Theorem ramub1lem1 16955

Description: Lemma for ramub1 16957. (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 3991	. . . . . . 7 ⊢ (𝜑 → 𝑉 ⊆ (𝑆 ∖ {𝑋}))
5	4	difss2d 4133	. . . . . 6 ⊢ (𝜑 → 𝑉 ⊆ 𝑆)
6		ramub1.x	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝑆)
7	6	snssd 4811	. . . . . 6 ⊢ (𝜑 → {𝑋} ⊆ 𝑆)
8	5, 7	unssd 4185	. . . . 5 ⊢ (𝜑 → (𝑉 ∪ {𝑋}) ⊆ 𝑆)
9	1, 8	sselpwd 5325	. . . 4 ⊢ (𝜑 → (𝑉 ∪ {𝑋}) ∈ 𝒫 𝑆)
10	9	adantr 482	. . 3 ⊢ ((𝜑 ∧ 𝐸 = 𝐷) → (𝑉 ∪ {𝑋}) ∈ 𝒫 𝑆)
11		iftrue 4533	. . . . . . 7 ⊢ (𝐸 = 𝐷 → if(𝐸 = 𝐷, ((𝐹‘𝐷) − 1), (𝐹‘𝐸)) = ((𝐹‘𝐷) − 1))
12	11	adantl 483	. . . . . 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 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝐸 = 𝐷) → if(𝐸 = 𝐷, ((𝐹‘𝐷) − 1), (𝐹‘𝐸)) ≤ (♯‘𝑉))
15	12, 14	eqbrtrrd 5171	. . . . 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 7083	. . . . . . . 8 ⊢ (𝜑 → (𝐹‘𝐷) ∈ ℕ)
19	18	adantr 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐸 = 𝐷) → (𝐹‘𝐷) ∈ ℕ)
20	19	nnred 12223	. . . . . 6 ⊢ ((𝜑 ∧ 𝐸 = 𝐷) → (𝐹‘𝐷) ∈ ℝ)
21		1red 11211	. . . . . 6 ⊢ ((𝜑 ∧ 𝐸 = 𝐷) → 1 ∈ ℝ)
22	1, 5	ssfid 9263	. . . . . . . 8 ⊢ (𝜑 → 𝑉 ∈ Fin)
23		hashcl 14312	. . . . . . . 8 ⊢ (𝑉 ∈ Fin → (♯‘𝑉) ∈ ℕ0)
24		nn0re 12477	. . . . . . . 8 ⊢ ((♯‘𝑉) ∈ ℕ0 → (♯‘𝑉) ∈ ℝ)
25	22, 23, 24	3syl 18	. . . . . . 7 ⊢ (𝜑 → (♯‘𝑉) ∈ ℝ)
26	25	adantr 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝐸 = 𝐷) → (♯‘𝑉) ∈ ℝ)
27	20, 21, 26	lesubaddd 11807	. . . . 5 ⊢ ((𝜑 ∧ 𝐸 = 𝐷) → (((𝐹‘𝐷) − 1) ≤ (♯‘𝑉) ↔ (𝐹‘𝐷) ≤ ((♯‘𝑉) + 1)))
28	15, 27	mpbid 231	. . . 4 ⊢ ((𝜑 ∧ 𝐸 = 𝐷) → (𝐹‘𝐷) ≤ ((♯‘𝑉) + 1))
29		fveq2 6888	. . . . 5 ⊢ (𝐸 = 𝐷 → (𝐹‘𝐸) = (𝐹‘𝐷))
30		snidg 4661	. . . . . . . 8 ⊢ (𝑋 ∈ 𝑆 → 𝑋 ∈ {𝑋})
31	6, 30	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ {𝑋})
32	4	sseld 3980	. . . . . . . 8 ⊢ (𝜑 → (𝑋 ∈ 𝑉 → 𝑋 ∈ (𝑆 ∖ {𝑋})))
33		eldifn 4126	. . . . . . . 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 14348	. . . . . . 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 698	. . . . 5 ⊢ (𝜑 → (♯‘(𝑉 ∪ {𝑋})) = ((♯‘𝑉) + 1))
39	29, 38	breqan12rd 5164	. . . 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 9040	. . . . . . 7 ⊢ {𝑋} ∈ Fin
42		unfi 9168	. . . . . . 7 ⊢ ((𝑉 ∈ Fin ∧ {𝑋} ∈ Fin) → (𝑉 ∪ {𝑋}) ∈ Fin)
43	22, 41, 42	sylancl 587	. . . . . 6 ⊢ (𝜑 → (𝑉 ∪ {𝑋}) ∈ Fin)
44		ramub1.m	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℕ)
45	44	nnnn0d 12528	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℕ0)
46		ramub1.3	. . . . . . 7 ⊢ 𝐶 = (𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})
47	46	hashbcval 16931	. . . . . 6 ⊢ (((𝑉 ∪ {𝑋}) ∈ Fin ∧ 𝑀 ∈ ℕ0) → ((𝑉 ∪ {𝑋})𝐶𝑀) = {𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∣ (♯‘𝑥) = 𝑀})
48	43, 45, 47	syl2anc 585	. . . . 5 ⊢ (𝜑 → ((𝑉 ∪ {𝑋})𝐶𝑀) = {𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∣ (♯‘𝑥) = 𝑀})
49	48	adantr 482	. . . 4 ⊢ ((𝜑 ∧ 𝐸 = 𝐷) → ((𝑉 ∪ {𝑋})𝐶𝑀) = {𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∣ (♯‘𝑥) = 𝑀})
50		simpl1l 1225	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ 𝑥 ∈ 𝒫 𝑉) → 𝜑)
51	46	hashbcval 16931	. . . . . . . . . 10 ⊢ ((𝑉 ∈ Fin ∧ 𝑀 ∈ ℕ0) → (𝑉𝐶𝑀) = {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 𝑀})
52	22, 45, 51	syl2anc 585	. . . . . . . . 9 ⊢ (𝜑 → (𝑉𝐶𝑀) = {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 𝑀})
53		ramub1.s	. . . . . . . . 9 ⊢ (𝜑 → (𝑉𝐶𝑀) ⊆ (◡𝐾 “ {𝐸}))
54	52, 53	eqsstrrd 4020	. . . . . . . 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 486	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ 𝑥 ∈ 𝒫 𝑉) → 𝑥 ∈ 𝒫 𝑉)
57		simpl3 1194	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ 𝑥 ∈ 𝒫 𝑉) → (♯‘𝑥) = 𝑀)
58		rabid 3453	. . . . . . . 8 ⊢ (𝑥 ∈ {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 𝑀} ↔ (𝑥 ∈ 𝒫 𝑉 ∧ (♯‘𝑥) = 𝑀))
59	56, 57, 58	sylanbrc 584	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ 𝑥 ∈ 𝒫 𝑉) → 𝑥 ∈ {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 𝑀})
60	55, 59	sseldd 3982	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ 𝑥 ∈ 𝒫 𝑉) → 𝑥 ∈ (◡𝐾 “ {𝐸}))
61		simpl2 1193	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}))
62	61	elpwid 4610	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → 𝑥 ⊆ (𝑉 ∪ {𝑋}))
63		simpl1l 1225	. . . . . . . . . . . 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 3991	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → 𝑥 ⊆ 𝑆)
66		vex 3479	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
67	66	elpw 4605	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝒫 𝑆 ↔ 𝑥 ⊆ 𝑆)
68	65, 67	sylibr 233	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → 𝑥 ∈ 𝒫 𝑆)
69		simpl3 1194	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → (♯‘𝑥) = 𝑀)
70		rabid 3453	. . . . . . . . 9 ⊢ (𝑥 ∈ {𝑥 ∈ 𝒫 𝑆 ∣ (♯‘𝑥) = 𝑀} ↔ (𝑥 ∈ 𝒫 𝑆 ∧ (♯‘𝑥) = 𝑀))
71	68, 69, 70	sylanbrc 584	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → 𝑥 ∈ {𝑥 ∈ 𝒫 𝑆 ∣ (♯‘𝑥) = 𝑀})
72	46	hashbcval 16931	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Fin ∧ 𝑀 ∈ ℕ0) → (𝑆𝐶𝑀) = {𝑥 ∈ 𝒫 𝑆 ∣ (♯‘𝑥) = 𝑀})
73	1, 45, 72	syl2anc 585	. . . . . . . . 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 2837	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → 𝑥 ∈ (𝑆𝐶𝑀))
76	3	difss2d 4133	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑊 ⊆ 𝑆)
77	1, 76	ssfid 9263	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑊 ∈ Fin)
78		nnm1nn0 12509	. . . . . . . . . . . . . . 15 ⊢ (𝑀 ∈ ℕ → (𝑀 − 1) ∈ ℕ0)
79	44, 78	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑀 − 1) ∈ ℕ0)
80	46	hashbcval 16931	. . . . . . . . . . . . . 14 ⊢ ((𝑊 ∈ Fin ∧ (𝑀 − 1) ∈ ℕ0) → (𝑊𝐶(𝑀 − 1)) = {𝑢 ∈ 𝒫 𝑊 ∣ (♯‘𝑢) = (𝑀 − 1)})
81	77, 79, 80	syl2anc 585	. . . . . . . . . . . . 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 4020	. . . . . . . . . . . 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 6897	. . . . . . . . . . . 12 ⊢ (𝑢 = (𝑥 ∖ {𝑋}) → ((♯‘𝑢) = (𝑀 − 1) ↔ (♯‘(𝑥 ∖ {𝑋})) = (𝑀 − 1)))
86		uncom 4152	. . . . . . . . . . . . . . . 16 ⊢ (𝑉 ∪ {𝑋}) = ({𝑋} ∪ 𝑉)
87	62, 86	sseqtrdi 4031	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → 𝑥 ⊆ ({𝑋} ∪ 𝑉))
88		ssundif 4486	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ⊆ ({𝑋} ∪ 𝑉) ↔ (𝑥 ∖ {𝑋}) ⊆ 𝑉)
89	87, 88	sylib 217	. . . . . . . . . . . . . 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 3991	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → (𝑥 ∖ {𝑋}) ⊆ 𝑊)
92	66	difexi 5327	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∖ {𝑋}) ∈ V
93	92	elpw 4605	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∖ {𝑋}) ∈ 𝒫 𝑊 ↔ (𝑥 ∖ {𝑋}) ⊆ 𝑊)
94	91, 93	sylibr 233	. . . . . . . . . . . 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 9263	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → 𝑥 ∈ Fin)
97		diffi 9175	. . . . . . . . . . . . . . . 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 14312	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∖ {𝑋}) ∈ Fin → (♯‘(𝑥 ∖ {𝑋})) ∈ ℕ0)
100		nn0cn 12478	. . . . . . . . . . . . . . 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 11164	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℂ
103		pncan 11462	. . . . . . . . . . . . . 14 ⊢ (((♯‘(𝑥 ∖ {𝑋})) ∈ ℂ ∧ 1 ∈ ℂ) → (((♯‘(𝑥 ∖ {𝑋})) + 1) − 1) = (♯‘(𝑥 ∖ {𝑋})))
104	101, 102, 103	sylancl 587	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → (((♯‘(𝑥 ∖ {𝑋})) + 1) − 1) = (♯‘(𝑥 ∖ {𝑋})))
105		neldifsnd 4795	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → ¬ 𝑋 ∈ (𝑥 ∖ {𝑋}))
106		hashunsng 14348	. . . . . . . . . . . . . . . . 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 698	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → (♯‘((𝑥 ∖ {𝑋}) ∪ {𝑋})) = ((♯‘(𝑥 ∖ {𝑋})) + 1))
109		undif1 4474	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∖ {𝑋}) ∪ {𝑋}) = (𝑥 ∪ {𝑋})
110		simpr 486	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → ¬ 𝑥 ∈ 𝒫 𝑉)
111	61, 110	eldifd 3958	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → 𝑥 ∈ (𝒫 (𝑉 ∪ {𝑋}) ∖ 𝒫 𝑉))
112		elpwunsn 4686	. . . . . . . . . . . . . . . . . . . . 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 4811	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → {𝑋} ⊆ 𝑥)
115		ssequn2 4182	. . . . . . . . . . . . . . . . . . 19 ⊢ ({𝑋} ⊆ 𝑥 ↔ (𝑥 ∪ {𝑋}) = 𝑥)
116	114, 115	sylib 217	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → (𝑥 ∪ {𝑋}) = 𝑥)
117	109, 116	eqtr2id 2786	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → 𝑥 = ((𝑥 ∖ {𝑋}) ∪ {𝑋}))
118	117	fveq2d 6892	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → (♯‘𝑥) = (♯‘((𝑥 ∖ {𝑋}) ∪ {𝑋})))
119	118, 69	eqtr3d 2775	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → (♯‘((𝑥 ∖ {𝑋}) ∪ {𝑋})) = 𝑀)
120	108, 119	eqtr3d 2775	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → ((♯‘(𝑥 ∖ {𝑋})) + 1) = 𝑀)
121	120	oveq1d 7419	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → (((♯‘(𝑥 ∖ {𝑋})) + 1) − 1) = (𝑀 − 1))
122	104, 121	eqtr3d 2775	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → (♯‘(𝑥 ∖ {𝑋})) = (𝑀 − 1))
123	85, 94, 122	elrabd 3684	. . . . . . . . . . 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 3982	. . . . . . . . . 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 6235	. . . . . . . . . . 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 2836	. . . . . . . . 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 4155	. . . . . . . . . . . 12 ⊢ (𝑢 = (𝑥 ∖ {𝑋}) → (𝑢 ∪ {𝑋}) = ((𝑥 ∖ {𝑋}) ∪ {𝑋}))
130	129	fveqeq2d 6896	. . . . . . . . . . 11 ⊢ (𝑢 = (𝑥 ∖ {𝑋}) → ((𝐾‘(𝑢 ∪ {𝑋})) = 𝐷 ↔ (𝐾‘((𝑥 ∖ {𝑋}) ∪ {𝑋})) = 𝐷))
131	130	elrab 3682	. . . . . . . . . 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 498	. . . . . . . . 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 6892	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → (𝐾‘𝑥) = (𝐾‘((𝑥 ∖ {𝑋}) ∪ {𝑋})))
135		simpl1r 1226	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → 𝐸 = 𝐷)
136	133, 134, 135	3eqtr4d 2783	. . . . . . 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 6715	. . . . . . . 8 ⊢ (𝜑 → 𝐾 Fn (𝑆𝐶𝑀))
139		fniniseg 7057	. . . . . . . 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 712	. . . . . 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 4071	. . . 4 ⊢ ((𝜑 ∧ 𝐸 = 𝐷) → {𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∣ (♯‘𝑥) = 𝑀} ⊆ (◡𝐾 “ {𝐸}))
144	49, 143	eqsstrd 4019	. . 3 ⊢ ((𝜑 ∧ 𝐸 = 𝐷) → ((𝑉 ∪ {𝑋})𝐶𝑀) ⊆ (◡𝐾 “ {𝐸}))
145		fveq2 6888	. . . . . 6 ⊢ (𝑧 = (𝑉 ∪ {𝑋}) → (♯‘𝑧) = (♯‘(𝑉 ∪ {𝑋})))
146	145	breq2d 5159	. . . . 5 ⊢ (𝑧 = (𝑉 ∪ {𝑋}) → ((𝐹‘𝐸) ≤ (♯‘𝑧) ↔ (𝐹‘𝐸) ≤ (♯‘(𝑉 ∪ {𝑋}))))
147		oveq1 7411	. . . . . 6 ⊢ (𝑧 = (𝑉 ∪ {𝑋}) → (𝑧𝐶𝑀) = ((𝑉 ∪ {𝑋})𝐶𝑀))
148	147	sseq1d 4012	. . . . 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 3612	. . 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 5325	. . . 4 ⊢ (𝜑 → 𝑉 ∈ 𝒫 𝑆)
153	152	adantr 482	. . 3 ⊢ ((𝜑 ∧ 𝐸 ≠ 𝐷) → 𝑉 ∈ 𝒫 𝑆)
154		ifnefalse 4539	. . . . 5 ⊢ (𝐸 ≠ 𝐷 → if(𝐸 = 𝐷, ((𝐹‘𝐷) − 1), (𝐹‘𝐸)) = (𝐹‘𝐸))
155	154	adantl 483	. . . 4 ⊢ ((𝜑 ∧ 𝐸 ≠ 𝐷) → if(𝐸 = 𝐷, ((𝐹‘𝐷) − 1), (𝐹‘𝐸)) = (𝐹‘𝐸))
156	13	adantr 482	. . . 4 ⊢ ((𝜑 ∧ 𝐸 ≠ 𝐷) → if(𝐸 = 𝐷, ((𝐹‘𝐷) − 1), (𝐹‘𝐸)) ≤ (♯‘𝑉))
157	155, 156	eqbrtrrd 5171	. . 3 ⊢ ((𝜑 ∧ 𝐸 ≠ 𝐷) → (𝐹‘𝐸) ≤ (♯‘𝑉))
158	53	adantr 482	. . 3 ⊢ ((𝜑 ∧ 𝐸 ≠ 𝐷) → (𝑉𝐶𝑀) ⊆ (◡𝐾 “ {𝐸}))
159		fveq2 6888	. . . . . 6 ⊢ (𝑧 = 𝑉 → (♯‘𝑧) = (♯‘𝑉))
160	159	breq2d 5159	. . . . 5 ⊢ (𝑧 = 𝑉 → ((𝐹‘𝐸) ≤ (♯‘𝑧) ↔ (𝐹‘𝐸) ≤ (♯‘𝑉)))
161		oveq1 7411	. . . . . 6 ⊢ (𝑧 = 𝑉 → (𝑧𝐶𝑀) = (𝑉𝐶𝑀))
162	161	sseq1d 4012	. . . . 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 3612	. . 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 3030	1 ⊢ (𝜑 → ∃𝑧 ∈ 𝒫 𝑆((𝐹‘𝐸) ≤ (♯‘𝑧) ∧ (𝑧𝐶𝑀) ⊆ (◡𝐾 “ {𝐸})))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   ≠ wne 2941  ∃wrex 3071  {crab 3433  Vcvv 3475   ∖ cdif 3944   ∪ cun 3945   ⊆ wss 3947  ifcif 4527  𝒫 cpw 4601  {csn 4627   class class class wbr 5147   ↦ cmpt 5230  ^◡ccnv 5674   “ cima 5678   Fn wfn 6535  ⟶wf 6536  ‘cfv 6540  (class class class)co 7404   ∈ cmpo 7406  Fincfn 8935  ℂcc 11104  ℝcr 11105  1c1 11107   + caddc 11109   ≤ cle 11245   − cmin 11440  ℕcn 12208  ℕ₀cn0 12468  ♯chash 14286   Ramsey cram 16928

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-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7720  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183

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-reu 3378  df-rab 3434  df-v 3477  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7360  df-ov 7407  df-oprab 7408  df-mpo 7409  df-om 7851  df-1st 7970  df-2nd 7971  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-1o 8461  df-oadd 8465  df-er 8699  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-dju 9892  df-card 9930  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-nn 12209  df-n0 12469  df-z 12555  df-uz 12819  df-fz 13481  df-hash 14287

This theorem is referenced by:  ramub1lem2  16956