Theorem ackbij1lem18 10194

Description: Lemma for ackbij1 10195. (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 4091	. . . 4 ⊢ (𝐴 ∖ ∩ (ω ∖ 𝐴)) ⊆ 𝐴
2		ackbij.f	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦)))
3	2	ackbij1lem11 10187	. . . 4 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∖ ∩ (ω ∖ 𝐴)) ⊆ 𝐴) → (𝐴 ∖ ∩ (ω ∖ 𝐴)) ∈ (𝒫 ω ∩ Fin))
4	1, 3	mpan2 701	. . 3 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → (𝐴 ∖ ∩ (ω ∖ 𝐴)) ∈ (𝒫 ω ∩ Fin))
5		difss 4091	. . . . . . 7 ⊢ (ω ∖ 𝐴) ⊆ ω
6		omsson 7852	. . . . . . 7 ⊢ ω ⊆ On
7	5, 6	sstri 3947	. . . . . 6 ⊢ (ω ∖ 𝐴) ⊆ On
8		ominf 9210	. . . . . . . 8 ⊢ ¬ ω ∈ Fin
9		elinel2 4156	. . . . . . . 8 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → 𝐴 ∈ Fin)
10		difinf 9257	. . . . . . . 8 ⊢ ((¬ ω ∈ Fin ∧ 𝐴 ∈ Fin) → ¬ (ω ∖ 𝐴) ∈ Fin)
11	8, 9, 10	sylancr 596	. . . . . . 7 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → ¬ (ω ∖ 𝐴) ∈ Fin)
12		0fi 9025	. . . . . . . . 9 ⊢ ∅ ∈ Fin
13		eleq1 2852	. . . . . . . . 9 ⊢ ((ω ∖ 𝐴) = ∅ → ((ω ∖ 𝐴) ∈ Fin ↔ ∅ ∈ Fin))
14	12, 13	mpbiri 260	. . . . . . . 8 ⊢ ((ω ∖ 𝐴) = ∅ → (ω ∖ 𝐴) ∈ Fin)
15	14	necon3bi 2985	. . . . . . 7 ⊢ (¬ (ω ∖ 𝐴) ∈ Fin → (ω ∖ 𝐴) ≠ ∅)
16	11, 15	syl 17	. . . . . 6 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → (ω ∖ 𝐴) ≠ ∅)
17		onint 7775	. . . . . 6 ⊢ (((ω ∖ 𝐴) ⊆ On ∧ (ω ∖ 𝐴) ≠ ∅) → ∩ (ω ∖ 𝐴) ∈ (ω ∖ 𝐴))
18	7, 16, 17	sylancr 596	. . . . 5 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → ∩ (ω ∖ 𝐴) ∈ (ω ∖ 𝐴))
19	18	eldifad 3918	. . . 4 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → ∩ (ω ∖ 𝐴) ∈ ω)
20		ackbij1lem4 10180	. . . 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 10182	. . 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 593	. 2 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → ((𝐴 ∖ ∩ (ω ∖ 𝐴)) ∪ {∩ (ω ∖ 𝐴)}) ∈ (𝒫 ω ∩ Fin))
24	18	eldifbd 3919	. . . . . 6 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → ¬ ∩ (ω ∖ 𝐴) ∈ 𝐴)
25		disjsn 4672	. . . . . 6 ⊢ ((𝐴 ∩ {∩ (ω ∖ 𝐴)}) = ∅ ↔ ¬ ∩ (ω ∖ 𝐴) ∈ 𝐴)
26	24, 25	sylibr 236	. . . . 5 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → (𝐴 ∩ {∩ (ω ∖ 𝐴)}) = ∅)
27		ssdisj 4416	. . . . 5 ⊢ (((𝐴 ∖ ∩ (ω ∖ 𝐴)) ⊆ 𝐴 ∧ (𝐴 ∩ {∩ (ω ∖ 𝐴)}) = ∅) → ((𝐴 ∖ ∩ (ω ∖ 𝐴)) ∩ {∩ (ω ∖ 𝐴)}) = ∅)
28	1, 26, 27	sylancr 596	. . . 4 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → ((𝐴 ∖ ∩ (ω ∖ 𝐴)) ∩ {∩ (ω ∖ 𝐴)}) = ∅)
29	2	ackbij1lem9 10185	. . . 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 1392	. . 3 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → (𝐹‘((𝐴 ∖ ∩ (ω ∖ 𝐴)) ∪ {∩ (ω ∖ 𝐴)})) = ((𝐹‘(𝐴 ∖ ∩ (ω ∖ 𝐴))) +o (𝐹‘{∩ (ω ∖ 𝐴)})))
31	2	ackbij1lem14 10190	. . . . 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 7414	. . 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 10186	. . . . . . 7 ⊢ 𝐹:(𝒫 ω ∩ Fin)⟶ω
35	34	ffvelcdmi 7066	. . . . . 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 10179	. . . . . . 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 7066	. . . . . 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 8578	. . . . 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 593	. . . 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 4429	. . . . . . . 8 ⊢ ((𝐴 ∖ ∩ (ω ∖ 𝐴)) ∩ ∩ (ω ∖ 𝐴)) = ∅
44	43	a1i 11	. . . . . . 7 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → ((𝐴 ∖ ∩ (ω ∖ 𝐴)) ∩ ∩ (ω ∖ 𝐴)) = ∅)
45	2	ackbij1lem9 10185	. . . . . . 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 1392	. . . . . 6 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → (𝐹‘((𝐴 ∖ ∩ (ω ∖ 𝐴)) ∪ ∩ (ω ∖ 𝐴))) = ((𝐹‘(𝐴 ∖ ∩ (ω ∖ 𝐴))) +o (𝐹‘∩ (ω ∖ 𝐴))))
47		uncom 4113	. . . . . . . 8 ⊢ ((𝐴 ∖ ∩ (ω ∖ 𝐴)) ∪ ∩ (ω ∖ 𝐴)) = (∩ (ω ∖ 𝐴) ∪ (𝐴 ∖ ∩ (ω ∖ 𝐴)))
48		onnmin 7783	. . . . . . . . . . . . . . 15 ⊢ (((ω ∖ 𝐴) ⊆ On ∧ 𝑎 ∈ (ω ∖ 𝐴)) → ¬ 𝑎 ∈ ∩ (ω ∖ 𝐴))
49	7, 48	mpan 700	. . . . . . . . . . . . . 14 ⊢ (𝑎 ∈ (ω ∖ 𝐴) → ¬ 𝑎 ∈ ∩ (ω ∖ 𝐴))
50	49	con2i 139	. . . . . . . . . . . . 13 ⊢ (𝑎 ∈ ∩ (ω ∖ 𝐴) → ¬ 𝑎 ∈ (ω ∖ 𝐴))
51	50	adantl 485	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝑎 ∈ ∩ (ω ∖ 𝐴)) → ¬ 𝑎 ∈ (ω ∖ 𝐴))
52		ordom 7858	. . . . . . . . . . . . . . 15 ⊢ Ord ω
53		ordelss 6364	. . . . . . . . . . . . . . 15 ⊢ ((Ord ω ∧ ∩ (ω ∖ 𝐴) ∈ ω) → ∩ (ω ∖ 𝐴) ⊆ ω)
54	52, 19, 53	sylancr 596	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → ∩ (ω ∖ 𝐴) ⊆ ω)
55	54	sselda 3938	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝑎 ∈ ∩ (ω ∖ 𝐴)) → 𝑎 ∈ ω)
56		eldif 3916	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 ∈ (ω ∖ 𝐴) ↔ (𝑎 ∈ ω ∧ ¬ 𝑎 ∈ 𝐴))
57	56	simplbi2 504	. . . . . . . . . . . . . . 15 ⊢ (𝑎 ∈ ω → (¬ 𝑎 ∈ 𝐴 → 𝑎 ∈ (ω ∖ 𝐴)))
58	57	orrd 874	. . . . . . . . . . . . . 14 ⊢ (𝑎 ∈ ω → (𝑎 ∈ 𝐴 ∨ 𝑎 ∈ (ω ∖ 𝐴)))
59	58	orcomd 882	. . . . . . . . . . . . 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 899	. . . . . . . . . . . 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 416	. . . . . . . . . 10 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → (𝑎 ∈ ∩ (ω ∖ 𝐴) → 𝑎 ∈ 𝐴))
64	63	ssrdv 3944	. . . . . . . . 9 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → ∩ (ω ∖ 𝐴) ⊆ 𝐴)
65		undif 4438	. . . . . . . . 9 ⊢ (∩ (ω ∖ 𝐴) ⊆ 𝐴 ↔ (∩ (ω ∖ 𝐴) ∪ (𝐴 ∖ ∩ (ω ∖ 𝐴))) = 𝐴)
66	64, 65	sylib 220	. . . . . . . 8 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → (∩ (ω ∖ 𝐴) ∪ (𝐴 ∖ ∩ (ω ∖ 𝐴))) = 𝐴)
67	47, 66	eqtrid 2811	. . . . . . 7 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → ((𝐴 ∖ ∩ (ω ∖ 𝐴)) ∪ ∩ (ω ∖ 𝐴)) = 𝐴)
68	67	fveq2d 6873	. . . . . 6 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → (𝐹‘((𝐴 ∖ ∩ (ω ∖ 𝐴)) ∪ ∩ (ω ∖ 𝐴))) = (𝐹‘𝐴))
69	46, 68	eqtr3d 2801	. . . . 5 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → ((𝐹‘(𝐴 ∖ ∩ (ω ∖ 𝐴))) +o (𝐹‘∩ (ω ∖ 𝐴))) = (𝐹‘𝐴))
70		suceq 6416	. . . . 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 2799	. . 3 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → ((𝐹‘(𝐴 ∖ ∩ (ω ∖ 𝐴))) +o suc (𝐹‘∩ (ω ∖ 𝐴))) = suc (𝐹‘𝐴))
73	30, 33, 72	3eqtrd 2803	. 2 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → (𝐹‘((𝐴 ∖ ∩ (ω ∖ 𝐴)) ∪ {∩ (ω ∖ 𝐴)})) = suc (𝐹‘𝐴))
74		fveqeq2 6878	. . 3 ⊢ (𝑏 = ((𝐴 ∖ ∩ (ω ∖ 𝐴)) ∪ {∩ (ω ∖ 𝐴)}) → ((𝐹‘𝑏) = suc (𝐹‘𝐴) ↔ (𝐹‘((𝐴 ∖ ∩ (ω ∖ 𝐴)) ∪ {∩ (ω ∖ 𝐴)})) = suc (𝐹‘𝐴)))
75	74	rspcev 3583	. 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 593	1 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → ∃𝑏 ∈ (𝒫 ω ∩ Fin)(𝐹‘𝑏) = suc (𝐹‘𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   ∨ wo 858   = wceq 1562   ∈ wcel 2144   ≠ wne 2959  ∃wrex 3088   ∖ cdif 3903   ∪ cun 3904   ∩ cin 3905   ⊆ wss 3906  ∅c0 4287  𝒫 cpw 4557  {csn 4584  ∩ cint 4907  ∪ ciun 4951   ↦ cmpt 5183   × cxp 5647  Ord word 6347  Oncon0 6348  suc csuc 6350  ‘cfv 6523  (class class class)co 7398  ωcom 7848   +_o coa 8436  Fincfn 8929  cardccrd 9895

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-sep 5248  ax-nul 5258  ax-pow 5324  ax-pr 5392  ax-un 7720

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3079  df-rex 3089  df-reu 3370  df-rab 3417  df-v 3458  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5544  df-eprel 5549  df-po 5557  df-so 5558  df-fr 5602  df-we 5604  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-pred 6290  df-ord 6351  df-on 6352  df-lim 6353  df-suc 6354  df-iota 6479  df-fun 6525  df-fn 6526  df-f 6527  df-f1 6528  df-fo 6529  df-f1o 6530  df-fv 6531  df-ov 7401  df-oprab 7402  df-mpo 7403  df-om 7849  df-1st 7972  df-2nd 7973  df-frecs 8264  df-wrecs 8295  df-recs 8344  df-rdg 8383  df-1o 8439  df-2o 8440  df-oadd 8443  df-er 8680  df-map 8812  df-en 8930  df-dom 8931  df-sdom 8932  df-fin 8933  df-dju 9861  df-card 9899

This theorem is referenced by:  ackbij1  10195