Theorem ackbij1lem18 10144

Description: Lemma for ackbij1 10145. (Contributed by Stefan O'Rear, 18-Nov-2014.)

Hypothesis

Ref	Expression
ackbij.f	⊢ 𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦)))

Assertion

Ref	Expression
ackbij1lem18	⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → ∃𝑏 ∈ (𝒫 ω ∩ Fin)(𝐹‘𝑏) = suc (𝐹‘𝐴))

Distinct variable groups:   𝐹,𝑏,𝑥,𝑦   𝐴,𝑏,𝑥,𝑦

Proof of Theorem ackbij1lem18

Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		difss 4086	. . . 4 ⊢ (𝐴 ∖ ∩ (ω ∖ 𝐴)) ⊆ 𝐴
2		ackbij.f	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦)))
3	2	ackbij1lem11 10137	. . . 4 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∖ ∩ (ω ∖ 𝐴)) ⊆ 𝐴) → (𝐴 ∖ ∩ (ω ∖ 𝐴)) ∈ (𝒫 ω ∩ Fin))
4	1, 3	mpan2 691	. . 3 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → (𝐴 ∖ ∩ (ω ∖ 𝐴)) ∈ (𝒫 ω ∩ Fin))
5		difss 4086	. . . . . . 7 ⊢ (ω ∖ 𝐴) ⊆ ω
6		omsson 7810	. . . . . . 7 ⊢ ω ⊆ On
7	5, 6	sstri 3941	. . . . . 6 ⊢ (ω ∖ 𝐴) ⊆ On
8		ominf 9162	. . . . . . . 8 ⊢ ¬ ω ∈ Fin
9		elinel2 4152	. . . . . . . 8 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → 𝐴 ∈ Fin)
10		difinf 9209	. . . . . . . 8 ⊢ ((¬ ω ∈ Fin ∧ 𝐴 ∈ Fin) → ¬ (ω ∖ 𝐴) ∈ Fin)
11	8, 9, 10	sylancr 587	. . . . . . 7 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → ¬ (ω ∖ 𝐴) ∈ Fin)
12		0fi 8977	. . . . . . . . 9 ⊢ ∅ ∈ Fin
13		eleq1 2822	. . . . . . . . 9 ⊢ ((ω ∖ 𝐴) = ∅ → ((ω ∖ 𝐴) ∈ Fin ↔ ∅ ∈ Fin))
14	12, 13	mpbiri 258	. . . . . . . 8 ⊢ ((ω ∖ 𝐴) = ∅ → (ω ∖ 𝐴) ∈ Fin)
15	14	necon3bi 2956	. . . . . . 7 ⊢ (¬ (ω ∖ 𝐴) ∈ Fin → (ω ∖ 𝐴) ≠ ∅)
16	11, 15	syl 17	. . . . . 6 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → (ω ∖ 𝐴) ≠ ∅)
17		onint 7733	. . . . . 6 ⊢ (((ω ∖ 𝐴) ⊆ On ∧ (ω ∖ 𝐴) ≠ ∅) → ∩ (ω ∖ 𝐴) ∈ (ω ∖ 𝐴))
18	7, 16, 17	sylancr 587	. . . . 5 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → ∩ (ω ∖ 𝐴) ∈ (ω ∖ 𝐴))
19	18	eldifad 3911	. . . 4 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → ∩ (ω ∖ 𝐴) ∈ ω)
20		ackbij1lem4 10130	. . . 4 ⊢ (∩ (ω ∖ 𝐴) ∈ ω → {∩ (ω ∖ 𝐴)} ∈ (𝒫 ω ∩ Fin))
21	19, 20	syl 17	. . 3 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → {∩ (ω ∖ 𝐴)} ∈ (𝒫 ω ∩ Fin))
22		ackbij1lem6 10132	. . 3 ⊢ (((𝐴 ∖ ∩ (ω ∖ 𝐴)) ∈ (𝒫 ω ∩ Fin) ∧ {∩ (ω ∖ 𝐴)} ∈ (𝒫 ω ∩ Fin)) → ((𝐴 ∖ ∩ (ω ∖ 𝐴)) ∪ {∩ (ω ∖ 𝐴)}) ∈ (𝒫 ω ∩ Fin))
23	4, 21, 22	syl2anc 584	. 2 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → ((𝐴 ∖ ∩ (ω ∖ 𝐴)) ∪ {∩ (ω ∖ 𝐴)}) ∈ (𝒫 ω ∩ Fin))
24	18	eldifbd 3912	. . . . . 6 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → ¬ ∩ (ω ∖ 𝐴) ∈ 𝐴)
25		disjsn 4666	. . . . . 6 ⊢ ((𝐴 ∩ {∩ (ω ∖ 𝐴)}) = ∅ ↔ ¬ ∩ (ω ∖ 𝐴) ∈ 𝐴)
26	24, 25	sylibr 234	. . . . 5 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → (𝐴 ∩ {∩ (ω ∖ 𝐴)}) = ∅)
27		ssdisj 4410	. . . . 5 ⊢ (((𝐴 ∖ ∩ (ω ∖ 𝐴)) ⊆ 𝐴 ∧ (𝐴 ∩ {∩ (ω ∖ 𝐴)}) = ∅) → ((𝐴 ∖ ∩ (ω ∖ 𝐴)) ∩ {∩ (ω ∖ 𝐴)}) = ∅)
28	1, 26, 27	sylancr 587	. . . 4 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → ((𝐴 ∖ ∩ (ω ∖ 𝐴)) ∩ {∩ (ω ∖ 𝐴)}) = ∅)
29	2	ackbij1lem9 10135	. . . 4 ⊢ (((𝐴 ∖ ∩ (ω ∖ 𝐴)) ∈ (𝒫 ω ∩ Fin) ∧ {∩ (ω ∖ 𝐴)} ∈ (𝒫 ω ∩ Fin) ∧ ((𝐴 ∖ ∩ (ω ∖ 𝐴)) ∩ {∩ (ω ∖ 𝐴)}) = ∅) → (𝐹‘((𝐴 ∖ ∩ (ω ∖ 𝐴)) ∪ {∩ (ω ∖ 𝐴)})) = ((𝐹‘(𝐴 ∖ ∩ (ω ∖ 𝐴))) +o (𝐹‘{∩ (ω ∖ 𝐴)})))
30	4, 21, 28, 29	syl3anc 1373	. . 3 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → (𝐹‘((𝐴 ∖ ∩ (ω ∖ 𝐴)) ∪ {∩ (ω ∖ 𝐴)})) = ((𝐹‘(𝐴 ∖ ∩ (ω ∖ 𝐴))) +o (𝐹‘{∩ (ω ∖ 𝐴)})))
31	2	ackbij1lem14 10140	. . . . 5 ⊢ (∩ (ω ∖ 𝐴) ∈ ω → (𝐹‘{∩ (ω ∖ 𝐴)}) = suc (𝐹‘∩ (ω ∖ 𝐴)))
32	19, 31	syl 17	. . . 4 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → (𝐹‘{∩ (ω ∖ 𝐴)}) = suc (𝐹‘∩ (ω ∖ 𝐴)))
33	32	oveq2d 7372	. . 3 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → ((𝐹‘(𝐴 ∖ ∩ (ω ∖ 𝐴))) +o (𝐹‘{∩ (ω ∖ 𝐴)})) = ((𝐹‘(𝐴 ∖ ∩ (ω ∖ 𝐴))) +o suc (𝐹‘∩ (ω ∖ 𝐴))))
34	2	ackbij1lem10 10136	. . . . . . 7 ⊢ 𝐹:(𝒫 ω ∩ Fin)⟶ω
35	34	ffvelcdmi 7026	. . . . . 6 ⊢ ((𝐴 ∖ ∩ (ω ∖ 𝐴)) ∈ (𝒫 ω ∩ Fin) → (𝐹‘(𝐴 ∖ ∩ (ω ∖ 𝐴))) ∈ ω)
36	4, 35	syl 17	. . . . 5 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → (𝐹‘(𝐴 ∖ ∩ (ω ∖ 𝐴))) ∈ ω)
37		ackbij1lem3 10129	. . . . . . 7 ⊢ (∩ (ω ∖ 𝐴) ∈ ω → ∩ (ω ∖ 𝐴) ∈ (𝒫 ω ∩ Fin))
38	19, 37	syl 17	. . . . . 6 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → ∩ (ω ∖ 𝐴) ∈ (𝒫 ω ∩ Fin))
39	34	ffvelcdmi 7026	. . . . . 6 ⊢ (∩ (ω ∖ 𝐴) ∈ (𝒫 ω ∩ Fin) → (𝐹‘∩ (ω ∖ 𝐴)) ∈ ω)
40	38, 39	syl 17	. . . . 5 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → (𝐹‘∩ (ω ∖ 𝐴)) ∈ ω)
41		nnasuc 8532	. . . . 5 ⊢ (((𝐹‘(𝐴 ∖ ∩ (ω ∖ 𝐴))) ∈ ω ∧ (𝐹‘∩ (ω ∖ 𝐴)) ∈ ω) → ((𝐹‘(𝐴 ∖ ∩ (ω ∖ 𝐴))) +o suc (𝐹‘∩ (ω ∖ 𝐴))) = suc ((𝐹‘(𝐴 ∖ ∩ (ω ∖ 𝐴))) +o (𝐹‘∩ (ω ∖ 𝐴))))
42	36, 40, 41	syl2anc 584	. . . 4 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → ((𝐹‘(𝐴 ∖ ∩ (ω ∖ 𝐴))) +o suc (𝐹‘∩ (ω ∖ 𝐴))) = suc ((𝐹‘(𝐴 ∖ ∩ (ω ∖ 𝐴))) +o (𝐹‘∩ (ω ∖ 𝐴))))
43		disjdifr 4423	. . . . . . . 8 ⊢ ((𝐴 ∖ ∩ (ω ∖ 𝐴)) ∩ ∩ (ω ∖ 𝐴)) = ∅
44	43	a1i 11	. . . . . . 7 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → ((𝐴 ∖ ∩ (ω ∖ 𝐴)) ∩ ∩ (ω ∖ 𝐴)) = ∅)
45	2	ackbij1lem9 10135	. . . . . . 7 ⊢ (((𝐴 ∖ ∩ (ω ∖ 𝐴)) ∈ (𝒫 ω ∩ Fin) ∧ ∩ (ω ∖ 𝐴) ∈ (𝒫 ω ∩ Fin) ∧ ((𝐴 ∖ ∩ (ω ∖ 𝐴)) ∩ ∩ (ω ∖ 𝐴)) = ∅) → (𝐹‘((𝐴 ∖ ∩ (ω ∖ 𝐴)) ∪ ∩ (ω ∖ 𝐴))) = ((𝐹‘(𝐴 ∖ ∩ (ω ∖ 𝐴))) +o (𝐹‘∩ (ω ∖ 𝐴))))
46	4, 38, 44, 45	syl3anc 1373	. . . . . 6 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → (𝐹‘((𝐴 ∖ ∩ (ω ∖ 𝐴)) ∪ ∩ (ω ∖ 𝐴))) = ((𝐹‘(𝐴 ∖ ∩ (ω ∖ 𝐴))) +o (𝐹‘∩ (ω ∖ 𝐴))))
47		uncom 4108	. . . . . . . 8 ⊢ ((𝐴 ∖ ∩ (ω ∖ 𝐴)) ∪ ∩ (ω ∖ 𝐴)) = (∩ (ω ∖ 𝐴) ∪ (𝐴 ∖ ∩ (ω ∖ 𝐴)))
48		onnmin 7741	. . . . . . . . . . . . . . 15 ⊢ (((ω ∖ 𝐴) ⊆ On ∧ 𝑎 ∈ (ω ∖ 𝐴)) → ¬ 𝑎 ∈ ∩ (ω ∖ 𝐴))
49	7, 48	mpan 690	. . . . . . . . . . . . . 14 ⊢ (𝑎 ∈ (ω ∖ 𝐴) → ¬ 𝑎 ∈ ∩ (ω ∖ 𝐴))
50	49	con2i 139	. . . . . . . . . . . . 13 ⊢ (𝑎 ∈ ∩ (ω ∖ 𝐴) → ¬ 𝑎 ∈ (ω ∖ 𝐴))
51	50	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝑎 ∈ ∩ (ω ∖ 𝐴)) → ¬ 𝑎 ∈ (ω ∖ 𝐴))
52		ordom 7816	. . . . . . . . . . . . . . 15 ⊢ Ord ω
53		ordelss 6331	. . . . . . . . . . . . . . 15 ⊢ ((Ord ω ∧ ∩ (ω ∖ 𝐴) ∈ ω) → ∩ (ω ∖ 𝐴) ⊆ ω)
54	52, 19, 53	sylancr 587	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → ∩ (ω ∖ 𝐴) ⊆ ω)
55	54	sselda 3931	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝑎 ∈ ∩ (ω ∖ 𝐴)) → 𝑎 ∈ ω)
56		eldif 3909	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 ∈ (ω ∖ 𝐴) ↔ (𝑎 ∈ ω ∧ ¬ 𝑎 ∈ 𝐴))
57	56	simplbi2 500	. . . . . . . . . . . . . . 15 ⊢ (𝑎 ∈ ω → (¬ 𝑎 ∈ 𝐴 → 𝑎 ∈ (ω ∖ 𝐴)))
58	57	orrd 863	. . . . . . . . . . . . . 14 ⊢ (𝑎 ∈ ω → (𝑎 ∈ 𝐴 ∨ 𝑎 ∈ (ω ∖ 𝐴)))
59	58	orcomd 871	. . . . . . . . . . . . 13 ⊢ (𝑎 ∈ ω → (𝑎 ∈ (ω ∖ 𝐴) ∨ 𝑎 ∈ 𝐴))
60	55, 59	syl 17	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝑎 ∈ ∩ (ω ∖ 𝐴)) → (𝑎 ∈ (ω ∖ 𝐴) ∨ 𝑎 ∈ 𝐴))
61		orel1 888	. . . . . . . . . . . 12 ⊢ (¬ 𝑎 ∈ (ω ∖ 𝐴) → ((𝑎 ∈ (ω ∖ 𝐴) ∨ 𝑎 ∈ 𝐴) → 𝑎 ∈ 𝐴))
62	51, 60, 61	sylc 65	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝑎 ∈ ∩ (ω ∖ 𝐴)) → 𝑎 ∈ 𝐴)
63	62	ex 412	. . . . . . . . . 10 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → (𝑎 ∈ ∩ (ω ∖ 𝐴) → 𝑎 ∈ 𝐴))
64	63	ssrdv 3937	. . . . . . . . 9 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → ∩ (ω ∖ 𝐴) ⊆ 𝐴)
65		undif 4432	. . . . . . . . 9 ⊢ (∩ (ω ∖ 𝐴) ⊆ 𝐴 ↔ (∩ (ω ∖ 𝐴) ∪ (𝐴 ∖ ∩ (ω ∖ 𝐴))) = 𝐴)
66	64, 65	sylib 218	. . . . . . . 8 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → (∩ (ω ∖ 𝐴) ∪ (𝐴 ∖ ∩ (ω ∖ 𝐴))) = 𝐴)
67	47, 66	eqtrid 2781	. . . . . . 7 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → ((𝐴 ∖ ∩ (ω ∖ 𝐴)) ∪ ∩ (ω ∖ 𝐴)) = 𝐴)
68	67	fveq2d 6836	. . . . . 6 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → (𝐹‘((𝐴 ∖ ∩ (ω ∖ 𝐴)) ∪ ∩ (ω ∖ 𝐴))) = (𝐹‘𝐴))
69	46, 68	eqtr3d 2771	. . . . 5 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → ((𝐹‘(𝐴 ∖ ∩ (ω ∖ 𝐴))) +o (𝐹‘∩ (ω ∖ 𝐴))) = (𝐹‘𝐴))
70		suceq 6383	. . . . 5 ⊢ (((𝐹‘(𝐴 ∖ ∩ (ω ∖ 𝐴))) +o (𝐹‘∩ (ω ∖ 𝐴))) = (𝐹‘𝐴) → suc ((𝐹‘(𝐴 ∖ ∩ (ω ∖ 𝐴))) +o (𝐹‘∩ (ω ∖ 𝐴))) = suc (𝐹‘𝐴))
71	69, 70	syl 17	. . . 4 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → suc ((𝐹‘(𝐴 ∖ ∩ (ω ∖ 𝐴))) +o (𝐹‘∩ (ω ∖ 𝐴))) = suc (𝐹‘𝐴))
72	42, 71	eqtrd 2769	. . 3 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → ((𝐹‘(𝐴 ∖ ∩ (ω ∖ 𝐴))) +o suc (𝐹‘∩ (ω ∖ 𝐴))) = suc (𝐹‘𝐴))
73	30, 33, 72	3eqtrd 2773	. 2 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → (𝐹‘((𝐴 ∖ ∩ (ω ∖ 𝐴)) ∪ {∩ (ω ∖ 𝐴)})) = suc (𝐹‘𝐴))
74		fveqeq2 6841	. . 3 ⊢ (𝑏 = ((𝐴 ∖ ∩ (ω ∖ 𝐴)) ∪ {∩ (ω ∖ 𝐴)}) → ((𝐹‘𝑏) = suc (𝐹‘𝐴) ↔ (𝐹‘((𝐴 ∖ ∩ (ω ∖ 𝐴)) ∪ {∩ (ω ∖ 𝐴)})) = suc (𝐹‘𝐴)))
75	74	rspcev 3574	. 2 ⊢ ((((𝐴 ∖ ∩ (ω ∖ 𝐴)) ∪ {∩ (ω ∖ 𝐴)}) ∈ (𝒫 ω ∩ Fin) ∧ (𝐹‘((𝐴 ∖ ∩ (ω ∖ 𝐴)) ∪ {∩ (ω ∖ 𝐴)})) = suc (𝐹‘𝐴)) → ∃𝑏 ∈ (𝒫 ω ∩ Fin)(𝐹‘𝑏) = suc (𝐹‘𝐴))
76	23, 73, 75	syl2anc 584	1 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → ∃𝑏 ∈ (𝒫 ω ∩ Fin)(𝐹‘𝑏) = suc (𝐹‘𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 847   = wceq 1541   ∈ wcel 2113   ≠ wne 2930  ∃wrex 3058   ∖ cdif 3896   ∪ cun 3897   ∩ cin 3898   ⊆ wss 3899  ∅c0 4283  𝒫 cpw 4552  {csn 4578  ∩ cint 4900  ∪ ciun 4944   ↦ cmpt 5177   × cxp 5620  Ord word 6314  Oncon0 6315  suc csuc 6317  ‘cfv 6490  (class class class)co 7356  ωcom 7806   +_o coa 8392  Fincfn 8881  cardccrd 9845

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-int 4901  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-tr 5204  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-2o 8396  df-oadd 8399  df-er 8633  df-map 8763  df-en 8882  df-dom 8883  df-sdom 8884  df-fin 8885  df-dju 9811  df-card 9849

This theorem is referenced by:  ackbij1  10145