♯‘(𝑉 ∪ {𝑋})) = ((♯‘𝑉) + 1)))
37	6, 36	syl 17	. . . . . 6 ⊢ (𝜑 → ((𝑉 ∈ Fin ∧ ¬ 𝑋 ∈ 𝑉) → (♯‘(𝑉 ∪ {𝑋})) = ((♯‘𝑉) + 1)))
38	22, 35, 37	mp2and 699	. . . . 5 ⊢ (𝜑 → (♯‘(𝑉 ∪ {𝑋})) = ((♯‘𝑉) + 1))
39	29, 38	breqan12rd 5142	. . . 4 ⊢ ((𝜑 ∧ 𝐸 = 𝐷) → ((𝐹‘𝐸) ≤ (♯‘(𝑉 ∪ {𝑋})) ↔ (𝐹‘𝐷) ≤ ((♯‘𝑉) + 1)))
40	28, 39	mpbird 257	. . 3 ⊢ ((𝜑 ∧ 𝐸 = 𝐷) → (𝐹‘𝐸) ≤ (♯‘(𝑉 ∪ {𝑋})))
41		snfi 9066	. . . . . . 7 ⊢ {𝑋} ∈ Fin
42		unfi 9194	. . . . . . 7 ⊢ ((𝑉 ∈ Fin ∧ {𝑋} ∈ Fin) → (𝑉 ∪ {𝑋}) ∈ Fin)
43	22, 41, 42	sylancl 586	. . . . . 6 ⊢ (𝜑 → (𝑉 ∪ {𝑋}) ∈ Fin)
44		ramub1.m	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℕ)
45	44	nnnn0d 12571	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℕ₀)
46		ramub1.3	. . . . . . 7 ⊢ 𝐶 = (𝑎 ∈ V, 𝑖 ∈ ℕ₀ ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})
47	46	hashbcval 17023	. . . . . 6 ⊢ (((𝑉 ∪ {𝑋}) ∈ Fin ∧ 𝑀 ∈ ℕ₀) → ((𝑉 ∪ {𝑋})𝐶𝑀) = {𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∣ (♯‘𝑥) = 𝑀})
48	43, 45, 47	syl2anc 584	. . . . 5 ⊢ (𝜑 → ((𝑉 ∪ {𝑋})𝐶𝑀) = {𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∣ (♯‘𝑥) = 𝑀})
49	48	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝐸 = 𝐷) → ((𝑉 ∪ {𝑋})𝐶𝑀) = {𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∣ (♯‘𝑥) = 𝑀})
50		simpl1l 1224	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ 𝑥 ∈ 𝒫 𝑉) → 𝜑)
51	46	hashbcval 17023	. . . . . . . . . 10 ⊢ ((𝑉 ∈ Fin ∧ 𝑀 ∈ ℕ₀) → (𝑉𝐶𝑀) = {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 𝑀})
52	22, 45, 51	syl2anc 584	. . . . . . . . 9 ⊢ (𝜑 → (𝑉𝐶𝑀) = {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 𝑀})
53		ramub1.s	. . . . . . . . 9 ⊢ (𝜑 → (𝑉𝐶𝑀) ⊆ (^◡𝐾 “ {𝐸}))
54	52, 53	eqsstrrd 4001	. . . . . . . 8 ⊢ (𝜑 → {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 𝑀} ⊆ (^◡𝐾 “ {𝐸}))
55	50, 54	syl 17	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ 𝑥 ∈ 𝒫 𝑉) → {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 𝑀} ⊆ (^◡𝐾 “ {𝐸}))
56		simpr 484	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ 𝑥 ∈ 𝒫 𝑉) → 𝑥 ∈ 𝒫 𝑉)
57		simpl3 1193	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ 𝑥 ∈ 𝒫 𝑉) → (♯‘𝑥) = 𝑀)
58		rabid 3442	. . . . . . . 8 ⊢ (𝑥 ∈ {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 𝑀} ↔ (𝑥 ∈ 𝒫 𝑉 ∧ (♯‘𝑥) = 𝑀))
59	56, 57, 58	sylanbrc 583	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ 𝑥 ∈ 𝒫 𝑉) → 𝑥 ∈ {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 𝑀})
60	55, 59	sseldd 3966	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ 𝑥 ∈ 𝒫 𝑉) → 𝑥 ∈ (^◡𝐾 “ {𝐸}))
61		simpl2 1192	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}))
62	61	elpwid 4591	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → 𝑥 ⊆ (𝑉 ∪ {𝑋}))
63		simpl1l 1224	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → 𝜑)
64	63, 8	syl 17	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → (𝑉 ∪ {𝑋}) ⊆ 𝑆)
65	62, 64	sstrd 3976	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → 𝑥 ⊆ 𝑆)
66		vex 3468	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
67	66	elpw 4586	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝒫 𝑆 ↔ 𝑥 ⊆ 𝑆)
68	65, 67	sylibr 234	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → 𝑥 ∈ 𝒫 𝑆)
69		simpl3 1193	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → (♯‘𝑥) = 𝑀)
70		rabid 3442	. . . . . . . . 9 ⊢ (𝑥 ∈ {𝑥 ∈ 𝒫 𝑆 ∣ (♯‘𝑥) = 𝑀} ↔ (𝑥 ∈ 𝒫 𝑆 ∧ (♯‘𝑥) = 𝑀))
71	68, 69, 70	sylanbrc 583	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → 𝑥 ∈ {𝑥 ∈ 𝒫 𝑆 ∣ (♯‘𝑥) = 𝑀})
72	46	hashbcval 17023	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Fin ∧ 𝑀 ∈ ℕ₀) → (𝑆𝐶𝑀) = {𝑥 ∈ 𝒫 𝑆 ∣ (♯‘𝑥) = 𝑀})
73	1, 45, 72	syl2anc 584	. . . . . . . . 9 ⊢ (𝜑 → (𝑆𝐶𝑀) = {𝑥 ∈ 𝒫 𝑆 ∣ (♯‘𝑥) = 𝑀})
74	63, 73	syl 17	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → (𝑆𝐶𝑀) = {𝑥 ∈ 𝒫 𝑆 ∣ (♯‘𝑥) = 𝑀})
75	71, 74	eleqtrrd 2836	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → 𝑥 ∈ (𝑆𝐶𝑀))
76	3	difss2d 4121	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑊 ⊆ 𝑆)
77	1, 76	ssfid 9284	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑊 ∈ Fin)
78		nnm1nn0 12551	. . . . . . . . . . . . . . 15 ⊢ (𝑀 ∈ ℕ → (𝑀 − 1) ∈ ℕ₀)
79	44, 78	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑀 − 1) ∈ ℕ₀)
80	46	hashbcval 17023	. . . . . . . . . . . . . 14 ⊢ ((𝑊 ∈ Fin ∧ (𝑀 − 1) ∈ ℕ₀) → (𝑊𝐶(𝑀 − 1)) = {𝑢 ∈ 𝒫 𝑊 ∣ (♯‘𝑢) = (𝑀 − 1)})
81	77, 79, 80	syl2anc 584	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑊𝐶(𝑀 − 1)) = {𝑢 ∈ 𝒫 𝑊 ∣ (♯‘𝑢) = (𝑀 − 1)})
82		ramub1.8	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑊𝐶(𝑀 − 1)) ⊆ (^◡𝐻 “ {𝐷}))
83	81, 82	eqsstrrd 4001	. . . . . . . . . . . 12 ⊢ (𝜑 → {𝑢 ∈ 𝒫 𝑊 ∣ (♯‘𝑢) = (𝑀 − 1)} ⊆ (^◡𝐻 “ {𝐷}))
84	63, 83	syl 17	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → {𝑢 ∈ 𝒫 𝑊 ∣ (♯‘𝑢) = (𝑀 − 1)} ⊆ (^◡𝐻 “ {𝐷}))
85		fveqeq2 6896	. . . . . . . . . . . 12 ⊢ (𝑢 = (𝑥 ∖ {𝑋}) → ((♯‘𝑢) = (𝑀 − 1) ↔ (♯‘(𝑥 ∖ {𝑋})) = (𝑀 − 1)))
86		uncom 4140	. . . . . . . . . . . . . . . 16 ⊢ (𝑉 ∪ {𝑋}) = ({𝑋} ∪ 𝑉)
87	62, 86	sseqtrdi 4006	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → 𝑥 ⊆ ({𝑋} ∪ 𝑉))
88		ssundif 4470	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ⊆ ({𝑋} ∪ 𝑉) ↔ (𝑥 ∖ {𝑋}) ⊆ 𝑉)
89	87, 88	sylib 218	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → (𝑥 ∖ {𝑋}) ⊆ 𝑉)
90	63, 2	syl 17	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → 𝑉 ⊆ 𝑊)
91	89, 90	sstrd 3976	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → (𝑥 ∖ {𝑋}) ⊆ 𝑊)
92	66	difexi 5312	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∖ {𝑋}) ∈ V
93	92	elpw 4586	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∖ {𝑋}) ∈ 𝒫 𝑊 ↔ (𝑥 ∖ {𝑋}) ⊆ 𝑊)
94	91, 93	sylibr 234	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → (𝑥 ∖ {𝑋}) ∈ 𝒫 𝑊)
95	63, 1	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → 𝑆 ∈ Fin)
96	95, 65	ssfid 9284	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → 𝑥 ∈ Fin)
97		diffi 9198	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ Fin → (𝑥 ∖ {𝑋}) ∈ Fin)
98	96, 97	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → (𝑥 ∖ {𝑋}) ∈ Fin)
99		hashcl 14378	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∖ {𝑋}) ∈ Fin → (♯‘(𝑥 ∖ {𝑋})) ∈ ℕ₀)
100		nn0cn 12520	. . . . . . . . . . . . . . 15 ⊢ ((♯‘(𝑥 ∖ {𝑋})) ∈ ℕ₀ → (♯‘(𝑥 ∖ {𝑋})) ∈ ℂ)
101	98, 99, 100	3syl 18	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → (♯‘(𝑥 ∖ {𝑋})) ∈ ℂ)
102		ax-1cn 11196	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℂ
103		pncan 11497	. . . . . . . . . . . . . 14 ⊢ (((♯‘(𝑥 ∖ {𝑋})) ∈ ℂ ∧ 1 ∈ ℂ) → (((♯‘(𝑥 ∖ {𝑋})) + 1) − 1) = (♯‘(𝑥 ∖ {𝑋})))
104	101, 102, 103	sylancl 586	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → (((♯‘(𝑥 ∖ {𝑋})) + 1) − 1) = (♯‘(𝑥 ∖ {𝑋})))
105		neldifsnd 4775	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → ¬ 𝑋 ∈ (𝑥 ∖ {𝑋}))
106		hashunsng 14414	. . . . . . . . . . . . . . . . 17 ⊢ (𝑋 ∈ 𝑆 → (((𝑥 ∖ {𝑋}) ∈ Fin ∧ ¬ 𝑋 ∈ (𝑥 ∖ {𝑋})) → (♯‘((𝑥 ∖ {𝑋}) ∪ {𝑋})) = ((♯‘(𝑥 ∖ {𝑋})) + 1)))
107	63, 6, 106	3syl 18	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → (((𝑥 ∖ {𝑋}) ∈ Fin ∧ ¬ 𝑋 ∈ (𝑥 ∖ {𝑋})) → (♯‘((𝑥 ∖ {𝑋}) ∪ {𝑋})) = ((♯‘(𝑥 ∖ {𝑋})) + 1)))
108	98, 105, 107	mp2and 699	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → (♯‘((𝑥 ∖ {𝑋}) ∪ {𝑋})) = ((♯‘(𝑥 ∖ {𝑋})) + 1))
109		undif1 4458	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∖ {𝑋}) ∪ {𝑋}) = (𝑥 ∪ {𝑋})
110		simpr 484	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → ¬ 𝑥 ∈ 𝒫 𝑉)
111	61, 110	eldifd 3944	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → 𝑥 ∈ (𝒫 (𝑉 ∪ {𝑋}) ∖ 𝒫 𝑉))
112		elpwunsn 4666	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ (𝒫 (𝑉 ∪ {𝑋}) ∖ 𝒫 𝑉) → 𝑋 ∈ 𝑥)
113	111, 112	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → 𝑋 ∈ 𝑥)
114	113	snssd 4791	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → {𝑋} ⊆ 𝑥)
115		ssequn2 4171	. . . . . . . . . . . . . . . . . . 19 ⊢ ({𝑋} ⊆ 𝑥 ↔ (𝑥 ∪ {𝑋}) = 𝑥)
116	114, 115	sylib 218	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → (𝑥 ∪ {𝑋}) = 𝑥)
117	109, 116	eqtr2id 2782	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → 𝑥 = ((𝑥 ∖ {𝑋}) ∪ {𝑋}))
118	117	fveq2d 6891	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → (♯‘𝑥) = (♯‘((𝑥 ∖ {𝑋}) ∪ {𝑋})))
119	118, 69	eqtr3d 2771	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → (♯‘((𝑥 ∖ {𝑋}) ∪ {𝑋})) = 𝑀)
120	108, 119	eqtr3d 2771	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → ((♯‘(𝑥 ∖ {𝑋})) + 1) = 𝑀)
121	120	oveq1d 7429	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → (((♯‘(𝑥 ∖ {𝑋})) + 1) − 1) = (𝑀 − 1))
122	104, 121	eqtr3d 2771	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → (♯‘(𝑥 ∖ {𝑋})) = (𝑀 − 1))
123	85, 94, 122	elrabd 3678	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → (𝑥 ∖ {𝑋}) ∈ {𝑢 ∈ 𝒫 𝑊 ∣ (♯‘𝑢) = (𝑀 − 1)})
124	84, 123	sseldd 3966	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → (𝑥 ∖ {𝑋}) ∈ (^◡𝐻 “ {𝐷}))
125		ramub1.h	. . . . . . . . . . . 12 ⊢ 𝐻 = (𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1)) ↦ (𝐾‘(𝑢 ∪ {𝑋})))
126	125	mptiniseg 6241	. . . . . . . . . . 11 ⊢ (𝐷 ∈ 𝑅 → (^◡𝐻 “ {𝐷}) = {𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1)) ∣ (𝐾‘(𝑢 ∪ {𝑋})) = 𝐷})
127	63, 17, 126	3syl 18	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → (^◡𝐻 “ {𝐷}) = {𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1)) ∣ (𝐾‘(𝑢 ∪ {𝑋})) = 𝐷})
128	124, 127	eleqtrd 2835	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → (𝑥 ∖ {𝑋}) ∈ {𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1)) ∣ (𝐾‘(𝑢 ∪ {𝑋})) = 𝐷})
129		uneq1 4143	. . . . . . . . . . . 12 ⊢ (𝑢 = (𝑥 ∖ {𝑋}) → (𝑢 ∪ {𝑋}) = ((𝑥 ∖ {𝑋}) ∪ {𝑋}))
130	129	fveqeq2d 6895	. . . . . . . . . . 11 ⊢ (𝑢 = (𝑥 ∖ {𝑋}) → ((𝐾‘(𝑢 ∪ {𝑋})) = 𝐷 ↔ (𝐾‘((𝑥 ∖ {𝑋}) ∪ {𝑋})) = 𝐷))
131	130	elrab 3676	. . . . . . . . . 10 ⊢ ((𝑥 ∖ {𝑋}) ∈ {𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1)) ∣ (𝐾‘(𝑢 ∪ {𝑋})) = 𝐷} ↔ ((𝑥 ∖ {𝑋}) ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1)) ∧ (𝐾‘((𝑥 ∖ {𝑋}) ∪ {𝑋})) = 𝐷))
132	131	simprbi 496	. . . . . . . . 9 ⊢ ((𝑥 ∖ {𝑋}) ∈ {𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1)) ∣ (𝐾‘(𝑢 ∪ {𝑋})) = 𝐷} → (𝐾‘((𝑥 ∖ {𝑋}) ∪ {𝑋})) = 𝐷)
133	128, 132	syl 17	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → (𝐾‘((𝑥 ∖ {𝑋}) ∪ {𝑋})) = 𝐷)
134	117	fveq2d 6891	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → (𝐾‘𝑥) = (𝐾‘((𝑥 ∖ {𝑋}) ∪ {𝑋})))
135		simpl1r 1225	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → 𝐸 = 𝐷)
136	133, 134, 135	3eqtr4d 2779	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → (𝐾‘𝑥) = 𝐸)
137		ramub1.6	. . . . . . . . 9 ⊢ (𝜑 → 𝐾:(𝑆𝐶𝑀)⟶𝑅)
138	137	ffnd 6718	. . . . . . . 8 ⊢ (𝜑 → 𝐾 Fn (𝑆𝐶𝑀))
139		fniniseg 7061	. . . . . . . 8 ⊢ (𝐾 Fn (𝑆𝐶𝑀) → (𝑥 ∈ (^◡𝐾 “ {𝐸}) ↔ (𝑥 ∈ (𝑆𝐶𝑀) ∧ (𝐾‘𝑥) = 𝐸)))
140	63, 138, 139	3syl 18	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → (𝑥 ∈ (^◡𝐾 “ {𝐸}) ↔ (𝑥 ∈ (𝑆𝐶𝑀) ∧ (𝐾‘𝑥) = 𝐸)))
141	75, 136, 140	mpbir2and 713	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → 𝑥 ∈ (^◡𝐾 “ {𝐸}))
142	60, 141	pm2.61dan 812	. . . . 5 ⊢ (((𝜑 ∧ 𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) → 𝑥 ∈ (^◡𝐾 “ {𝐸}))
143	142	rabssdv 4057	. . . 4 ⊢ ((𝜑 ∧ 𝐸 = 𝐷) → {𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∣ (♯‘𝑥) = 𝑀} ⊆ (^◡𝐾 “ {𝐸}))
144	49, 143	eqsstrd 4000	. . 3 ⊢ ((𝜑 ∧ 𝐸 = 𝐷) → ((𝑉 ∪ {𝑋})𝐶𝑀) ⊆ (^◡𝐾 “ {𝐸}))
145		fveq2 6887	. . . . . 6 ⊢ (𝑧 = (𝑉 ∪ {𝑋}) → (♯‘𝑧) = (♯‘(𝑉 ∪ {𝑋})))
146	145	breq2d 5137	. . . . 5 ⊢ (𝑧 = (𝑉 ∪ {𝑋}) → ((𝐹‘𝐸) ≤ (♯‘𝑧) ↔ (𝐹‘𝐸) ≤ (♯‘(𝑉 ∪ {𝑋}))))
147		oveq1 7421	. . . . . 6 ⊢ (𝑧 = (𝑉 ∪ {𝑋}) → (𝑧𝐶𝑀) = ((𝑉 ∪ {𝑋})𝐶𝑀))
148	147	sseq1d 3997	. . . . 5 ⊢ (𝑧 = (𝑉 ∪ {𝑋}) → ((𝑧𝐶𝑀) ⊆ (^◡𝐾 “ {𝐸}) ↔ ((𝑉 ∪ {𝑋})𝐶𝑀) ⊆ (^◡𝐾 “ {𝐸})))
149	146, 148	anbi12d 632	. . . 4 ⊢ (𝑧 = (𝑉 ∪ {𝑋}) → (((𝐹‘𝐸) ≤ (♯‘𝑧) ∧ (𝑧𝐶𝑀) ⊆ (^◡𝐾 “ {𝐸})) ↔ ((𝐹‘𝐸) ≤ (♯‘(𝑉 ∪ {𝑋})) ∧ ((𝑉 ∪ {𝑋})𝐶𝑀) ⊆ (^◡𝐾 “ {𝐸}))))
150	149	rspcev 3606	. . 3 ⊢ (((𝑉 ∪ {𝑋}) ∈ 𝒫 𝑆 ∧ ((𝐹‘𝐸) ≤ (♯‘(𝑉 ∪ {𝑋})) ∧ ((𝑉 ∪ {𝑋})𝐶𝑀) ⊆ (^◡𝐾 “ {𝐸}))) → ∃𝑧 ∈ 𝒫 𝑆((𝐹‘𝐸) ≤ (♯‘𝑧) ∧ (𝑧𝐶𝑀) ⊆ (^◡𝐾 “ {𝐸})))
151	10, 40, 144, 150	syl12anc 836	. 2 ⊢ ((𝜑 ∧ 𝐸 = 𝐷) → ∃𝑧 ∈ 𝒫 𝑆((𝐹‘𝐸) ≤ (♯‘𝑧) ∧ (𝑧𝐶𝑀) ⊆ (^◡𝐾 “ {𝐸})))
152	1, 5	sselpwd 5310	. . . 4 ⊢ (𝜑 → 𝑉 ∈ 𝒫 𝑆)
153	152	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝐸 ≠ 𝐷) → 𝑉 ∈ 𝒫 𝑆)
154		ifnefalse 4519	. . . . 5 ⊢ (𝐸 ≠ 𝐷 → if(𝐸 = 𝐷, ((𝐹‘𝐷) − 1), (𝐹‘𝐸)) = (𝐹‘𝐸))
155	154	adantl 481	. . . 4 ⊢ ((𝜑 ∧ 𝐸 ≠ 𝐷) → if(𝐸 = 𝐷, ((𝐹‘𝐷) − 1), (𝐹‘𝐸)) = (𝐹‘𝐸))
156	13	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝐸 ≠ 𝐷) → if(𝐸 = 𝐷, ((𝐹‘𝐷) − 1), (𝐹‘𝐸)) ≤ (♯‘𝑉))
157	155, 156	eqbrtrrd 5149	. . 3 ⊢ ((𝜑 ∧ 𝐸 ≠ 𝐷) → (𝐹‘𝐸) ≤ (♯‘𝑉))
158	53	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝐸 ≠ 𝐷) → (𝑉𝐶𝑀) ⊆ (^◡𝐾 “ {𝐸}))
159		fveq2 6887	. . . . . 6 ⊢ (𝑧 = 𝑉 → (♯‘𝑧) = (♯‘𝑉))
160	159	breq2d 5137	. . . . 5 ⊢ (𝑧 = 𝑉 → ((𝐹‘𝐸) ≤ (♯‘𝑧) ↔ (𝐹‘𝐸) ≤ (♯‘𝑉)))
161		oveq1 7421	. . . . . 6 ⊢ (𝑧 = 𝑉 → (𝑧𝐶𝑀) = (𝑉𝐶𝑀))
162	161	sseq1d 3997	. . . . 5 ⊢ (𝑧 = 𝑉 → ((𝑧𝐶𝑀) ⊆ (^◡𝐾 “ {𝐸}) ↔ (𝑉𝐶𝑀) ⊆ (^◡𝐾 “ {𝐸})))
163	160, 162	anbi12d 632	. . . 4 ⊢ (𝑧 = 𝑉 → (((𝐹‘𝐸) ≤ (♯‘𝑧) ∧ (𝑧𝐶𝑀) ⊆ (^◡𝐾 “ {𝐸})) ↔ ((𝐹‘𝐸) ≤ (♯‘𝑉) ∧ (𝑉𝐶𝑀) ⊆ (^◡𝐾 “ {𝐸}))))
164	163	rspcev 3606	. . 3 ⊢ ((𝑉 ∈ 𝒫 𝑆 ∧ ((𝐹‘𝐸) ≤ (♯‘𝑉) ∧ (𝑉𝐶𝑀) ⊆ (^◡𝐾 “ {𝐸}))) → ∃𝑧 ∈ 𝒫 𝑆((𝐹‘𝐸) ≤ (♯‘𝑧) ∧ (𝑧𝐶𝑀) ⊆ (^◡𝐾 “ {𝐸})))
165	153, 157, 158, 164	syl12anc 836	. 2 ⊢ ((𝜑 ∧ 𝐸 ≠ 𝐷) → ∃𝑧 ∈ 𝒫 𝑆((𝐹‘𝐸) ≤ (♯‘𝑧) ∧ (𝑧𝐶𝑀) ⊆ (^◡𝐾 “ {𝐸})))
166	151, 165	pm2.61dane 3018	1 ⊢ (𝜑 → ∃𝑧 ∈ 𝒫 𝑆((𝐹‘𝐸) ≤ (♯‘𝑧) ∧ (𝑧𝐶𝑀) ⊆ (^◡𝐾 “ {𝐸})))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1539 ∈ wcel 2107 ≠ wne 2931 ∃wrex 3059 {crab 3420 Vcvv 3464 ∖ cdif 3930 ∪ cun 3931 ⊆ wss 3933 ifcif 4507 𝒫 cpw 4582 {csn 4608 class class class wbr 5125 ↦ cmpt 5207 ^◡ccnv 5666 “ cima 5670 Fn wfn 6537 ⟶wf 6538 ‘cfv 6542 (class class class)co 7414 ∈ cmpo 7416 Fincfn 8968 ℂcc 11136 ℝcr 11137 1c1 11139 + caddc 11141 ≤ cle 11279 − cmin 11475 ℕcn 12249 ℕ₀cn0 12510 ♯chash 14352 Ramsey cram 17020

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2706 ax-sep 5278 ax-nul 5288 ax-pow 5347 ax-pr 5414 ax-un 7738 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2808 df-nfc 2884 df-ne 2932 df-nel 3036 df-ral 3051 df-rex 3060 df-reu 3365 df-rab 3421 df-v 3466 df-sbc 3773 df-csb 3882 df-dif 3936 df-un 3938 df-in 3940 df-ss 3950 df-pss 3953 df-nul 4316 df-if 4508 df-pw 4584 df-sn 4609 df-pr 4611 df-op 4615 df-uni 4890 df-int 4929 df-iun 4975 df-br 5126 df-opab 5188 df-mpt 5208 df-tr 5242 df-id 5560 df-eprel 5566 df-po 5574 df-so 5575 df-fr 5619 df-we 5621 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 6303 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6495 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7371 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7871 df-1st 7997 df-2nd 7998 df-frecs 8289 df-wrecs 8320 df-recs 8394 df-rdg 8433 df-1o 8489 df-oadd 8493 df-er 8728 df-en 8969 df-dom 8970 df-sdom 8971 df-fin 8972 df-dju 9924 df-card 9962 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11477 df-neg 11478 df-nn 12250 df-n0 12511 df-z 12598 df-uz 12862 df-fz 13531 df-hash 14353

This theorem is referenced by: ramub1lem2 17048

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