Theorem ramub1lem1 16997

Description: Lemma for ramub1 16999. (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 3957	. . . . . . 7 ⊢ (𝜑 → 𝑉 ⊆ (𝑆 ∖ {𝑋}))
5	4	difss2d 4102	. . . . . 6 ⊢ (𝜑 → 𝑉 ⊆ 𝑆)
6		ramub1.x	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝑆)
7	6	snssd 4773	. . . . . 6 ⊢ (𝜑 → {𝑋} ⊆ 𝑆)
8	5, 7	unssd 4155	. . . . 5 ⊢ (𝜑 → (𝑉 ∪ {𝑋}) ⊆ 𝑆)
9	1, 8	sselpwd 5283	. . . 4 ⊢ (𝜑 → (𝑉 ∪ {𝑋}) ∈ 𝒫 𝑆)
10	9	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝐸 = 𝐷) → (𝑉 ∪ {𝑋}) ∈ 𝒫 𝑆)
11		iftrue 4494	. . . . . . 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 5131	. . . . 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 7057	. . . . . . . 8 ⊢ (𝜑 → (𝐹‘𝐷) ∈ ℕ)
19	18	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐸 = 𝐷) → (𝐹‘𝐷) ∈ ℕ)
20	19	nnred 12201	. . . . . 6 ⊢ ((𝜑 ∧ 𝐸 = 𝐷) → (𝐹‘𝐷) ∈ ℝ)
21		1red 11175	. . . . . 6 ⊢ ((𝜑 ∧ 𝐸 = 𝐷) → 1 ∈ ℝ)
22	1, 5	ssfid 9212	. . . . . . . 8 ⊢ (𝜑 → 𝑉 ∈ Fin)
23		hashcl 14321	. . . . . . . 8 ⊢ (𝑉 ∈ Fin → (♯‘𝑉) ∈ ℕ0)
24		nn0re 12451	. . . . . . . 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 11775	. . . . 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 6858	. . . . 5 ⊢ (𝐸 = 𝐷 → (𝐹‘𝐸) = (𝐹‘𝐷))
30		snidg 4624	. . . . . . . 8 ⊢ (𝑋 ∈ 𝑆 → 𝑋 ∈ {𝑋})
31	6, 30	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ {𝑋})
32	4	sseld 3945	. . . . . . . 8 ⊢ (𝜑 → (𝑋 ∈ 𝑉 → 𝑋 ∈ (𝑆 ∖ {𝑋})))
33		eldifn 4095	. . . . . . . 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 14357	. . . . . . 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 5124	. . . 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 9014	. . . . . . 7 ⊢ {𝑋} ∈ Fin
42		unfi 9135	. . . . . . 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 12503	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℕ0)
46		ramub1.3	. . . . . . 7 ⊢ 𝐶 = (𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})
47	46	hashbcval 16973	. . . . . 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 1225	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ 𝑥 ∈ 𝒫 𝑉) → 𝜑)
51	46	hashbcval 16973	. . . . . . . . . 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 3982	. . . . . . . 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 1194	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ 𝑥 ∈ 𝒫 𝑉) → (♯‘𝑥) = 𝑀)
58		rabid 3427	. . . . . . . 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 3947	. . . . . 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 4572	. . . . . . . . . . 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 3957	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → 𝑥 ⊆ 𝑆)
66		vex 3451	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
67	66	elpw 4567	. . . . . . . . . 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 1194	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → (♯‘𝑥) = 𝑀)
70		rabid 3427	. . . . . . . . 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 16973	. . . . . . . . . 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 2831	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → 𝑥 ∈ (𝑆𝐶𝑀))
76	3	difss2d 4102	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑊 ⊆ 𝑆)
77	1, 76	ssfid 9212	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑊 ∈ Fin)
78		nnm1nn0 12483	. . . . . . . . . . . . . . 15 ⊢ (𝑀 ∈ ℕ → (𝑀 − 1) ∈ ℕ0)
79	44, 78	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑀 − 1) ∈ ℕ0)
80	46	hashbcval 16973	. . . . . . . . . . . . . 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 3982	. . . . . . . . . . . 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 6867	. . . . . . . . . . . 12 ⊢ (𝑢 = (𝑥 ∖ {𝑋}) → ((♯‘𝑢) = (𝑀 − 1) ↔ (♯‘(𝑥 ∖ {𝑋})) = (𝑀 − 1)))
86		uncom 4121	. . . . . . . . . . . . . . . 16 ⊢ (𝑉 ∪ {𝑋}) = ({𝑋} ∪ 𝑉)
87	62, 86	sseqtrdi 3987	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → 𝑥 ⊆ ({𝑋} ∪ 𝑉))
88		ssundif 4451	. . . . . . . . . . . . . . 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 3957	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → (𝑥 ∖ {𝑋}) ⊆ 𝑊)
92	66	difexi 5285	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∖ {𝑋}) ∈ V
93	92	elpw 4567	. . . . . . . . . . . . 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 9212	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → 𝑥 ∈ Fin)
97		diffi 9139	. . . . . . . . . . . . . . . 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 14321	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∖ {𝑋}) ∈ Fin → (♯‘(𝑥 ∖ {𝑋})) ∈ ℕ0)
100		nn0cn 12452	. . . . . . . . . . . . . . 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 11126	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℂ
103		pncan 11427	. . . . . . . . . . . . . 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 4757	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → ¬ 𝑋 ∈ (𝑥 ∖ {𝑋}))
106		hashunsng 14357	. . . . . . . . . . . . . . . . 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 4439	. . . . . . . . . . . . . . . . . 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 3925	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → 𝑥 ∈ (𝒫 (𝑉 ∪ {𝑋}) ∖ 𝒫 𝑉))
112		elpwunsn 4648	. . . . . . . . . . . . . . . . . . . . 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 4773	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → {𝑋} ⊆ 𝑥)
115		ssequn2 4152	. . . . . . . . . . . . . . . . . . 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 2777	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → 𝑥 = ((𝑥 ∖ {𝑋}) ∪ {𝑋}))
118	117	fveq2d 6862	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → (♯‘𝑥) = (♯‘((𝑥 ∖ {𝑋}) ∪ {𝑋})))
119	118, 69	eqtr3d 2766	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → (♯‘((𝑥 ∖ {𝑋}) ∪ {𝑋})) = 𝑀)
120	108, 119	eqtr3d 2766	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → ((♯‘(𝑥 ∖ {𝑋})) + 1) = 𝑀)
121	120	oveq1d 7402	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → (((♯‘(𝑥 ∖ {𝑋})) + 1) − 1) = (𝑀 − 1))
122	104, 121	eqtr3d 2766	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → (♯‘(𝑥 ∖ {𝑋})) = (𝑀 − 1))
123	85, 94, 122	elrabd 3661	. . . . . . . . . . 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 3947	. . . . . . . . . 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 6212	. . . . . . . . . . 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 2830	. . . . . . . . 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 4124	. . . . . . . . . . . 12 ⊢ (𝑢 = (𝑥 ∖ {𝑋}) → (𝑢 ∪ {𝑋}) = ((𝑥 ∖ {𝑋}) ∪ {𝑋}))
130	129	fveqeq2d 6866	. . . . . . . . . . 11 ⊢ (𝑢 = (𝑥 ∖ {𝑋}) → ((𝐾‘(𝑢 ∪ {𝑋})) = 𝐷 ↔ (𝐾‘((𝑥 ∖ {𝑋}) ∪ {𝑋})) = 𝐷))
131	130	elrab 3659	. . . . . . . . . 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 6862	. . . . . . . 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 2774	. . . . . . 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 6689	. . . . . . . 8 ⊢ (𝜑 → 𝐾 Fn (𝑆𝐶𝑀))
139		fniniseg 7032	. . . . . . . 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 4038	. . . 4 ⊢ ((𝜑 ∧ 𝐸 = 𝐷) → {𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∣ (♯‘𝑥) = 𝑀} ⊆ (◡𝐾 “ {𝐸}))
144	49, 143	eqsstrd 3981	. . 3 ⊢ ((𝜑 ∧ 𝐸 = 𝐷) → ((𝑉 ∪ {𝑋})𝐶𝑀) ⊆ (◡𝐾 “ {𝐸}))
145		fveq2 6858	. . . . . 6 ⊢ (𝑧 = (𝑉 ∪ {𝑋}) → (♯‘𝑧) = (♯‘(𝑉 ∪ {𝑋})))
146	145	breq2d 5119	. . . . 5 ⊢ (𝑧 = (𝑉 ∪ {𝑋}) → ((𝐹‘𝐸) ≤ (♯‘𝑧) ↔ (𝐹‘𝐸) ≤ (♯‘(𝑉 ∪ {𝑋}))))
147		oveq1 7394	. . . . . 6 ⊢ (𝑧 = (𝑉 ∪ {𝑋}) → (𝑧𝐶𝑀) = ((𝑉 ∪ {𝑋})𝐶𝑀))
148	147	sseq1d 3978	. . . . 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 3588	. . 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 5283	. . . 4 ⊢ (𝜑 → 𝑉 ∈ 𝒫 𝑆)
153	152	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝐸 ≠ 𝐷) → 𝑉 ∈ 𝒫 𝑆)
154		ifnefalse 4500	. . . . 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 5131	. . 3 ⊢ ((𝜑 ∧ 𝐸 ≠ 𝐷) → (𝐹‘𝐸) ≤ (♯‘𝑉))
158	53	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝐸 ≠ 𝐷) → (𝑉𝐶𝑀) ⊆ (◡𝐾 “ {𝐸}))
159		fveq2 6858	. . . . . 6 ⊢ (𝑧 = 𝑉 → (♯‘𝑧) = (♯‘𝑉))
160	159	breq2d 5119	. . . . 5 ⊢ (𝑧 = 𝑉 → ((𝐹‘𝐸) ≤ (♯‘𝑧) ↔ (𝐹‘𝐸) ≤ (♯‘𝑉)))
161		oveq1 7394	. . . . . 6 ⊢ (𝑧 = 𝑉 → (𝑧𝐶𝑀) = (𝑉𝐶𝑀))
162	161	sseq1d 3978	. . . . 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 3588	. . 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 3012	1 ⊢ (𝜑 → ∃𝑧 ∈ 𝒫 𝑆((𝐹‘𝐸) ≤ (♯‘𝑧) ∧ (𝑧𝐶𝑀) ⊆ (◡𝐾 “ {𝐸})))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  ∃wrex 3053  {crab 3405  Vcvv 3447   ∖ cdif 3911   ∪ cun 3912   ⊆ wss 3914  ifcif 4488  𝒫 cpw 4563  {csn 4589   class class class wbr 5107   ↦ cmpt 5188  ^◡ccnv 5637   “ cima 5641   Fn wfn 6506  ⟶wf 6507  ‘cfv 6511  (class class class)co 7387   ∈ cmpo 7389  Fincfn 8918  ℂcc 11066  ℝcr 11067  1c1 11069   + caddc 11071   ≤ cle 11209   − cmin 11405  ℕcn 12186  ℕ₀cn0 12442  ♯chash 14295   Ramsey cram 16970

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 2701  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-cnex 11124  ax-resscn 11125  ax-1cn 11126  ax-icn 11127  ax-addcl 11128  ax-addrcl 11129  ax-mulcl 11130  ax-mulrcl 11131  ax-mulcom 11132  ax-addass 11133  ax-mulass 11134  ax-distr 11135  ax-i2m1 11136  ax-1ne0 11137  ax-1rid 11138  ax-rnegex 11139  ax-rrecex 11140  ax-cnre 11141  ax-pre-lttri 11142  ax-pre-lttrn 11143  ax-pre-ltadd 11144  ax-pre-mulgt0 11145

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-int 4911  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-1st 7968  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-1o 8434  df-oadd 8438  df-er 8671  df-en 8919  df-dom 8920  df-sdom 8921  df-fin 8922  df-dju 9854  df-card 9892  df-pnf 11210  df-mnf 11211  df-xr 11212  df-ltxr 11213  df-le 11214  df-sub 11407  df-neg 11408  df-nn 12187  df-n0 12443  df-z 12530  df-uz 12794  df-fz 13469  df-hash 14296

This theorem is referenced by:  ramub1lem2  16998