Theorem fin1a2lem7 10408

Description: Lemma for fin1a2 10417. Split a III-infinite set in two pieces. (Contributed by Stefan O'Rear, 7-Nov-2014.)

Hypotheses

Ref	Expression
fin1a2lem.b	⊢ 𝐸 = (𝑥 ∈ ω ↦ (2o ·o 𝑥))
fin1a2lem.aa	⊢ 𝑆 = (𝑥 ∈ On ↦ suc 𝑥)

Assertion

Ref	Expression
fin1a2lem7	⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝒫 𝐴(𝑦 ∈ FinIII ∨ (𝐴 ∖ 𝑦) ∈ FinIII)) → 𝐴 ∈ FinIII)

Distinct variable groups:   𝑦,𝐴   𝑦,𝐸

Allowed substitution hints:   𝐴(𝑥)   𝑆(𝑥, 𝑦)   𝐸(𝑥)   𝑉(𝑥, 𝑦)

Proof of Theorem fin1a2lem7

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		peano1 7894	. . . . . 6 ⊢ ∅ ∈ ω
2		ne0i 4297	. . . . . 6 ⊢ (∅ ∈ ω → ω ≠ ∅)
3		brwdomn0 9541	. . . . . 6 ⊢ (ω ≠ ∅ → (ω ≼* 𝐴 ↔ ∃𝑓 𝑓:𝐴–onto→ω))
4	1, 2, 3	mp2b 10	. . . . 5 ⊢ (ω ≼* 𝐴 ↔ ∃𝑓 𝑓:𝐴–onto→ω)
5		vex 3462	. . . . . . . . . 10 ⊢ 𝑓 ∈ V
6		fof 6799	. . . . . . . . . 10 ⊢ (𝑓:𝐴–onto→ω → 𝑓:𝐴⟶ω)
7		dmfex 7911	. . . . . . . . . 10 ⊢ ((𝑓 ∈ V ∧ 𝑓:𝐴⟶ω) → 𝐴 ∈ V)
8	5, 6, 7	sylancr 599	. . . . . . . . 9 ⊢ (𝑓:𝐴–onto→ω → 𝐴 ∈ V)
9		cnvimass 6089	. . . . . . . . . 10 ⊢ (◡𝑓 “ ran 𝐸) ⊆ dom 𝑓
10	9, 6	fssdm 6732	. . . . . . . . 9 ⊢ (𝑓:𝐴–onto→ω → (◡𝑓 “ ran 𝐸) ⊆ 𝐴)
11	8, 10	sselpwd 5304	. . . . . . . 8 ⊢ (𝑓:𝐴–onto→ω → (◡𝑓 “ ran 𝐸) ∈ 𝒫 𝐴)
12		fin1a2lem.b	. . . . . . . . . . . . . 14 ⊢ 𝐸 = (𝑥 ∈ ω ↦ (2o ·o 𝑥))
13	12	fin1a2lem4 10405	. . . . . . . . . . . . 13 ⊢ 𝐸:ω–1-1→ω
14		f1cnv 6852	. . . . . . . . . . . . 13 ⊢ (𝐸:ω–1-1→ω → ◡𝐸:ran 𝐸–1-1-onto→ω)
15		f1ofo 6835	. . . . . . . . . . . . 13 ⊢ (◡𝐸:ran 𝐸–1-1-onto→ω → ◡𝐸:ran 𝐸–onto→ω)
16	13, 14, 15	mp2b 10	. . . . . . . . . . . 12 ⊢ ◡𝐸:ran 𝐸–onto→ω
17		fofun 6800	. . . . . . . . . . . 12 ⊢ (◡𝐸:ran 𝐸–onto→ω → Fun ◡𝐸)
18	16, 17	ax-mp 5	. . . . . . . . . . 11 ⊢ Fun ◡𝐸
19	5	resex 6033	. . . . . . . . . . 11 ⊢ (𝑓 ↾ (◡𝑓 “ ran 𝐸)) ∈ V
20		cofunexg 7955	. . . . . . . . . . 11 ⊢ ((Fun ◡𝐸 ∧ (𝑓 ↾ (◡𝑓 “ ran 𝐸)) ∈ V) → (◡𝐸 ∘ (𝑓 ↾ (◡𝑓 “ ran 𝐸))) ∈ V)
21	18, 19, 20	mp2an 705	. . . . . . . . . 10 ⊢ (◡𝐸 ∘ (𝑓 ↾ (◡𝑓 “ ran 𝐸))) ∈ V
22		fofun 6800	. . . . . . . . . . . . 13 ⊢ (𝑓:𝐴–onto→ω → Fun 𝑓)
23		fores 6809	. . . . . . . . . . . . 13 ⊢ ((Fun 𝑓 ∧ (◡𝑓 “ ran 𝐸) ⊆ dom 𝑓) → (𝑓 ↾ (◡𝑓 “ ran 𝐸)):(◡𝑓 “ ran 𝐸)–onto→(𝑓 “ (◡𝑓 “ ran 𝐸)))
24	22, 9, 23	sylancl 598	. . . . . . . . . . . 12 ⊢ (𝑓:𝐴–onto→ω → (𝑓 ↾ (◡𝑓 “ ran 𝐸)):(◡𝑓 “ ran 𝐸)–onto→(𝑓 “ (◡𝑓 “ ran 𝐸)))
25		f1f 6781	. . . . . . . . . . . . . . 15 ⊢ (𝐸:ω–1-1→ω → 𝐸:ω⟶ω)
26		frn 6720	. . . . . . . . . . . . . . 15 ⊢ (𝐸:ω⟶ω → ran 𝐸 ⊆ ω)
27	13, 25, 26	mp2b 10	. . . . . . . . . . . . . 14 ⊢ ran 𝐸 ⊆ ω
28		foimacnv 6845	. . . . . . . . . . . . . 14 ⊢ ((𝑓:𝐴–onto→ω ∧ ran 𝐸 ⊆ ω) → (𝑓 “ (◡𝑓 “ ran 𝐸)) = ran 𝐸)
29	27, 28	mpan2 704	. . . . . . . . . . . . 13 ⊢ (𝑓:𝐴–onto→ω → (𝑓 “ (◡𝑓 “ ran 𝐸)) = ran 𝐸)
30		foeq3 6797	. . . . . . . . . . . . 13 ⊢ ((𝑓 “ (◡𝑓 “ ran 𝐸)) = ran 𝐸 → ((𝑓 ↾ (◡𝑓 “ ran 𝐸)):(◡𝑓 “ ran 𝐸)–onto→(𝑓 “ (◡𝑓 “ ran 𝐸)) ↔ (𝑓 ↾ (◡𝑓 “ ran 𝐸)):(◡𝑓 “ ran 𝐸)–onto→ran 𝐸))
31	29, 30	syl 18	. . . . . . . . . . . 12 ⊢ (𝑓:𝐴–onto→ω → ((𝑓 ↾ (◡𝑓 “ ran 𝐸)):(◡𝑓 “ ran 𝐸)–onto→(𝑓 “ (◡𝑓 “ ran 𝐸)) ↔ (𝑓 ↾ (◡𝑓 “ ran 𝐸)):(◡𝑓 “ ran 𝐸)–onto→ran 𝐸))
32	24, 31	mpbid 235	. . . . . . . . . . 11 ⊢ (𝑓:𝐴–onto→ω → (𝑓 ↾ (◡𝑓 “ ran 𝐸)):(◡𝑓 “ ran 𝐸)–onto→ran 𝐸)
33		foco 6813	. . . . . . . . . . 11 ⊢ ((◡𝐸:ran 𝐸–onto→ω ∧ (𝑓 ↾ (◡𝑓 “ ran 𝐸)):(◡𝑓 “ ran 𝐸)–onto→ran 𝐸) → (◡𝐸 ∘ (𝑓 ↾ (◡𝑓 “ ran 𝐸))):(◡𝑓 “ ran 𝐸)–onto→ω)
34	16, 32, 33	sylancr 599	. . . . . . . . . 10 ⊢ (𝑓:𝐴–onto→ω → (◡𝐸 ∘ (𝑓 ↾ (◡𝑓 “ ran 𝐸))):(◡𝑓 “ ran 𝐸)–onto→ω)
35		fowdom 9543	. . . . . . . . . 10 ⊢ (((◡𝐸 ∘ (𝑓 ↾ (◡𝑓 “ ran 𝐸))) ∈ V ∧ (◡𝐸 ∘ (𝑓 ↾ (◡𝑓 “ ran 𝐸))):(◡𝑓 “ ran 𝐸)–onto→ω) → ω ≼* (◡𝑓 “ ran 𝐸))
36	21, 34, 35	sylancr 599	. . . . . . . . 9 ⊢ (𝑓:𝐴–onto→ω → ω ≼* (◡𝑓 “ ran 𝐸))
37	5	cnvex 7931	. . . . . . . . . . . 12 ⊢ ◡𝑓 ∈ V
38	37	imaex 7920	. . . . . . . . . . 11 ⊢ (◡𝑓 “ ran 𝐸) ∈ V
39		isfin3-2 10369	. . . . . . . . . . 11 ⊢ ((◡𝑓 “ ran 𝐸) ∈ V → ((◡𝑓 “ ran 𝐸) ∈ FinIII ↔ ¬ ω ≼* (◡𝑓 “ ran 𝐸)))
40	38, 39	ax-mp 5	. . . . . . . . . 10 ⊢ ((◡𝑓 “ ran 𝐸) ∈ FinIII ↔ ¬ ω ≼* (◡𝑓 “ ran 𝐸))
41	40	con2bii 360	. . . . . . . . 9 ⊢ (ω ≼* (◡𝑓 “ ran 𝐸) ↔ ¬ (◡𝑓 “ ran 𝐸) ∈ FinIII)
42	36, 41	sylib 221	. . . . . . . 8 ⊢ (𝑓:𝐴–onto→ω → ¬ (◡𝑓 “ ran 𝐸) ∈ FinIII)
43		fin1a2lem.aa	. . . . . . . . . . . . . . 15 ⊢ 𝑆 = (𝑥 ∈ On ↦ suc 𝑥)
44	12, 43	fin1a2lem6 10407	. . . . . . . . . . . . . 14 ⊢ (𝑆 ↾ ran 𝐸):ran 𝐸–1-1-onto→(ω ∖ ran 𝐸)
45		f1ocnv 6840	. . . . . . . . . . . . . 14 ⊢ ((𝑆 ↾ ran 𝐸):ran 𝐸–1-1-onto→(ω ∖ ran 𝐸) → ◡(𝑆 ↾ ran 𝐸):(ω ∖ ran 𝐸)–1-1-onto→ran 𝐸)
46		f1ofo 6835	. . . . . . . . . . . . . 14 ⊢ (◡(𝑆 ↾ ran 𝐸):(ω ∖ ran 𝐸)–1-1-onto→ran 𝐸 → ◡(𝑆 ↾ ran 𝐸):(ω ∖ ran 𝐸)–onto→ran 𝐸)
47	44, 45, 46	mp2b 10	. . . . . . . . . . . . 13 ⊢ ◡(𝑆 ↾ ran 𝐸):(ω ∖ ran 𝐸)–onto→ran 𝐸
48		foco 6813	. . . . . . . . . . . . 13 ⊢ ((◡𝐸:ran 𝐸–onto→ω ∧ ◡(𝑆 ↾ ran 𝐸):(ω ∖ ran 𝐸)–onto→ran 𝐸) → (◡𝐸 ∘ ◡(𝑆 ↾ ran 𝐸)):(ω ∖ ran 𝐸)–onto→ω)
49	16, 47, 48	mp2an 705	. . . . . . . . . . . 12 ⊢ (◡𝐸 ∘ ◡(𝑆 ↾ ran 𝐸)):(ω ∖ ran 𝐸)–onto→ω
50		fofun 6800	. . . . . . . . . . . 12 ⊢ ((◡𝐸 ∘ ◡(𝑆 ↾ ran 𝐸)):(ω ∖ ran 𝐸)–onto→ω → Fun (◡𝐸 ∘ ◡(𝑆 ↾ ran 𝐸)))
51	49, 50	ax-mp 5	. . . . . . . . . . 11 ⊢ Fun (◡𝐸 ∘ ◡(𝑆 ↾ ran 𝐸))
52	5	resex 6033	. . . . . . . . . . 11 ⊢ (𝑓 ↾ (𝐴 ∖ (◡𝑓 “ ran 𝐸))) ∈ V
53		cofunexg 7955	. . . . . . . . . . 11 ⊢ ((Fun (◡𝐸 ∘ ◡(𝑆 ↾ ran 𝐸)) ∧ (𝑓 ↾ (𝐴 ∖ (◡𝑓 “ ran 𝐸))) ∈ V) → ((◡𝐸 ∘ ◡(𝑆 ↾ ran 𝐸)) ∘ (𝑓 ↾ (𝐴 ∖ (◡𝑓 “ ran 𝐸)))) ∈ V)
54	51, 52, 53	mp2an 705	. . . . . . . . . 10 ⊢ ((◡𝐸 ∘ ◡(𝑆 ↾ ran 𝐸)) ∘ (𝑓 ↾ (𝐴 ∖ (◡𝑓 “ ran 𝐸)))) ∈ V
55		difss 4093	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∖ (◡𝑓 “ ran 𝐸)) ⊆ 𝐴
56	6	fdmd 6723	. . . . . . . . . . . . . 14 ⊢ (𝑓:𝐴–onto→ω → dom 𝑓 = 𝐴)
57	55, 56	sseqtrrid 3983	. . . . . . . . . . . . 13 ⊢ (𝑓:𝐴–onto→ω → (𝐴 ∖ (◡𝑓 “ ran 𝐸)) ⊆ dom 𝑓)
58		fores 6809	. . . . . . . . . . . . 13 ⊢ ((Fun 𝑓 ∧ (𝐴 ∖ (◡𝑓 “ ran 𝐸)) ⊆ dom 𝑓) → (𝑓 ↾ (𝐴 ∖ (◡𝑓 “ ran 𝐸))):(𝐴 ∖ (◡𝑓 “ ran 𝐸))–onto→(𝑓 “ (𝐴 ∖ (◡𝑓 “ ran 𝐸))))
59	22, 57, 58	syl2anc 596	. . . . . . . . . . . 12 ⊢ (𝑓:𝐴–onto→ω → (𝑓 ↾ (𝐴 ∖ (◡𝑓 “ ran 𝐸))):(𝐴 ∖ (◡𝑓 “ ran 𝐸))–onto→(𝑓 “ (𝐴 ∖ (◡𝑓 “ ran 𝐸))))
60		funcnvcnv 6610	. . . . . . . . . . . . . . . 16 ⊢ (Fun 𝑓 → Fun ◡◡𝑓)
61		imadif 6627	. . . . . . . . . . . . . . . 16 ⊢ (Fun ◡◡𝑓 → (◡𝑓 “ (ω ∖ ran 𝐸)) = ((◡𝑓 “ ω) ∖ (◡𝑓 “ ran 𝐸)))
62	22, 60, 61	3syl 19	. . . . . . . . . . . . . . 15 ⊢ (𝑓:𝐴–onto→ω → (◡𝑓 “ (ω ∖ ran 𝐸)) = ((◡𝑓 “ ω) ∖ (◡𝑓 “ ran 𝐸)))
63	62	imaeq2d 6067	. . . . . . . . . . . . . 14 ⊢ (𝑓:𝐴–onto→ω → (𝑓 “ (◡𝑓 “ (ω ∖ ran 𝐸))) = (𝑓 “ ((◡𝑓 “ ω) ∖ (◡𝑓 “ ran 𝐸))))
64		difss 4093	. . . . . . . . . . . . . . 15 ⊢ (ω ∖ ran 𝐸) ⊆ ω
65		foimacnv 6845	. . . . . . . . . . . . . . 15 ⊢ ((𝑓:𝐴–onto→ω ∧ (ω ∖ ran 𝐸) ⊆ ω) → (𝑓 “ (◡𝑓 “ (ω ∖ ran 𝐸))) = (ω ∖ ran 𝐸))
66	64, 65	mpan2 704	. . . . . . . . . . . . . 14 ⊢ (𝑓:𝐴–onto→ω → (𝑓 “ (◡𝑓 “ (ω ∖ ran 𝐸))) = (ω ∖ ran 𝐸))
67		fimacnv 6735	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓:𝐴⟶ω → (◡𝑓 “ ω) = 𝐴)
68	6, 67	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝑓:𝐴–onto→ω → (◡𝑓 “ ω) = 𝐴)
69	68	difeq1d 4083	. . . . . . . . . . . . . . 15 ⊢ (𝑓:𝐴–onto→ω → ((◡𝑓 “ ω) ∖ (◡𝑓 “ ran 𝐸)) = (𝐴 ∖ (◡𝑓 “ ran 𝐸)))
70	69	imaeq2d 6067	. . . . . . . . . . . . . 14 ⊢ (𝑓:𝐴–onto→ω → (𝑓 “ ((◡𝑓 “ ω) ∖ (◡𝑓 “ ran 𝐸))) = (𝑓 “ (𝐴 ∖ (◡𝑓 “ ran 𝐸))))
71	63, 66, 70	3eqtr3rd 2810	. . . . . . . . . . . . 13 ⊢ (𝑓:𝐴–onto→ω → (𝑓 “ (𝐴 ∖ (◡𝑓 “ ran 𝐸))) = (ω ∖ ran 𝐸))
72		foeq3 6797	. . . . . . . . . . . . 13 ⊢ ((𝑓 “ (𝐴 ∖ (◡𝑓 “ ran 𝐸))) = (ω ∖ ran 𝐸) → ((𝑓 ↾ (𝐴 ∖ (◡𝑓 “ ran 𝐸))):(𝐴 ∖ (◡𝑓 “ ran 𝐸))–onto→(𝑓 “ (𝐴 ∖ (◡𝑓 “ ran 𝐸))) ↔ (𝑓 ↾ (𝐴 ∖ (◡𝑓 “ ran 𝐸))):(𝐴 ∖ (◡𝑓 “ ran 𝐸))–onto→(ω ∖ ran 𝐸)))
73	71, 72	syl 18	. . . . . . . . . . . 12 ⊢ (𝑓:𝐴–onto→ω → ((𝑓 ↾ (𝐴 ∖ (◡𝑓 “ ran 𝐸))):(𝐴 ∖ (◡𝑓 “ ran 𝐸))–onto→(𝑓 “ (𝐴 ∖ (◡𝑓 “ ran 𝐸))) ↔ (𝑓 ↾ (𝐴 ∖ (◡𝑓 “ ran 𝐸))):(𝐴 ∖ (◡𝑓 “ ran 𝐸))–onto→(ω ∖ ran 𝐸)))
74	59, 73	mpbid 235	. . . . . . . . . . 11 ⊢ (𝑓:𝐴–onto→ω → (𝑓 ↾ (𝐴 ∖ (◡𝑓 “ ran 𝐸))):(𝐴 ∖ (◡𝑓 “ ran 𝐸))–onto→(ω ∖ ran 𝐸))
75		foco 6813	. . . . . . . . . . 11 ⊢ (((◡𝐸 ∘ ◡(𝑆 ↾ ran 𝐸)):(ω ∖ ran 𝐸)–onto→ω ∧ (𝑓 ↾ (𝐴 ∖ (◡𝑓 “ ran 𝐸))):(𝐴 ∖ (◡𝑓 “ ran 𝐸))–onto→(ω ∖ ran 𝐸)) → ((◡𝐸 ∘ ◡(𝑆 ↾ ran 𝐸)) ∘ (𝑓 ↾ (𝐴 ∖ (◡𝑓 “ ran 𝐸)))):(𝐴 ∖ (◡𝑓 “ ran 𝐸))–onto→ω)
76	49, 74, 75	sylancr 599	. . . . . . . . . 10 ⊢ (𝑓:𝐴–onto→ω → ((◡𝐸 ∘ ◡(𝑆 ↾ ran 𝐸)) ∘ (𝑓 ↾ (𝐴 ∖ (◡𝑓 “ ran 𝐸)))):(𝐴 ∖ (◡𝑓 “ ran 𝐸))–onto→ω)
77		fowdom 9543	. . . . . . . . . 10 ⊢ ((((◡𝐸 ∘ ◡(𝑆 ↾ ran 𝐸)) ∘ (𝑓 ↾ (𝐴 ∖ (◡𝑓 “ ran 𝐸)))) ∈ V ∧ ((◡𝐸 ∘ ◡(𝑆 ↾ ran 𝐸)) ∘ (𝑓 ↾ (𝐴 ∖ (◡𝑓 “ ran 𝐸)))):(𝐴 ∖ (◡𝑓 “ ran 𝐸))–onto→ω) → ω ≼* (𝐴 ∖ (◡𝑓 “ ran 𝐸)))
78	54, 76, 77	sylancr 599	. . . . . . . . 9 ⊢ (𝑓:𝐴–onto→ω → ω ≼* (𝐴 ∖ (◡𝑓 “ ran 𝐸)))
79		difexg 5305	. . . . . . . . . . 11 ⊢ (𝐴 ∈ V → (𝐴 ∖ (◡𝑓 “ ran 𝐸)) ∈ V)
80		isfin3-2 10369	. . . . . . . . . . 11 ⊢ ((𝐴 ∖ (◡𝑓 “ ran 𝐸)) ∈ V → ((𝐴 ∖ (◡𝑓 “ ran 𝐸)) ∈ FinIII ↔ ¬ ω ≼* (𝐴 ∖ (◡𝑓 “ ran 𝐸))))
81	8, 79, 80	3syl 19	. . . . . . . . . 10 ⊢ (𝑓:𝐴–onto→ω → ((𝐴 ∖ (◡𝑓 “ ran 𝐸)) ∈ FinIII ↔ ¬ ω ≼* (𝐴 ∖ (◡𝑓 “ ran 𝐸))))
82	81	con2bid 357	. . . . . . . . 9 ⊢ (𝑓:𝐴–onto→ω → (ω ≼* (𝐴 ∖ (◡𝑓 “ ran 𝐸)) ↔ ¬ (𝐴 ∖ (◡𝑓 “ ran 𝐸)) ∈ FinIII))
83	78, 82	mpbid 235	. . . . . . . 8 ⊢ (𝑓:𝐴–onto→ω → ¬ (𝐴 ∖ (◡𝑓 “ ran 𝐸)) ∈ FinIII)
84		eleq1 2854	. . . . . . . . . . . 12 ⊢ (𝑦 = (◡𝑓 “ ran 𝐸) → (𝑦 ∈ FinIII ↔ (◡𝑓 “ ran 𝐸) ∈ FinIII))
85		difeq2 4078	. . . . . . . . . . . . 13 ⊢ (𝑦 = (◡𝑓 “ ran 𝐸) → (𝐴 ∖ 𝑦) = (𝐴 ∖ (◡𝑓 “ ran 𝐸)))
86	85	eleq1d 2851	. . . . . . . . . . . 12 ⊢ (𝑦 = (◡𝑓 “ ran 𝐸) → ((𝐴 ∖ 𝑦) ∈ FinIII ↔ (𝐴 ∖ (◡𝑓 “ ran 𝐸)) ∈ FinIII))
87	84, 86	orbi12d 932	. . . . . . . . . . 11 ⊢ (𝑦 = (◡𝑓 “ ran 𝐸) → ((𝑦 ∈ FinIII ∨ (𝐴 ∖ 𝑦) ∈ FinIII) ↔ ((◡𝑓 “ ran 𝐸) ∈ FinIII ∨ (𝐴 ∖ (◡𝑓 “ ran 𝐸)) ∈ FinIII)))
88	87	notbid 321	. . . . . . . . . 10 ⊢ (𝑦 = (◡𝑓 “ ran 𝐸) → (¬ (𝑦 ∈ FinIII ∨ (𝐴 ∖ 𝑦) ∈ FinIII) ↔ ¬ ((◡𝑓 “ ran 𝐸) ∈ FinIII ∨ (𝐴 ∖ (◡𝑓 “ ran 𝐸)) ∈ FinIII)))
89		ioran 999	. . . . . . . . . 10 ⊢ (¬ ((◡𝑓 “ ran 𝐸) ∈ FinIII ∨ (𝐴 ∖ (◡𝑓 “ ran 𝐸)) ∈ FinIII) ↔ (¬ (◡𝑓 “ ran 𝐸) ∈ FinIII ∧ ¬ (𝐴 ∖ (◡𝑓 “ ran 𝐸)) ∈ FinIII))
90	88, 89	bitrdi 290	. . . . . . . . 9 ⊢ (𝑦 = (◡𝑓 “ ran 𝐸) → (¬ (𝑦 ∈ FinIII ∨ (𝐴 ∖ 𝑦) ∈ FinIII) ↔ (¬ (◡𝑓 “ ran 𝐸) ∈ FinIII ∧ ¬ (𝐴 ∖ (◡𝑓 “ ran 𝐸)) ∈ FinIII)))
91	90	rspcev 3584	. . . . . . . 8 ⊢ (((◡𝑓 “ ran 𝐸) ∈ 𝒫 𝐴 ∧ (¬ (◡𝑓 “ ran 𝐸) ∈ FinIII ∧ ¬ (𝐴 ∖ (◡𝑓 “ ran 𝐸)) ∈ FinIII)) → ∃𝑦 ∈ 𝒫 𝐴 ¬ (𝑦 ∈ FinIII ∨ (𝐴 ∖ 𝑦) ∈ FinIII))
92	11, 42, 83, 91	syl12anc 850	. . . . . . 7 ⊢ (𝑓:𝐴–onto→ω → ∃𝑦 ∈ 𝒫 𝐴 ¬ (𝑦 ∈ FinIII ∨ (𝐴 ∖ 𝑦) ∈ FinIII))
93		rexnal 3120	. . . . . . 7 ⊢ (∃𝑦 ∈ 𝒫 𝐴 ¬ (𝑦 ∈ FinIII ∨ (𝐴 ∖ 𝑦) ∈ FinIII) ↔ ¬ ∀𝑦 ∈ 𝒫 𝐴(𝑦 ∈ FinIII ∨ (𝐴 ∖ 𝑦) ∈ FinIII))
94	92, 93	sylib 221	. . . . . 6 ⊢ (𝑓:𝐴–onto→ω → ¬ ∀𝑦 ∈ 𝒫 𝐴(𝑦 ∈ FinIII ∨ (𝐴 ∖ 𝑦) ∈ FinIII))
95	94	exlimiv 1963	. . . . 5 ⊢ (∃𝑓 𝑓:𝐴–onto→ω → ¬ ∀𝑦 ∈ 𝒫 𝐴(𝑦 ∈ FinIII ∨ (𝐴 ∖ 𝑦) ∈ FinIII))
96	4, 95	sylbi 220	. . . 4 ⊢ (ω ≼* 𝐴 → ¬ ∀𝑦 ∈ 𝒫 𝐴(𝑦 ∈ FinIII ∨ (𝐴 ∖ 𝑦) ∈ FinIII))
97	96	con2i 140	. . 3 ⊢ (∀𝑦 ∈ 𝒫 𝐴(𝑦 ∈ FinIII ∨ (𝐴 ∖ 𝑦) ∈ FinIII) → ¬ ω ≼* 𝐴)
98		isfin3-2 10369	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ FinIII ↔ ¬ ω ≼* 𝐴))
99	97, 98	imbitrrid 249	. 2 ⊢ (𝐴 ∈ 𝑉 → (∀𝑦 ∈ 𝒫 𝐴(𝑦 ∈ FinIII ∨ (𝐴 ∖ 𝑦) ∈ FinIII) → 𝐴 ∈ FinIII))
100	99	imp 412	1 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝒫 𝐴(𝑦 ∈ FinIII ∨ (𝐴 ∖ 𝑦) ∈ FinIII)) → 𝐴 ∈ FinIII)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1570  ∃wex 1812   ∈ wcel 2146   ≠ wne 2961  ∀wral 3082  ∃wrex 3092  Vcvv 3458   ∖ cdif 3905   ⊆ wss 3908  ∅c0 4289  𝒫 cpw 4567   class class class wbr 5114   ↦ cmpt 5197  ^◡ccnv 5665  dom cdm 5666  ran crn 5667   ↾ cres 5668   “ cima 5669   ∘ ccom 5670  Oncon0 6367  suc csuc 6369  Fun wfun 6537  ⟶wf 6539  –1-1→wf1 6540  –onto→wfo 6541  –1-1-onto→wf1o 6542  (class class class)co 7423  ωcom 7871  2_oc2o 8456   ·_o comu 8460   ≼^* cwdom 9536  Fin^IIIcfin3 10283

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-rep 5243  ax-sep 5262  ax-nul 5274  ax-pow 5341  ax-pr 5409  ax-un 7745

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-rmo 3372  df-reu 3373  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3928  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-int 4918  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-tr 5224  df-id 5561  df-eprel 5566  df-po 5574  df-so 5575  df-fr 5619  df-se 5620  df-we 5621  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6309  df-ord 6370  df-on 6371  df-lim 6372  df-suc 6373  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7380  df-ov 7426  df-oprab 7427  df-mpo 7428  df-om 7872  df-1st 7995  df-2nd 7996  df-frecs 8287  df-wrecs 8318  df-recs 8367  df-rdg 8406  df-seqom 8444  df-1o 8462  df-2o 8463  df-oadd 8466  df-omul 8467  df-er 8703  df-map 8835  df-en 8953  df-dom 8954  df-sdom 8955  df-fin 8956  df-wdom 9537  df-card 9944  df-fin4 10289  df-fin3 10290

This theorem is used by:  fin1a2lem8  10409