Theorem ackbij1lem18 10221

Description: Lemma for ackbij1 10222. (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 4098	. . . 4 ⊢ (𝐴 ∖ ∩ (ω ∖ 𝐴)) ⊆ 𝐴
2		ackbij.f	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦)))
3	2	ackbij1lem11 10214	. . . 4 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∖ ∩ (ω ∖ 𝐴)) ⊆ 𝐴) → (𝐴 ∖ ∩ (ω ∖ 𝐴)) ∈ (𝒫 ω ∩ Fin))
4	1, 3	mpan2 703	. . 3 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → (𝐴 ∖ ∩ (ω ∖ 𝐴)) ∈ (𝒫 ω ∩ Fin))
5		difss 4098	. . . . . . 7 ⊢ (ω ∖ 𝐴) ⊆ ω
6		omsson 7868	. . . . . . 7 ⊢ ω ⊆ On
7	5, 6	sstri 3954	. . . . . 6 ⊢ (ω ∖ 𝐴) ⊆ On
8		ominf 9226	. . . . . . . 8 ⊢ ¬ ω ∈ Fin
9		elinel2 4163	. . . . . . . 8 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → 𝐴 ∈ Fin)
10		difinf 9273	. . . . . . . 8 ⊢ ((¬ ω ∈ Fin ∧ 𝐴 ∈ Fin) → ¬ (ω ∖ 𝐴) ∈ Fin)
11	8, 9, 10	sylancr 598	. . . . . . 7 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → ¬ (ω ∖ 𝐴) ∈ Fin)
12		0fi 9041	. . . . . . . . 9 ⊢ ∅ ∈ Fin
13		eleq1 2857	. . . . . . . . 9 ⊢ ((ω ∖ 𝐴) = ∅ → ((ω ∖ 𝐴) ∈ Fin ↔ ∅ ∈ Fin))
14	12, 13	mpbiri 261	. . . . . . . 8 ⊢ ((ω ∖ 𝐴) = ∅ → (ω ∖ 𝐴) ∈ Fin)
15	14	necon3bi 2990	. . . . . . 7 ⊢ (¬ (ω ∖ 𝐴) ∈ Fin → (ω ∖ 𝐴) ≠ ∅)
16	11, 15	syl 18	. . . . . 6 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → (ω ∖ 𝐴) ≠ ∅)
17		onint 7791	. . . . . 6 ⊢ (((ω ∖ 𝐴) ⊆ On ∧ (ω ∖ 𝐴) ≠ ∅) → ∩ (ω ∖ 𝐴) ∈ (ω ∖ 𝐴))
18	7, 16, 17	sylancr 598	. . . . 5 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → ∩ (ω ∖ 𝐴) ∈ (ω ∖ 𝐴))
19	18	eldifad 3925	. . . 4 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → ∩ (ω ∖ 𝐴) ∈ ω)
20		ackbij1lem4 10207	. . . 4 ⊢ (∩ (ω ∖ 𝐴) ∈ ω → {∩ (ω ∖ 𝐴)} ∈ (𝒫 ω ∩ Fin))
21	19, 20	syl 18	. . 3 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → {∩ (ω ∖ 𝐴)} ∈ (𝒫 ω ∩ Fin))
22		ackbij1lem6 10209	. . 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 595	. 2 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → ((𝐴 ∖ ∩ (ω ∖ 𝐴)) ∪ {∩ (ω ∖ 𝐴)}) ∈ (𝒫 ω ∩ Fin))
24	18	eldifbd 3926	. . . . . 6 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → ¬ ∩ (ω ∖ 𝐴) ∈ 𝐴)
25		disjsn 4682	. . . . . 6 ⊢ ((𝐴 ∩ {∩ (ω ∖ 𝐴)}) = ∅ ↔ ¬ ∩ (ω ∖ 𝐴) ∈ 𝐴)
26	24, 25	sylibr 237	. . . . 5 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → (𝐴 ∩ {∩ (ω ∖ 𝐴)}) = ∅)
27		ssdisj 4426	. . . . 5 ⊢ (((𝐴 ∖ ∩ (ω ∖ 𝐴)) ⊆ 𝐴 ∧ (𝐴 ∩ {∩ (ω ∖ 𝐴)}) = ∅) → ((𝐴 ∖ ∩ (ω ∖ 𝐴)) ∩ {∩ (ω ∖ 𝐴)}) = ∅)
28	1, 26, 27	sylancr 598	. . . 4 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → ((𝐴 ∖ ∩ (ω ∖ 𝐴)) ∩ {∩ (ω ∖ 𝐴)}) = ∅)
29	2	ackbij1lem9 10212	. . . 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 1396	. . 3 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → (𝐹‘((𝐴 ∖ ∩ (ω ∖ 𝐴)) ∪ {∩ (ω ∖ 𝐴)})) = ((𝐹‘(𝐴 ∖ ∩ (ω ∖ 𝐴))) +o (𝐹‘{∩ (ω ∖ 𝐴)})))
31	2	ackbij1lem14 10217	. . . . 5 ⊢ (∩ (ω ∖ 𝐴) ∈ ω → (𝐹‘{∩ (ω ∖ 𝐴)}) = suc (𝐹‘∩ (ω ∖ 𝐴)))
32	19, 31	syl 18	. . . 4 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → (𝐹‘{∩ (ω ∖ 𝐴)}) = suc (𝐹‘∩ (ω ∖ 𝐴)))
33	32	oveq2d 7429	. . 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 10213	. . . . . . 7 ⊢ 𝐹:(𝒫 ω ∩ Fin)⟶ω
35	34	ffvelcdmi 7081	. . . . . 6 ⊢ ((𝐴 ∖ ∩ (ω ∖ 𝐴)) ∈ (𝒫 ω ∩ Fin) → (𝐹‘(𝐴 ∖ ∩ (ω ∖ 𝐴))) ∈ ω)
36	4, 35	syl 18	. . . . 5 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → (𝐹‘(𝐴 ∖ ∩ (ω ∖ 𝐴))) ∈ ω)
37		ackbij1lem3 10206	. . . . . . 7 ⊢ (∩ (ω ∖ 𝐴) ∈ ω → ∩ (ω ∖ 𝐴) ∈ (𝒫 ω ∩ Fin))
38	19, 37	syl 18	. . . . . 6 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → ∩ (ω ∖ 𝐴) ∈ (𝒫 ω ∩ Fin))
39	34	ffvelcdmi 7081	. . . . . 6 ⊢ (∩ (ω ∖ 𝐴) ∈ (𝒫 ω ∩ Fin) → (𝐹‘∩ (ω ∖ 𝐴)) ∈ ω)
40	38, 39	syl 18	. . . . 5 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → (𝐹‘∩ (ω ∖ 𝐴)) ∈ ω)
41		nnasuc 8594	. . . . 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 595	. . . 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 4439	. . . . . . . 8 ⊢ ((𝐴 ∖ ∩ (ω ∖ 𝐴)) ∩ ∩ (ω ∖ 𝐴)) = ∅
44	43	a1i 11	. . . . . . 7 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → ((𝐴 ∖ ∩ (ω ∖ 𝐴)) ∩ ∩ (ω ∖ 𝐴)) = ∅)
45	2	ackbij1lem9 10212	. . . . . . 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 1396	. . . . . 6 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → (𝐹‘((𝐴 ∖ ∩ (ω ∖ 𝐴)) ∪ ∩ (ω ∖ 𝐴))) = ((𝐹‘(𝐴 ∖ ∩ (ω ∖ 𝐴))) +o (𝐹‘∩ (ω ∖ 𝐴))))
47		uncom 4120	. . . . . . . 8 ⊢ ((𝐴 ∖ ∩ (ω ∖ 𝐴)) ∪ ∩ (ω ∖ 𝐴)) = (∩ (ω ∖ 𝐴) ∪ (𝐴 ∖ ∩ (ω ∖ 𝐴)))
48		onnmin 7799	. . . . . . . . . . . . . . 15 ⊢ (((ω ∖ 𝐴) ⊆ On ∧ 𝑎 ∈ (ω ∖ 𝐴)) → ¬ 𝑎 ∈ ∩ (ω ∖ 𝐴))
49	7, 48	mpan 702	. . . . . . . . . . . . . 14 ⊢ (𝑎 ∈ (ω ∖ 𝐴) → ¬ 𝑎 ∈ ∩ (ω ∖ 𝐴))
50	49	con2i 140	. . . . . . . . . . . . 13 ⊢ (𝑎 ∈ ∩ (ω ∖ 𝐴) → ¬ 𝑎 ∈ (ω ∖ 𝐴))
51	50	adantl 486	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝑎 ∈ ∩ (ω ∖ 𝐴)) → ¬ 𝑎 ∈ (ω ∖ 𝐴))
52		ordom 7874	. . . . . . . . . . . . . . 15 ⊢ Ord ω
53		ordelss 6379	. . . . . . . . . . . . . . 15 ⊢ ((Ord ω ∧ ∩ (ω ∖ 𝐴) ∈ ω) → ∩ (ω ∖ 𝐴) ⊆ ω)
54	52, 19, 53	sylancr 598	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → ∩ (ω ∖ 𝐴) ⊆ ω)
55	54	sselda 3945	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝑎 ∈ ∩ (ω ∖ 𝐴)) → 𝑎 ∈ ω)
56		eldif 3923	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 ∈ (ω ∖ 𝐴) ↔ (𝑎 ∈ ω ∧ ¬ 𝑎 ∈ 𝐴))
57	56	simplbi2 505	. . . . . . . . . . . . . . 15 ⊢ (𝑎 ∈ ω → (¬ 𝑎 ∈ 𝐴 → 𝑎 ∈ (ω ∖ 𝐴)))
58	57	orrd 876	. . . . . . . . . . . . . 14 ⊢ (𝑎 ∈ ω → (𝑎 ∈ 𝐴 ∨ 𝑎 ∈ (ω ∖ 𝐴)))
59	58	orcomd 884	. . . . . . . . . . . . 13 ⊢ (𝑎 ∈ ω → (𝑎 ∈ (ω ∖ 𝐴) ∨ 𝑎 ∈ 𝐴))
60	55, 59	syl 18	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝑎 ∈ ∩ (ω ∖ 𝐴)) → (𝑎 ∈ (ω ∖ 𝐴) ∨ 𝑎 ∈ 𝐴))
61		orel1 901	. . . . . . . . . . . 12 ⊢ (¬ 𝑎 ∈ (ω ∖ 𝐴) → ((𝑎 ∈ (ω ∖ 𝐴) ∨ 𝑎 ∈ 𝐴) → 𝑎 ∈ 𝐴))
62	51, 60, 61	sylc 66	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝑎 ∈ ∩ (ω ∖ 𝐴)) → 𝑎 ∈ 𝐴)
63	62	ex 417	. . . . . . . . . 10 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → (𝑎 ∈ ∩ (ω ∖ 𝐴) → 𝑎 ∈ 𝐴))
64	63	ssrdv 3951	. . . . . . . . 9 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → ∩ (ω ∖ 𝐴) ⊆ 𝐴)
65		undif 4448	. . . . . . . . 9 ⊢ (∩ (ω ∖ 𝐴) ⊆ 𝐴 ↔ (∩ (ω ∖ 𝐴) ∪ (𝐴 ∖ ∩ (ω ∖ 𝐴))) = 𝐴)
66	64, 65	sylib 221	. . . . . . . 8 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → (∩ (ω ∖ 𝐴) ∪ (𝐴 ∖ ∩ (ω ∖ 𝐴))) = 𝐴)
67	47, 66	eqtrid 2816	. . . . . . 7 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → ((𝐴 ∖ ∩ (ω ∖ 𝐴)) ∪ ∩ (ω ∖ 𝐴)) = 𝐴)
68	67	fveq2d 6888	. . . . . 6 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → (𝐹‘((𝐴 ∖ ∩ (ω ∖ 𝐴)) ∪ ∩ (ω ∖ 𝐴))) = (𝐹‘𝐴))
69	46, 68	eqtr3d 2806	. . . . 5 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → ((𝐹‘(𝐴 ∖ ∩ (ω ∖ 𝐴))) +o (𝐹‘∩ (ω ∖ 𝐴))) = (𝐹‘𝐴))
70		suceq 6432	. . . . 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 18	. . . 4 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → suc ((𝐹‘(𝐴 ∖ ∩ (ω ∖ 𝐴))) +o (𝐹‘∩ (ω ∖ 𝐴))) = suc (𝐹‘𝐴))
72	42, 71	eqtrd 2804	. . 3 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → ((𝐹‘(𝐴 ∖ ∩ (ω ∖ 𝐴))) +o suc (𝐹‘∩ (ω ∖ 𝐴))) = suc (𝐹‘𝐴))
73	30, 33, 72	3eqtrd 2808	. 2 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → (𝐹‘((𝐴 ∖ ∩ (ω ∖ 𝐴)) ∪ {∩ (ω ∖ 𝐴)})) = suc (𝐹‘𝐴))
74		fveqeq2 6893	. . 3 ⊢ (𝑏 = ((𝐴 ∖ ∩ (ω ∖ 𝐴)) ∪ {∩ (ω ∖ 𝐴)}) → ((𝐹‘𝑏) = suc (𝐹‘𝐴) ↔ (𝐹‘((𝐴 ∖ ∩ (ω ∖ 𝐴)) ∪ {∩ (ω ∖ 𝐴)})) = suc (𝐹‘𝐴)))
75	74	rspcev 3590	. 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 595	1 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → ∃𝑏 ∈ (𝒫 ω ∩ Fin)(𝐹‘𝑏) = suc (𝐹‘𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   ∨ wo 860   = wceq 1567   ∈ wcel 2149   ≠ wne 2964  ∃wrex 3095   ∖ cdif 3910   ∪ cun 3911   ∩ cin 3912   ⊆ wss 3913  ∅c0 4294  𝒫 cpw 4567  {csn 4594  ∩ cint 4916  ∪ ciun 4960   ↦ cmpt 5196   × cxp 5662  Ord word 6362  Oncon0 6363  suc csuc 6365  ‘cfv 6539  (class class class)co 7413  ωcom 7864   +_o coa 8452  Fincfn 8945  cardccrd 9923

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5273  ax-pow 5339  ax-pr 5407  ax-un 7735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-int 4917  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5559  df-eprel 5564  df-po 5572  df-so 5573  df-fr 5617  df-we 5619  df-xp 5670  df-rel 5671  df-cnv 5672  df-co 5673  df-dm 5674  df-rn 5675  df-res 5676  df-ima 5677  df-pred 6305  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6495  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7865  df-1st 7988  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-rdg 8399  df-1o 8455  df-2o 8456  df-oadd 8459  df-er 8696  df-map 8828  df-en 8946  df-dom 8947  df-sdom 8948  df-fin 8949  df-dju 9889  df-card 9927

This theorem is referenced by:  ackbij1  10222