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Theorem hashfibclem 11199

Description: Lemma for hashfibc 11200: inductive step. (Contributed by Mario Carneiro, 13-Jul-2014.)

**Hypotheses**
Ref	Expression
hashbc.1	⊢ (𝜑 → 𝐴 ∈ Fin)
hashbc.2	⊢ (𝜑 → ¬ 𝑧 ∈ 𝐴)
hashfibc.3	⊢ (𝜑 → ∀𝑗 ∈ ℤ ((♯‘𝐴)C𝑗) = (♯‘{𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∣ (♯‘𝑥) = 𝑗}))
hashbc.4	⊢ (𝜑 → 𝐾 ∈ ℤ)

**Assertion**
Ref	Expression
hashfibclem	⊢ (𝜑 → ((♯‘(𝐴 ∪ {𝑧}))C𝐾) = (♯‘{𝑥 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ∣ (♯‘𝑥) = 𝐾}))

Distinct variable groups: 𝐴,𝑗,𝑥,𝑧 𝑗,𝐾,𝑥 𝜑,𝑥 Allowed substitution hints: 𝜑(𝑧,𝑗) 𝐾(𝑧)

Proof of Theorem hashfibclem Dummy variables 𝑢 𝑣 𝑟 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq2 6057	. . . . . 6 ⊢ (𝑗 = 𝐾 → ((♯‘𝐴)C𝑗) = ((♯‘𝐴)C𝐾))
2		eqeq2 2242	. . . . . . . 8 ⊢ (𝑗 = 𝐾 → ((♯‘𝑥) = 𝑗 ↔ (♯‘𝑥) = 𝐾))
3	2	rabbidv 2801	. . . . . . 7 ⊢ (𝑗 = 𝐾 → {𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∣ (♯‘𝑥) = 𝑗} = {𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∣ (♯‘𝑥) = 𝐾})
4	3	fveq2d 5673	. . . . . 6 ⊢ (𝑗 = 𝐾 → (♯‘{𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∣ (♯‘𝑥) = 𝑗}) = (♯‘{𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∣ (♯‘𝑥) = 𝐾}))
5	1, 4	eqeq12d 2247	. . . . 5 ⊢ (𝑗 = 𝐾 → (((♯‘𝐴)C𝑗) = (♯‘{𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∣ (♯‘𝑥) = 𝑗}) ↔ ((♯‘𝐴)C𝐾) = (♯‘{𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∣ (♯‘𝑥) = 𝐾})))
6		hashfibc.3	. . . . 5 ⊢ (𝜑 → ∀𝑗 ∈ ℤ ((♯‘𝐴)C𝑗) = (♯‘{𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∣ (♯‘𝑥) = 𝑗}))
7		hashbc.4	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ ℤ)
8	5, 6, 7	rspcdva 2925	. . . 4 ⊢ (𝜑 → ((♯‘𝐴)C𝐾) = (♯‘{𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∣ (♯‘𝑥) = 𝐾}))
9		ssun1 3381	. . . . . . . . . . . . . . 15 ⊢ 𝐴 ⊆ (𝐴 ∪ {𝑧})
10	9	sspwi 3682	. . . . . . . . . . . . . 14 ⊢ 𝒫 𝐴 ⊆ 𝒫 (𝐴 ∪ {𝑧})
11	10	sseli 3233	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝒫 𝐴 → 𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}))
12	11	anim1i 340	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝒫 𝐴 ∧ 𝑥 ∈ Fin) → (𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ 𝑥 ∈ Fin))
13		elin 3401	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↔ (𝑥 ∈ 𝒫 𝐴 ∧ 𝑥 ∈ Fin))
14		elin 3401	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ↔ (𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ 𝑥 ∈ Fin))
15	12, 13, 14	3imtr4i 201	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝒫 𝐴 ∩ Fin) → 𝑥 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin))
16	15	adantl 277	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝐴 ∩ Fin)) → 𝑥 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin))
17		elinel1 3404	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝒫 𝐴 ∩ Fin) → 𝑥 ∈ 𝒫 𝐴)
18		hashbc.2	. . . . . . . . . . . 12 ⊢ (𝜑 → ¬ 𝑧 ∈ 𝐴)
19		elpwi 3677	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝒫 𝐴 → 𝑥 ⊆ 𝐴)
20	19	ssneld 3239	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝒫 𝐴 → (¬ 𝑧 ∈ 𝐴 → ¬ 𝑧 ∈ 𝑥))
21	18, 20	mpan9 281	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝒫 𝐴) → ¬ 𝑧 ∈ 𝑥)
22	17, 21	sylan2 286	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝐴 ∩ Fin)) → ¬ 𝑧 ∈ 𝑥)
23	16, 22	jca 306	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝐴 ∩ Fin)) → (𝑥 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ∧ ¬ 𝑧 ∈ 𝑥))
24		elinel1 3404	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) → 𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}))
25		elpwi 3677	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) → 𝑥 ⊆ (𝐴 ∪ {𝑧}))
26		uncom 3362	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∪ {𝑧}) = ({𝑧} ∪ 𝐴)
27	25, 26	sseqtrdi 3285	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) → 𝑥 ⊆ ({𝑧} ∪ 𝐴))
28	27	adantr 276	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ ¬ 𝑧 ∈ 𝑥) → 𝑥 ⊆ ({𝑧} ∪ 𝐴))
29		simpr 110	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ ¬ 𝑧 ∈ 𝑥) → ¬ 𝑧 ∈ 𝑥)
30		disjsn 3750	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧 ∈ 𝑥)
31	29, 30	sylibr 134	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ ¬ 𝑧 ∈ 𝑥) → (𝑥 ∩ {𝑧}) = ∅)
32		disjssun 3571	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∩ {𝑧}) = ∅ → (𝑥 ⊆ ({𝑧} ∪ 𝐴) ↔ 𝑥 ⊆ 𝐴))
33	31, 32	syl 14	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ ¬ 𝑧 ∈ 𝑥) → (𝑥 ⊆ ({𝑧} ∪ 𝐴) ↔ 𝑥 ⊆ 𝐴))
34	28, 33	mpbid 147	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ ¬ 𝑧 ∈ 𝑥) → 𝑥 ⊆ 𝐴)
35		vex 2815	. . . . . . . . . . . . . 14 ⊢ 𝑥 ∈ V
36	35	elpw 3674	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴)
37	34, 36	sylibr 134	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ ¬ 𝑧 ∈ 𝑥) → 𝑥 ∈ 𝒫 𝐴)
38	24, 37	sylan 283	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ∧ ¬ 𝑧 ∈ 𝑥) → 𝑥 ∈ 𝒫 𝐴)
39		elinel2 3405	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) → 𝑥 ∈ Fin)
40	39	adantr 276	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ∧ ¬ 𝑧 ∈ 𝑥) → 𝑥 ∈ Fin)
41	38, 40	elind 3403	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ∧ ¬ 𝑧 ∈ 𝑥) → 𝑥 ∈ (𝒫 𝐴 ∩ Fin))
42	41	adantl 277	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ∧ ¬ 𝑧 ∈ 𝑥)) → 𝑥 ∈ (𝒫 𝐴 ∩ Fin))
43	23, 42	impbida 600	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↔ (𝑥 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ∧ ¬ 𝑧 ∈ 𝑥)))
44	43	anbi1d 465	. . . . . . 7 ⊢ (𝜑 → ((𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∧ (♯‘𝑥) = 𝐾) ↔ ((𝑥 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ∧ ¬ 𝑧 ∈ 𝑥) ∧ (♯‘𝑥) = 𝐾)))
45		anass 401	. . . . . . 7 ⊢ (((𝑥 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ∧ ¬ 𝑧 ∈ 𝑥) ∧ (♯‘𝑥) = 𝐾) ↔ (𝑥 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ∧ (¬ 𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)))
46	44, 45	bitrdi 196	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∧ (♯‘𝑥) = 𝐾) ↔ (𝑥 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ∧ (¬ 𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾))))
47	46	rabbidva2 2796	. . . . 5 ⊢ (𝜑 → {𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∣ (♯‘𝑥) = 𝐾} = {𝑥 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ∣ (¬ 𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)})
48	47	fveq2d 5673	. . . 4 ⊢ (𝜑 → (♯‘{𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∣ (♯‘𝑥) = 𝐾}) = (♯‘{𝑥 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ∣ (¬ 𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)}))
49	8, 48	eqtrd 2265	. . 3 ⊢ (𝜑 → ((♯‘𝐴)C𝐾) = (♯‘{𝑥 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ∣ (¬ 𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)}))
50		oveq2 6057	. . . . . 6 ⊢ (𝑗 = (𝐾 − 1) → ((♯‘𝐴)C𝑗) = ((♯‘𝐴)C(𝐾 − 1)))
51		eqeq2 2242	. . . . . . . 8 ⊢ (𝑗 = (𝐾 − 1) → ((♯‘𝑥) = 𝑗 ↔ (♯‘𝑥) = (𝐾 − 1)))
52	51	rabbidv 2801	. . . . . . 7 ⊢ (𝑗 = (𝐾 − 1) → {𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∣ (♯‘𝑥) = 𝑗} = {𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∣ (♯‘𝑥) = (𝐾 − 1)})
53	52	fveq2d 5673	. . . . . 6 ⊢ (𝑗 = (𝐾 − 1) → (♯‘{𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∣ (♯‘𝑥) = 𝑗}) = (♯‘{𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∣ (♯‘𝑥) = (𝐾 − 1)}))
54	50, 53	eqeq12d 2247	. . . . 5 ⊢ (𝑗 = (𝐾 − 1) → (((♯‘𝐴)C𝑗) = (♯‘{𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∣ (♯‘𝑥) = 𝑗}) ↔ ((♯‘𝐴)C(𝐾 − 1)) = (♯‘{𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∣ (♯‘𝑥) = (𝐾 − 1)})))
55		peano2zm 9611	. . . . . 6 ⊢ (𝐾 ∈ ℤ → (𝐾 − 1) ∈ ℤ)
56	7, 55	syl 14	. . . . 5 ⊢ (𝜑 → (𝐾 − 1) ∈ ℤ)
57	54, 6, 56	rspcdva 2925	. . . 4 ⊢ (𝜑 → ((♯‘𝐴)C(𝐾 − 1)) = (♯‘{𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∣ (♯‘𝑥) = (𝐾 − 1)}))
58		eqid 2232	. . . . . . 7 ⊢ {𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∣ (♯‘𝑥) = (𝐾 − 1)} = {𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∣ (♯‘𝑥) = (𝐾 − 1)}
59		hashbc.1	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ Fin)
60	59	pwexd 4293	. . . . . . . 8 ⊢ (𝜑 → 𝒫 𝐴 ∈ V)
61		inss1 3440	. . . . . . . . 9 ⊢ (𝒫 𝐴 ∩ Fin) ⊆ 𝒫 𝐴
62	61	a1i 9	. . . . . . . 8 ⊢ (𝜑 → (𝒫 𝐴 ∩ Fin) ⊆ 𝒫 𝐴)
63	60, 62	ssexd 4249	. . . . . . 7 ⊢ (𝜑 → (𝒫 𝐴 ∩ Fin) ∈ V)
64	58, 63	rabexd 4256	. . . . . 6 ⊢ (𝜑 → {𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∣ (♯‘𝑥) = (𝐾 − 1)} ∈ V)
65		eqid 2232	. . . . . . 7 ⊢ {𝑥 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ∣ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)} = {𝑥 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ∣ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)}
66		vsnex 4323	. . . . . . . . . 10 ⊢ {𝑧} ∈ V
67		unexg 4563	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Fin ∧ {𝑧} ∈ V) → (𝐴 ∪ {𝑧}) ∈ V)
68	59, 66, 67	sylancl 413	. . . . . . . . 9 ⊢ (𝜑 → (𝐴 ∪ {𝑧}) ∈ V)
69	68	pwexd 4293	. . . . . . . 8 ⊢ (𝜑 → 𝒫 (𝐴 ∪ {𝑧}) ∈ V)
70		inex1g 4245	. . . . . . . 8 ⊢ (𝒫 (𝐴 ∪ {𝑧}) ∈ V → (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ∈ V)
71	69, 70	syl 14	. . . . . . 7 ⊢ (𝜑 → (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ∈ V)
72	65, 71	rabexd 4256	. . . . . 6 ⊢ (𝜑 → {𝑥 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ∣ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)} ∈ V)
73		fveqeq2 5678	. . . . . . . 8 ⊢ (𝑥 = 𝑢 → ((♯‘𝑥) = (𝐾 − 1) ↔ (♯‘𝑢) = (𝐾 − 1)))
74	73	elrab 2972	. . . . . . 7 ⊢ (𝑢 ∈ {𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∣ (♯‘𝑥) = (𝐾 − 1)} ↔ (𝑢 ∈ (𝒫 𝐴 ∩ Fin) ∧ (♯‘𝑢) = (𝐾 − 1)))
75		eleq2 2296	. . . . . . . . . 10 ⊢ (𝑥 = (𝑢 ∪ {𝑧}) → (𝑧 ∈ 𝑥 ↔ 𝑧 ∈ (𝑢 ∪ {𝑧})))
76		fveqeq2 5678	. . . . . . . . . 10 ⊢ (𝑥 = (𝑢 ∪ {𝑧}) → ((♯‘𝑥) = 𝐾 ↔ (♯‘(𝑢 ∪ {𝑧})) = 𝐾))
77	75, 76	anbi12d 473	. . . . . . . . 9 ⊢ (𝑥 = (𝑢 ∪ {𝑧}) → ((𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾) ↔ (𝑧 ∈ (𝑢 ∪ {𝑧}) ∧ (♯‘(𝑢 ∪ {𝑧})) = 𝐾)))
78		elinel1 3404	. . . . . . . . . . 11 ⊢ (𝑢 ∈ (𝒫 𝐴 ∩ Fin) → 𝑢 ∈ 𝒫 𝐴)
79		elpwi 3677	. . . . . . . . . . . . . 14 ⊢ (𝑢 ∈ 𝒫 𝐴 → 𝑢 ⊆ 𝐴)
80	79	ad2antrl 490	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1))) → 𝑢 ⊆ 𝐴)
81		unss1 3387	. . . . . . . . . . . . 13 ⊢ (𝑢 ⊆ 𝐴 → (𝑢 ∪ {𝑧}) ⊆ (𝐴 ∪ {𝑧}))
82	80, 81	syl 14	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1))) → (𝑢 ∪ {𝑧}) ⊆ (𝐴 ∪ {𝑧}))
83		vex 2815	. . . . . . . . . . . . . 14 ⊢ 𝑢 ∈ V
84	83, 66	unex 4561	. . . . . . . . . . . . 13 ⊢ (𝑢 ∪ {𝑧}) ∈ V
85	84	elpw 3674	. . . . . . . . . . . 12 ⊢ ((𝑢 ∪ {𝑧}) ∈ 𝒫 (𝐴 ∪ {𝑧}) ↔ (𝑢 ∪ {𝑧}) ⊆ (𝐴 ∪ {𝑧}))
86	82, 85	sylibr 134	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1))) → (𝑢 ∪ {𝑧}) ∈ 𝒫 (𝐴 ∪ {𝑧}))
87	78, 86	sylanr1 404	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑢 ∈ (𝒫 𝐴 ∩ Fin) ∧ (♯‘𝑢) = (𝐾 − 1))) → (𝑢 ∪ {𝑧}) ∈ 𝒫 (𝐴 ∪ {𝑧}))
88		simprl 531	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑢 ∈ (𝒫 𝐴 ∩ Fin) ∧ (♯‘𝑢) = (𝐾 − 1))) → 𝑢 ∈ (𝒫 𝐴 ∩ Fin))
89	88	elin2d 3408	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑢 ∈ (𝒫 𝐴 ∩ Fin) ∧ (♯‘𝑢) = (𝐾 − 1))) → 𝑢 ∈ Fin)
90		vex 2815	. . . . . . . . . . . 12 ⊢ 𝑧 ∈ V
91	90	a1i 9	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑢 ∈ (𝒫 𝐴 ∩ Fin) ∧ (♯‘𝑢) = (𝐾 − 1))) → 𝑧 ∈ V)
92	18	adantr 276	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1))) → ¬ 𝑧 ∈ 𝐴)
93	80, 92	ssneldd 3240	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1))) → ¬ 𝑧 ∈ 𝑢)
94	78, 93	sylanr1 404	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑢 ∈ (𝒫 𝐴 ∩ Fin) ∧ (♯‘𝑢) = (𝐾 − 1))) → ¬ 𝑧 ∈ 𝑢)
95		unsnfi 7178	. . . . . . . . . . 11 ⊢ ((𝑢 ∈ Fin ∧ 𝑧 ∈ V ∧ ¬ 𝑧 ∈ 𝑢) → (𝑢 ∪ {𝑧}) ∈ Fin)
96	89, 91, 94, 95	syl3anc 1274	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑢 ∈ (𝒫 𝐴 ∩ Fin) ∧ (♯‘𝑢) = (𝐾 − 1))) → (𝑢 ∪ {𝑧}) ∈ Fin)
97	87, 96	elind 3403	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑢 ∈ (𝒫 𝐴 ∩ Fin) ∧ (♯‘𝑢) = (𝐾 − 1))) → (𝑢 ∪ {𝑧}) ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin))
98		snfig 7055	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ V → {𝑧} ∈ Fin)
99	98	elv 2816	. . . . . . . . . . . . 13 ⊢ {𝑧} ∈ Fin
100	99	a1i 9	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑢 ∈ (𝒫 𝐴 ∩ Fin) ∧ (♯‘𝑢) = (𝐾 − 1))) → {𝑧} ∈ Fin)
101		disjsn 3750	. . . . . . . . . . . . 13 ⊢ ((𝑢 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧 ∈ 𝑢)
102	94, 101	sylibr 134	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑢 ∈ (𝒫 𝐴 ∩ Fin) ∧ (♯‘𝑢) = (𝐾 − 1))) → (𝑢 ∩ {𝑧}) = ∅)
103		hashun 11164	. . . . . . . . . . . 12 ⊢ ((𝑢 ∈ Fin ∧ {𝑧} ∈ Fin ∧ (𝑢 ∩ {𝑧}) = ∅) → (♯‘(𝑢 ∪ {𝑧})) = ((♯‘𝑢) + (♯‘{𝑧})))
104	89, 100, 102, 103	syl3anc 1274	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑢 ∈ (𝒫 𝐴 ∩ Fin) ∧ (♯‘𝑢) = (𝐾 − 1))) → (♯‘(𝑢 ∪ {𝑧})) = ((♯‘𝑢) + (♯‘{𝑧})))
105		simprr 533	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑢 ∈ (𝒫 𝐴 ∩ Fin) ∧ (♯‘𝑢) = (𝐾 − 1))) → (♯‘𝑢) = (𝐾 − 1))
106		hashsng 11156	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ V → (♯‘{𝑧}) = 1)
107	106	elv 2816	. . . . . . . . . . . . 13 ⊢ (♯‘{𝑧}) = 1
108	107	a1i 9	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑢 ∈ (𝒫 𝐴 ∩ Fin) ∧ (♯‘𝑢) = (𝐾 − 1))) → (♯‘{𝑧}) = 1)
109	105, 108	oveq12d 6067	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑢 ∈ (𝒫 𝐴 ∩ Fin) ∧ (♯‘𝑢) = (𝐾 − 1))) → ((♯‘𝑢) + (♯‘{𝑧})) = ((𝐾 − 1) + 1))
110	7	adantr 276	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1))) → 𝐾 ∈ ℤ)
111	110	zcnd 9697	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1))) → 𝐾 ∈ ℂ)
112	78, 111	sylanr1 404	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑢 ∈ (𝒫 𝐴 ∩ Fin) ∧ (♯‘𝑢) = (𝐾 − 1))) → 𝐾 ∈ ℂ)
113		ax-1cn 8216	. . . . . . . . . . . 12 ⊢ 1 ∈ ℂ
114		npcan 8478	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝐾 − 1) + 1) = 𝐾)
115	112, 113, 114	sylancl 413	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑢 ∈ (𝒫 𝐴 ∩ Fin) ∧ (♯‘𝑢) = (𝐾 − 1))) → ((𝐾 − 1) + 1) = 𝐾)
116	104, 109, 115	3eqtrd 2269	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑢 ∈ (𝒫 𝐴 ∩ Fin) ∧ (♯‘𝑢) = (𝐾 − 1))) → (♯‘(𝑢 ∪ {𝑧})) = 𝐾)
117		ssun2 3382	. . . . . . . . . . 11 ⊢ {𝑧} ⊆ (𝑢 ∪ {𝑧})
118	90	snss 3828	. . . . . . . . . . 11 ⊢ (𝑧 ∈ (𝑢 ∪ {𝑧}) ↔ {𝑧} ⊆ (𝑢 ∪ {𝑧}))
119	117, 118	mpbir 146	. . . . . . . . . 10 ⊢ 𝑧 ∈ (𝑢 ∪ {𝑧})
120	116, 119	jctil 312	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑢 ∈ (𝒫 𝐴 ∩ Fin) ∧ (♯‘𝑢) = (𝐾 − 1))) → (𝑧 ∈ (𝑢 ∪ {𝑧}) ∧ (♯‘(𝑢 ∪ {𝑧})) = 𝐾))
121	77, 97, 120	elrabd 2974	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑢 ∈ (𝒫 𝐴 ∩ Fin) ∧ (♯‘𝑢) = (𝐾 − 1))) → (𝑢 ∪ {𝑧}) ∈ {𝑥 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ∣ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)})
122	121	ex 115	. . . . . . 7 ⊢ (𝜑 → ((𝑢 ∈ (𝒫 𝐴 ∩ Fin) ∧ (♯‘𝑢) = (𝐾 − 1)) → (𝑢 ∪ {𝑧}) ∈ {𝑥 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ∣ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)}))
123	74, 122	biimtrid 152	. . . . . 6 ⊢ (𝜑 → (𝑢 ∈ {𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∣ (♯‘𝑥) = (𝐾 − 1)} → (𝑢 ∪ {𝑧}) ∈ {𝑥 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ∣ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)}))
124		eleq2 2296	. . . . . . . . 9 ⊢ (𝑥 = 𝑣 → (𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑣))
125		fveqeq2 5678	. . . . . . . . 9 ⊢ (𝑥 = 𝑣 → ((♯‘𝑥) = 𝐾 ↔ (♯‘𝑣) = 𝐾))
126	124, 125	anbi12d 473	. . . . . . . 8 ⊢ (𝑥 = 𝑣 → ((𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾) ↔ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾)))
127	126	elrab 2972	. . . . . . 7 ⊢ (𝑣 ∈ {𝑥 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ∣ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)} ↔ (𝑣 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾)))
128		fveqeq2 5678	. . . . . . . . 9 ⊢ (𝑥 = (𝑣 ∖ {𝑧}) → ((♯‘𝑥) = (𝐾 − 1) ↔ (♯‘(𝑣 ∖ {𝑧})) = (𝐾 − 1)))
129		elinel1 3404	. . . . . . . . . . . 12 ⊢ (𝑣 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) → 𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}))
130	129	ad2antrl 490	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑣 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾))) → 𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}))
131		elpwi 3677	. . . . . . . . . . . . . 14 ⊢ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) → 𝑣 ⊆ (𝐴 ∪ {𝑧}))
132	131, 26	sseqtrdi 3285	. . . . . . . . . . . . 13 ⊢ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) → 𝑣 ⊆ ({𝑧} ∪ 𝐴))
133		ssundifim 3592	. . . . . . . . . . . . 13 ⊢ (𝑣 ⊆ ({𝑧} ∪ 𝐴) → (𝑣 ∖ {𝑧}) ⊆ 𝐴)
134	132, 133	syl 14	. . . . . . . . . . . 12 ⊢ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) → (𝑣 ∖ {𝑧}) ⊆ 𝐴)
135		vex 2815	. . . . . . . . . . . . . 14 ⊢ 𝑣 ∈ V
136	135	difexi 4252	. . . . . . . . . . . . 13 ⊢ (𝑣 ∖ {𝑧}) ∈ V
137	136	elpw 3674	. . . . . . . . . . . 12 ⊢ ((𝑣 ∖ {𝑧}) ∈ 𝒫 𝐴 ↔ (𝑣 ∖ {𝑧}) ⊆ 𝐴)
138	134, 137	sylibr 134	. . . . . . . . . . 11 ⊢ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) → (𝑣 ∖ {𝑧}) ∈ 𝒫 𝐴)
139	130, 138	syl 14	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑣 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾))) → (𝑣 ∖ {𝑧}) ∈ 𝒫 𝐴)
140		elinel2 3405	. . . . . . . . . . . 12 ⊢ (𝑣 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) → 𝑣 ∈ Fin)
141	140	ad2antrl 490	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑣 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾))) → 𝑣 ∈ Fin)
142		simprrl 541	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑣 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾))) → 𝑧 ∈ 𝑣)
143		diffisn 7149	. . . . . . . . . . 11 ⊢ ((𝑣 ∈ Fin ∧ 𝑧 ∈ 𝑣) → (𝑣 ∖ {𝑧}) ∈ Fin)
144	141, 142, 143	syl2anc 411	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑣 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾))) → (𝑣 ∖ {𝑧}) ∈ Fin)
145	139, 144	elind 3403	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑣 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾))) → (𝑣 ∖ {𝑧}) ∈ (𝒫 𝐴 ∩ Fin))
146		hashcl 11139	. . . . . . . . . . . . 13 ⊢ ((𝑣 ∖ {𝑧}) ∈ Fin → (♯‘(𝑣 ∖ {𝑧})) ∈ ℕ₀)
147	144, 146	syl 14	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑣 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾))) → (♯‘(𝑣 ∖ {𝑧})) ∈ ℕ₀)
148	147	nn0cnd 9551	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑣 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾))) → (♯‘(𝑣 ∖ {𝑧})) ∈ ℂ)
149		pncan 8475	. . . . . . . . . . 11 ⊢ (((♯‘(𝑣 ∖ {𝑧})) ∈ ℂ ∧ 1 ∈ ℂ) → (((♯‘(𝑣 ∖ {𝑧})) + 1) − 1) = (♯‘(𝑣 ∖ {𝑧})))
150	148, 113, 149	sylancl 413	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑣 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾))) → (((♯‘(𝑣 ∖ {𝑧})) + 1) − 1) = (♯‘(𝑣 ∖ {𝑧})))
151		uncom 3362	. . . . . . . . . . . . . 14 ⊢ ((𝑣 ∖ {𝑧}) ∪ {𝑧}) = ({𝑧} ∪ (𝑣 ∖ {𝑧}))
152	99	a1i 9	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑣 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾))) → {𝑧} ∈ Fin)
153	142	snssd 3838	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑣 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾))) → {𝑧} ⊆ 𝑣)
154		undiffi 7184	. . . . . . . . . . . . . . 15 ⊢ ((𝑣 ∈ Fin ∧ {𝑧} ∈ Fin ∧ {𝑧} ⊆ 𝑣) → 𝑣 = ({𝑧} ∪ (𝑣 ∖ {𝑧})))
155	141, 152, 153, 154	syl3anc 1274	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑣 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾))) → 𝑣 = ({𝑧} ∪ (𝑣 ∖ {𝑧})))
156	151, 155	eqtr4id 2284	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑣 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾))) → ((𝑣 ∖ {𝑧}) ∪ {𝑧}) = 𝑣)
157	156	fveq2d 5673	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑣 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾))) → (♯‘((𝑣 ∖ {𝑧}) ∪ {𝑧})) = (♯‘𝑣))
158		disjdifr 3581	. . . . . . . . . . . . . . 15 ⊢ ((𝑣 ∖ {𝑧}) ∩ {𝑧}) = ∅
159	158	a1i 9	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑣 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾))) → ((𝑣 ∖ {𝑧}) ∩ {𝑧}) = ∅)
160		hashun 11164	. . . . . . . . . . . . . 14 ⊢ (((𝑣 ∖ {𝑧}) ∈ Fin ∧ {𝑧} ∈ Fin ∧ ((𝑣 ∖ {𝑧}) ∩ {𝑧}) = ∅) → (♯‘((𝑣 ∖ {𝑧}) ∪ {𝑧})) = ((♯‘(𝑣 ∖ {𝑧})) + (♯‘{𝑧})))
161	144, 152, 159, 160	syl3anc 1274	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑣 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾))) → (♯‘((𝑣 ∖ {𝑧}) ∪ {𝑧})) = ((♯‘(𝑣 ∖ {𝑧})) + (♯‘{𝑧})))
162	107	oveq2i 6060	. . . . . . . . . . . . 13 ⊢ ((♯‘(𝑣 ∖ {𝑧})) + (♯‘{𝑧})) = ((♯‘(𝑣 ∖ {𝑧})) + 1)
163	161, 162	eqtrdi 2281	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑣 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾))) → (♯‘((𝑣 ∖ {𝑧}) ∪ {𝑧})) = ((♯‘(𝑣 ∖ {𝑧})) + 1))
164		simprrr 542	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑣 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾))) → (♯‘𝑣) = 𝐾)
165	157, 163, 164	3eqtr3d 2273	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑣 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾))) → ((♯‘(𝑣 ∖ {𝑧})) + 1) = 𝐾)
166	165	oveq1d 6064	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑣 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾))) → (((♯‘(𝑣 ∖ {𝑧})) + 1) − 1) = (𝐾 − 1))
167	150, 166	eqtr3d 2267	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑣 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾))) → (♯‘(𝑣 ∖ {𝑧})) = (𝐾 − 1))
168	128, 145, 167	elrabd 2974	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑣 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾))) → (𝑣 ∖ {𝑧}) ∈ {𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∣ (♯‘𝑥) = (𝐾 − 1)})
169	168	ex 115	. . . . . . 7 ⊢ (𝜑 → ((𝑣 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾)) → (𝑣 ∖ {𝑧}) ∈ {𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∣ (♯‘𝑥) = (𝐾 − 1)}))
170	127, 169	biimtrid 152	. . . . . 6 ⊢ (𝜑 → (𝑣 ∈ {𝑥 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ∣ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)} → (𝑣 ∖ {𝑧}) ∈ {𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∣ (♯‘𝑥) = (𝐾 − 1)}))
171	74, 127	anbi12i 460	. . . . . . 7 ⊢ ((𝑢 ∈ {𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∣ (♯‘𝑥) = (𝐾 − 1)} ∧ 𝑣 ∈ {𝑥 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ∣ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)}) ↔ ((𝑢 ∈ (𝒫 𝐴 ∩ Fin) ∧ (♯‘𝑢) = (𝐾 − 1)) ∧ (𝑣 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾))))
172		uncom 3362	. . . . . . . . . 10 ⊢ ({𝑧} ∪ (𝑣 ∖ {𝑧})) = ((𝑣 ∖ {𝑧}) ∪ {𝑧})
173	141	adantrl 478	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝑢 ∈ (𝒫 𝐴 ∩ Fin) ∧ (♯‘𝑢) = (𝐾 − 1)) ∧ (𝑣 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾)))) → 𝑣 ∈ Fin)
174	99	a1i 9	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝑢 ∈ (𝒫 𝐴 ∩ Fin) ∧ (♯‘𝑢) = (𝐾 − 1)) ∧ (𝑣 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾)))) → {𝑧} ∈ Fin)
175	153	adantrl 478	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝑢 ∈ (𝒫 𝐴 ∩ Fin) ∧ (♯‘𝑢) = (𝐾 − 1)) ∧ (𝑣 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾)))) → {𝑧} ⊆ 𝑣)
176	173, 174, 175, 154	syl3anc 1274	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝑢 ∈ (𝒫 𝐴 ∩ Fin) ∧ (♯‘𝑢) = (𝐾 − 1)) ∧ (𝑣 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾)))) → 𝑣 = ({𝑧} ∪ (𝑣 ∖ {𝑧})))
177	176	adantr 276	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((𝑢 ∈ (𝒫 𝐴 ∩ Fin) ∧ (♯‘𝑢) = (𝐾 − 1)) ∧ (𝑣 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾)))) ∧ 𝑢 = (𝑣 ∖ {𝑧})) → 𝑣 = ({𝑧} ∪ (𝑣 ∖ {𝑧})))
178		simpr 110	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((𝑢 ∈ (𝒫 𝐴 ∩ Fin) ∧ (♯‘𝑢) = (𝐾 − 1)) ∧ (𝑣 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾)))) ∧ 𝑢 = (𝑣 ∖ {𝑧})) → 𝑢 = (𝑣 ∖ {𝑧}))
179	178	uneq1d 3371	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((𝑢 ∈ (𝒫 𝐴 ∩ Fin) ∧ (♯‘𝑢) = (𝐾 − 1)) ∧ (𝑣 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾)))) ∧ 𝑢 = (𝑣 ∖ {𝑧})) → (𝑢 ∪ {𝑧}) = ((𝑣 ∖ {𝑧}) ∪ {𝑧}))
180	172, 177, 179	3eqtr4a 2291	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((𝑢 ∈ (𝒫 𝐴 ∩ Fin) ∧ (♯‘𝑢) = (𝐾 − 1)) ∧ (𝑣 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾)))) ∧ 𝑢 = (𝑣 ∖ {𝑧})) → 𝑣 = (𝑢 ∪ {𝑧}))
181		simpr 110	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ((𝑢 ∈ (𝒫 𝐴 ∩ Fin) ∧ (♯‘𝑢) = (𝐾 − 1)) ∧ (𝑣 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾)))) ∧ 𝑣 = (𝑢 ∪ {𝑧})) → 𝑣 = (𝑢 ∪ {𝑧}))
182		uncom 3362	. . . . . . . . . . . 12 ⊢ (𝑢 ∪ {𝑧}) = ({𝑧} ∪ 𝑢)
183	181, 182	eqtr2di 2282	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((𝑢 ∈ (𝒫 𝐴 ∩ Fin) ∧ (♯‘𝑢) = (𝐾 − 1)) ∧ (𝑣 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾)))) ∧ 𝑣 = (𝑢 ∪ {𝑧})) → ({𝑧} ∪ 𝑢) = 𝑣)
184		simpl 109	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((𝑢 ∈ (𝒫 𝐴 ∩ Fin) ∧ (♯‘𝑢) = (𝐾 − 1)) ∧ (𝑣 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾)))) → 𝜑)
185	78	adantr 276	. . . . . . . . . . . . . . 15 ⊢ ((𝑢 ∈ (𝒫 𝐴 ∩ Fin) ∧ (♯‘𝑢) = (𝐾 − 1)) → 𝑢 ∈ 𝒫 𝐴)
186	185	ad2antrl 490	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((𝑢 ∈ (𝒫 𝐴 ∩ Fin) ∧ (♯‘𝑢) = (𝐾 − 1)) ∧ (𝑣 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾)))) → 𝑢 ∈ 𝒫 𝐴)
187		simprlr 540	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((𝑢 ∈ (𝒫 𝐴 ∩ Fin) ∧ (♯‘𝑢) = (𝐾 − 1)) ∧ (𝑣 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾)))) → (♯‘𝑢) = (𝐾 − 1))
188		incom 3410	. . . . . . . . . . . . . . 15 ⊢ ({𝑧} ∩ 𝑢) = (𝑢 ∩ {𝑧})
189	93, 101	sylibr 134	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1))) → (𝑢 ∩ {𝑧}) = ∅)
190	188, 189	eqtrid 2277	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1))) → ({𝑧} ∩ 𝑢) = ∅)
191	184, 186, 187, 190	syl12anc 1272	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑢 ∈ (𝒫 𝐴 ∩ Fin) ∧ (♯‘𝑢) = (𝐾 − 1)) ∧ (𝑣 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾)))) → ({𝑧} ∩ 𝑢) = ∅)
192		uneqdifeqim 3594	. . . . . . . . . . . . 13 ⊢ (({𝑧} ⊆ 𝑣 ∧ ({𝑧} ∩ 𝑢) = ∅) → (({𝑧} ∪ 𝑢) = 𝑣 → (𝑣 ∖ {𝑧}) = 𝑢))
193	175, 191, 192	syl2anc 411	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝑢 ∈ (𝒫 𝐴 ∩ Fin) ∧ (♯‘𝑢) = (𝐾 − 1)) ∧ (𝑣 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾)))) → (({𝑧} ∪ 𝑢) = 𝑣 → (𝑣 ∖ {𝑧}) = 𝑢))
194	193	adantr 276	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((𝑢 ∈ (𝒫 𝐴 ∩ Fin) ∧ (♯‘𝑢) = (𝐾 − 1)) ∧ (𝑣 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾)))) ∧ 𝑣 = (𝑢 ∪ {𝑧})) → (({𝑧} ∪ 𝑢) = 𝑣 → (𝑣 ∖ {𝑧}) = 𝑢))
195	183, 194	mpd 13	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((𝑢 ∈ (𝒫 𝐴 ∩ Fin) ∧ (♯‘𝑢) = (𝐾 − 1)) ∧ (𝑣 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾)))) ∧ 𝑣 = (𝑢 ∪ {𝑧})) → (𝑣 ∖ {𝑧}) = 𝑢)
196	195	eqcomd 2238	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((𝑢 ∈ (𝒫 𝐴 ∩ Fin) ∧ (♯‘𝑢) = (𝐾 − 1)) ∧ (𝑣 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾)))) ∧ 𝑣 = (𝑢 ∪ {𝑧})) → 𝑢 = (𝑣 ∖ {𝑧}))
197	180, 196	impbida 600	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑢 ∈ (𝒫 𝐴 ∩ Fin) ∧ (♯‘𝑢) = (𝐾 − 1)) ∧ (𝑣 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾)))) → (𝑢 = (𝑣 ∖ {𝑧}) ↔ 𝑣 = (𝑢 ∪ {𝑧})))
198	197	ex 115	. . . . . . 7 ⊢ (𝜑 → (((𝑢 ∈ (𝒫 𝐴 ∩ Fin) ∧ (♯‘𝑢) = (𝐾 − 1)) ∧ (𝑣 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾))) → (𝑢 = (𝑣 ∖ {𝑧}) ↔ 𝑣 = (𝑢 ∪ {𝑧}))))
199	171, 198	biimtrid 152	. . . . . 6 ⊢ (𝜑 → ((𝑢 ∈ {𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∣ (♯‘𝑥) = (𝐾 − 1)} ∧ 𝑣 ∈ {𝑥 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ∣ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)}) → (𝑢 = (𝑣 ∖ {𝑧}) ↔ 𝑣 = (𝑢 ∪ {𝑧}))))
200	64, 72, 123, 170, 199	en3d 7007	. . . . 5 ⊢ (𝜑 → {𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∣ (♯‘𝑥) = (𝐾 − 1)} ≈ {𝑥 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ∣ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)})
201		fipwfi 7271	. . . . . . . 8 ⊢ (𝐴 ∈ Fin → (𝒫 𝐴 ∩ Fin) ∈ Fin)
202	59, 201	syl 14	. . . . . . 7 ⊢ (𝜑 → (𝒫 𝐴 ∩ Fin) ∈ Fin)
203		elinel2 3405	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝒫 𝐴 ∩ Fin) → 𝑥 ∈ Fin)
204		hashcl 11139	. . . . . . . . . . 11 ⊢ (𝑥 ∈ Fin → (♯‘𝑥) ∈ ℕ₀)
205	203, 204	syl 14	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝒫 𝐴 ∩ Fin) → (♯‘𝑥) ∈ ℕ₀)
206	205	nn0zd 9694	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝒫 𝐴 ∩ Fin) → (♯‘𝑥) ∈ ℤ)
207		zdceq 9649	. . . . . . . . 9 ⊢ (((♯‘𝑥) ∈ ℤ ∧ (𝐾 − 1) ∈ ℤ) → DECID (♯‘𝑥) = (𝐾 − 1))
208	206, 56, 207	syl2anr 290	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝐴 ∩ Fin)) → DECID (♯‘𝑥) = (𝐾 − 1))
209	208	ralrimiva 2615	. . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ (𝒫 𝐴 ∩ Fin)DECID (♯‘𝑥) = (𝐾 − 1))
210	202, 209	ssfirab 7196	. . . . . 6 ⊢ (𝜑 → {𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∣ (♯‘𝑥) = (𝐾 − 1)} ∈ Fin)
211	90	a1i 9	. . . . . . . . 9 ⊢ (𝜑 → 𝑧 ∈ V)
212		unsnfi 7178	. . . . . . . . 9 ⊢ ((𝐴 ∈ Fin ∧ 𝑧 ∈ V ∧ ¬ 𝑧 ∈ 𝐴) → (𝐴 ∪ {𝑧}) ∈ Fin)
213	59, 211, 18, 212	syl3anc 1274	. . . . . . . 8 ⊢ (𝜑 → (𝐴 ∪ {𝑧}) ∈ Fin)
214		fipwfi 7271	. . . . . . . 8 ⊢ ((𝐴 ∪ {𝑧}) ∈ Fin → (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ∈ Fin)
215	213, 214	syl 14	. . . . . . 7 ⊢ (𝜑 → (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ∈ Fin)
216		elequ1 2207	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑧 → (𝑟 ∈ 𝑥 ↔ 𝑧 ∈ 𝑥))
217	216	dcbid 846	. . . . . . . . . 10 ⊢ (𝑟 = 𝑧 → (DECID 𝑟 ∈ 𝑥 ↔ DECID 𝑧 ∈ 𝑥))
218	24	adantl 277	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin)) → 𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}))
219	218, 25	syl 14	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin)) → 𝑥 ⊆ (𝐴 ∪ {𝑧}))
220	213	adantr 276	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin)) → (𝐴 ∪ {𝑧}) ∈ Fin)
221	39	adantl 277	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin)) → 𝑥 ∈ Fin)
222		fissfi 7215	. . . . . . . . . . 11 ⊢ ((𝑥 ⊆ (𝐴 ∪ {𝑧}) ∧ (𝐴 ∪ {𝑧}) ∈ Fin ∧ 𝑥 ∈ Fin) → ∀𝑟 ∈ (𝐴 ∪ {𝑧})DECID 𝑟 ∈ 𝑥)
223	219, 220, 221, 222	syl3anc 1274	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin)) → ∀𝑟 ∈ (𝐴 ∪ {𝑧})DECID 𝑟 ∈ 𝑥)
224		ssun2 3382	. . . . . . . . . . . 12 ⊢ {𝑧} ⊆ (𝐴 ∪ {𝑧})
225		vsnid 3720	. . . . . . . . . . . 12 ⊢ 𝑧 ∈ {𝑧}
226	224, 225	sselii 3234	. . . . . . . . . . 11 ⊢ 𝑧 ∈ (𝐴 ∪ {𝑧})
227	226	a1i 9	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin)) → 𝑧 ∈ (𝐴 ∪ {𝑧}))
228	217, 223, 227	rspcdva 2925	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin)) → DECID 𝑧 ∈ 𝑥)
229	39, 204	syl 14	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) → (♯‘𝑥) ∈ ℕ₀)
230	229	nn0zd 9694	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) → (♯‘𝑥) ∈ ℤ)
231		zdceq 9649	. . . . . . . . . 10 ⊢ (((♯‘𝑥) ∈ ℤ ∧ 𝐾 ∈ ℤ) → DECID (♯‘𝑥) = 𝐾)
232	230, 7, 231	syl2anr 290	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin)) → DECID (♯‘𝑥) = 𝐾)
233	228, 232	dcand 941	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin)) → DECID (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾))
234	233	ralrimiva 2615	. . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin)DECID (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾))
235	215, 234	ssfirab 7196	. . . . . 6 ⊢ (𝜑 → {𝑥 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ∣ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)} ∈ Fin)
236		hashen 11142	. . . . . 6 ⊢ (({𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∣ (♯‘𝑥) = (𝐾 − 1)} ∈ Fin ∧ {𝑥 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ∣ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)} ∈ Fin) → ((♯‘{𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∣ (♯‘𝑥) = (𝐾 − 1)}) = (♯‘{𝑥 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ∣ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)}) ↔ {𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∣ (♯‘𝑥) = (𝐾 − 1)} ≈ {𝑥 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ∣ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)}))
237	210, 235, 236	syl2anc 411	. . . . 5 ⊢ (𝜑 → ((♯‘{𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∣ (♯‘𝑥) = (𝐾 − 1)}) = (♯‘{𝑥 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ∣ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)}) ↔ {𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∣ (♯‘𝑥) = (𝐾 − 1)} ≈ {𝑥 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ∣ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)}))
238	200, 237	mpbird 167	. . . 4 ⊢ (𝜑 → (♯‘{𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∣ (♯‘𝑥) = (𝐾 − 1)}) = (♯‘{𝑥 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ∣ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)}))
239	57, 238	eqtrd 2265	. . 3 ⊢ (𝜑 → ((♯‘𝐴)C(𝐾 − 1)) = (♯‘{𝑥 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ∣ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)}))
240	49, 239	oveq12d 6067	. 2 ⊢ (𝜑 → (((♯‘𝐴)C𝐾) + ((♯‘𝐴)C(𝐾 − 1))) = ((♯‘{𝑥 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ∣ (¬ 𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)}) + (♯‘{𝑥 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ∣ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)})))
241	99	a1i 9	. . . . . 6 ⊢ (𝜑 → {𝑧} ∈ Fin)
242		disjsn 3750	. . . . . . 7 ⊢ ((𝐴 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧 ∈ 𝐴)
243	18, 242	sylibr 134	. . . . . 6 ⊢ (𝜑 → (𝐴 ∩ {𝑧}) = ∅)
244		hashun 11164	. . . . . 6 ⊢ ((𝐴 ∈ Fin ∧ {𝑧} ∈ Fin ∧ (𝐴 ∩ {𝑧}) = ∅) → (♯‘(𝐴 ∪ {𝑧})) = ((♯‘𝐴) + (♯‘{𝑧})))
245	59, 241, 243, 244	syl3anc 1274	. . . . 5 ⊢ (𝜑 → (♯‘(𝐴 ∪ {𝑧})) = ((♯‘𝐴) + (♯‘{𝑧})))
246	107	oveq2i 6060	. . . . 5 ⊢ ((♯‘𝐴) + (♯‘{𝑧})) = ((♯‘𝐴) + 1)
247	245, 246	eqtrdi 2281	. . . 4 ⊢ (𝜑 → (♯‘(𝐴 ∪ {𝑧})) = ((♯‘𝐴) + 1))
248	247	oveq1d 6064	. . 3 ⊢ (𝜑 → ((♯‘(𝐴 ∪ {𝑧}))C𝐾) = (((♯‘𝐴) + 1)C𝐾))
249		hashcl 11139	. . . . 5 ⊢ (𝐴 ∈ Fin → (♯‘𝐴) ∈ ℕ₀)
250	59, 249	syl 14	. . . 4 ⊢ (𝜑 → (♯‘𝐴) ∈ ℕ₀)
251		bcpasc 11124	. . . 4 ⊢ (((♯‘𝐴) ∈ ℕ₀ ∧ 𝐾 ∈ ℤ) → (((♯‘𝐴)C𝐾) + ((♯‘𝐴)C(𝐾 − 1))) = (((♯‘𝐴) + 1)C𝐾))
252	250, 7, 251	syl2anc 411	. . 3 ⊢ (𝜑 → (((♯‘𝐴)C𝐾) + ((♯‘𝐴)C(𝐾 − 1))) = (((♯‘𝐴) + 1)C𝐾))
253	248, 252	eqtr4d 2268	. 2 ⊢ (𝜑 → ((♯‘(𝐴 ∪ {𝑧}))C𝐾) = (((♯‘𝐴)C𝐾) + ((♯‘𝐴)C(𝐾 − 1))))
254		pm2.1dc 845	. . . . . . . . 9 ⊢ (DECID 𝑧 ∈ 𝑥 → (¬ 𝑧 ∈ 𝑥 ∨ 𝑧 ∈ 𝑥))
255	228, 254	syl 14	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin)) → (¬ 𝑧 ∈ 𝑥 ∨ 𝑧 ∈ 𝑥))
256	255	biantrurd 305	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin)) → ((♯‘𝑥) = 𝐾 ↔ ((¬ 𝑧 ∈ 𝑥 ∨ 𝑧 ∈ 𝑥) ∧ (♯‘𝑥) = 𝐾)))
257		andir 827	. . . . . . 7 ⊢ (((¬ 𝑧 ∈ 𝑥 ∨ 𝑧 ∈ 𝑥) ∧ (♯‘𝑥) = 𝐾) ↔ ((¬ 𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾) ∨ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)))
258	256, 257	bitrdi 196	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin)) → ((♯‘𝑥) = 𝐾 ↔ ((¬ 𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾) ∨ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾))))
259	258	rabbidva 2800	. . . . 5 ⊢ (𝜑 → {𝑥 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ∣ (♯‘𝑥) = 𝐾} = {𝑥 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ∣ ((¬ 𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾) ∨ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾))})
260		unrab 3491	. . . . 5 ⊢ ({𝑥 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ∣ (¬ 𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)} ∪ {𝑥 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ∣ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)}) = {𝑥 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ∣ ((¬ 𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾) ∨ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾))}
261	259, 260	eqtr4di 2283	. . . 4 ⊢ (𝜑 → {𝑥 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ∣ (♯‘𝑥) = 𝐾} = ({𝑥 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ∣ (¬ 𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)} ∪ {𝑥 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ∣ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)}))
262	261	fveq2d 5673	. . 3 ⊢ (𝜑 → (♯‘{𝑥 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ∣ (♯‘𝑥) = 𝐾}) = (♯‘({𝑥 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ∣ (¬ 𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)} ∪ {𝑥 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ∣ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)})))
263		dcn 850	. . . . . . . 8 ⊢ (DECID 𝑧 ∈ 𝑥 → DECID ¬ 𝑧 ∈ 𝑥)
264	228, 263	syl 14	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin)) → DECID ¬ 𝑧 ∈ 𝑥)
265	264, 232	dcand 941	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin)) → DECID (¬ 𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾))
266	265	ralrimiva 2615	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin)DECID (¬ 𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾))
267	215, 266	ssfirab 7196	. . . 4 ⊢ (𝜑 → {𝑥 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ∣ (¬ 𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)} ∈ Fin)
268		inrab 3492	. . . . . 6 ⊢ ({𝑥 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ∣ (¬ 𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)} ∩ {𝑥 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ∣ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)}) = {𝑥 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ∣ ((¬ 𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾) ∧ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾))}
269		simprl 531	. . . . . . . . 9 ⊢ (((¬ 𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾) ∧ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)) → 𝑧 ∈ 𝑥)
270		simpll 527	. . . . . . . . 9 ⊢ (((¬ 𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾) ∧ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)) → ¬ 𝑧 ∈ 𝑥)
271	269, 270	pm2.65i 644	. . . . . . . 8 ⊢ ¬ ((¬ 𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾) ∧ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾))
272	271	rgenw 2597	. . . . . . 7 ⊢ ∀𝑥 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ¬ ((¬ 𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾) ∧ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾))
273		rabeq0 3537	. . . . . . 7 ⊢ ({𝑥 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ∣ ((¬ 𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾) ∧ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾))} = ∅ ↔ ∀𝑥 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ¬ ((¬ 𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾) ∧ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)))
274	272, 273	mpbir 146	. . . . . 6 ⊢ {𝑥 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ∣ ((¬ 𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾) ∧ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾))} = ∅
275	268, 274	eqtri 2253	. . . . 5 ⊢ ({𝑥 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ∣ (¬ 𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)} ∩ {𝑥 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ∣ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)}) = ∅
276	275	a1i 9	. . . 4 ⊢ (𝜑 → ({𝑥 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ∣ (¬ 𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)} ∩ {𝑥 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ∣ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)}) = ∅)
277		hashun 11164	. . . 4 ⊢ (({𝑥 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ∣ (¬ 𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)} ∈ Fin ∧ {𝑥 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ∣ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)} ∈ Fin ∧ ({𝑥 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ∣ (¬ 𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)} ∩ {𝑥 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ∣ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)}) = ∅) → (♯‘({𝑥 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ∣ (¬ 𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)} ∪ {𝑥 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ∣ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)})) = ((♯‘{𝑥 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ∣ (¬ 𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)}) + (♯‘{𝑥 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ∣ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)})))
278	267, 235, 276, 277	syl3anc 1274	. . 3 ⊢ (𝜑 → (♯‘({𝑥 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ∣ (¬ 𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)} ∪ {𝑥 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ∣ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)})) = ((♯‘{𝑥 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ∣ (¬ 𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)}) + (♯‘{𝑥 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ∣ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)})))
279	262, 278	eqtrd 2265	. 2 ⊢ (𝜑 → (♯‘{𝑥 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ∣ (♯‘𝑥) = 𝐾}) = ((♯‘{𝑥 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ∣ (¬ 𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)}) + (♯‘{𝑥 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ∣ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)})))
280	240, 253, 279	3eqtr4d 2275	1 ⊢ (𝜑 → ((♯‘(𝐴 ∪ {𝑧}))C𝐾) = (♯‘{𝑥 ∈ (𝒫 (𝐴 ∪ {𝑧}) ∩ Fin) ∣ (♯‘𝑥) = 𝐾}))

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 ∨ wo 716 DECID wdc 842 = wceq 1398 ∈ wcel 2203 ∀wral 2520 {crab 2524 Vcvv 2812 ∖ cdif 3207 ∪ cun 3208 ∩ cin 3209 ⊆ wss 3210 ∅c0 3507 𝒫 cpw 3668 {csn 3688 class class class wbr 4108 ‘cfv 5351 (class class class)co 6049 ≈ cen 6972 Fincfn 6974 ℂcc 8121 1c1 8124 + caddc 8126 − cmin 8440 ℕ₀cn0 9492 ℤcz 9573 Ccbc 11105 ♯chash 11133

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2205 ax-14 2206 ax-ext 2214 ax-coll 4224 ax-sep 4227 ax-nul 4235 ax-pow 4286 ax-pr 4321 ax-un 4553 ax-setind 4658 ax-iinf 4709 ax-cnex 8214 ax-resscn 8215 ax-1cn 8216 ax-1re 8217 ax-icn 8218 ax-addcl 8219 ax-addrcl 8220 ax-mulcl 8221 ax-mulrcl 8222 ax-addcom 8223 ax-mulcom 8224 ax-addass 8225 ax-mulass 8226 ax-distr 8227 ax-i2m1 8228 ax-0lt1 8229 ax-1rid 8230 ax-0id 8231 ax-rnegex 8232 ax-precex 8233 ax-cnre 8234 ax-pre-ltirr 8235 ax-pre-ltwlin 8236 ax-pre-lttrn 8237 ax-pre-apti 8238 ax-pre-ltadd 8239 ax-pre-mulgt0 8240 ax-pre-mulext 8241

This theorem depends on definitions: df-bi 117 df-dc 843 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1812 df-eu 2083 df-mo 2084 df-clab 2219 df-cleq 2225 df-clel 2228 df-nfc 2373 df-ne 2413 df-nel 2508 df-ral 2525 df-rex 2526 df-reu 2527 df-rmo 2528 df-rab 2529 df-v 2814 df-sbc 3042 df-csb 3138 df-dif 3212 df-un 3214 df-in 3216 df-ss 3223 df-nul 3508 df-if 3620 df-pw 3670 df-sn 3694 df-pr 3695 df-op 3697 df-uni 3914 df-int 3949 df-iun 3992 df-br 4109 df-opab 4171 df-mpt 4172 df-tr 4208 df-id 4413 df-po 4416 df-iso 4417 df-iord 4486 df-on 4488 df-ilim 4489 df-suc 4491 df-iom 4712 df-xp 4754 df-rel 4755 df-cnv 4756 df-co 4757 df-dm 4758 df-rn 4759 df-res 4760 df-ima 4761 df-iota 5311 df-fun 5353 df-fn 5354 df-f 5355 df-f1 5356 df-fo 5357 df-f1o 5358 df-fv 5359 df-riota 6002 df-ov 6052 df-oprab 6053 df-mpo 6054 df-1st 6333 df-2nd 6334 df-recs 6535 df-irdg 6600 df-frec 6621 df-1o 6646 df-2o 6647 df-oadd 6650 df-er 6766 df-map 6883 df-en 6975 df-dom 6976 df-fin 6977 df-pnf 8306 df-mnf 8307 df-xr 8308 df-ltxr 8309 df-le 8310 df-sub 8442 df-neg 8443 df-reap 8845 df-ap 8852 df-div 8943 df-inn 9234 df-n0 9493 df-z 9574 df-uz 9850 df-q 9948 df-rp 9983 df-fz 10339 df-seqfrec 10806 df-fac 11084 df-bc 11106 df-ihash 11134

This theorem is referenced by: hashfibc 11200

W3C validator