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Theorem fin1a2lem7 10443
Description: Lemma for fin1a2 10452. 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 ((𝐴𝑉 ∧ ∀𝑦 ∈ 𝒫 𝐴(𝑦 ∈ FinIII ∨ (𝐴𝑦) ∈ 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.
StepHypRef Expression
1 peano1 7910 . . . . . 6 ∅ ∈ ω
2 ne0i 4346 . . . . . 6 (∅ ∈ ω → ω ≠ ∅)
3 brwdomn0 9606 . . . . . 6 (ω ≠ ∅ → (ω ≼* 𝐴 ↔ ∃𝑓 𝑓:𝐴onto→ω))
41, 2, 3mp2b 10 . . . . 5 (ω ≼* 𝐴 ↔ ∃𝑓 𝑓:𝐴onto→ω)
5 vex 3481 . . . . . . . . . 10 𝑓 ∈ V
6 fof 6820 . . . . . . . . . 10 (𝑓:𝐴onto→ω → 𝑓:𝐴⟶ω)
7 dmfex 7927 . . . . . . . . . 10 ((𝑓 ∈ V ∧ 𝑓:𝐴⟶ω) → 𝐴 ∈ V)
85, 6, 7sylancr 587 . . . . . . . . 9 (𝑓:𝐴onto→ω → 𝐴 ∈ V)
9 cnvimass 6101 . . . . . . . . . 10 (𝑓 “ ran 𝐸) ⊆ dom 𝑓
109, 6fssdm 6755 . . . . . . . . 9 (𝑓:𝐴onto→ω → (𝑓 “ ran 𝐸) ⊆ 𝐴)
118, 10sselpwd 5333 . . . . . . . 8 (𝑓:𝐴onto→ω → (𝑓 “ ran 𝐸) ∈ 𝒫 𝐴)
12 fin1a2lem.b . . . . . . . . . . . . . 14 𝐸 = (𝑥 ∈ ω ↦ (2o ·o 𝑥))
1312fin1a2lem4 10440 . . . . . . . . . . . . 13 𝐸:ω–1-1→ω
14 f1cnv 6872 . . . . . . . . . . . . 13 (𝐸:ω–1-1→ω → 𝐸:ran 𝐸1-1-onto→ω)
15 f1ofo 6855 . . . . . . . . . . . . 13 (𝐸:ran 𝐸1-1-onto→ω → 𝐸:ran 𝐸onto→ω)
1613, 14, 15mp2b 10 . . . . . . . . . . . 12 𝐸:ran 𝐸onto→ω
17 fofun 6821 . . . . . . . . . . . 12 (𝐸:ran 𝐸onto→ω → Fun 𝐸)
1816, 17ax-mp 5 . . . . . . . . . . 11 Fun 𝐸
195resex 6048 . . . . . . . . . . 11 (𝑓 ↾ (𝑓 “ ran 𝐸)) ∈ V
20 cofunexg 7971 . . . . . . . . . . 11 ((Fun 𝐸 ∧ (𝑓 ↾ (𝑓 “ ran 𝐸)) ∈ V) → (𝐸 ∘ (𝑓 ↾ (𝑓 “ ran 𝐸))) ∈ V)
2118, 19, 20mp2an 692 . . . . . . . . . 10 (𝐸 ∘ (𝑓 ↾ (𝑓 “ ran 𝐸))) ∈ V
22 fofun 6821 . . . . . . . . . . . . 13 (𝑓:𝐴onto→ω → Fun 𝑓)
23 fores 6830 . . . . . . . . . . . . 13 ((Fun 𝑓 ∧ (𝑓 “ ran 𝐸) ⊆ dom 𝑓) → (𝑓 ↾ (𝑓 “ ran 𝐸)):(𝑓 “ ran 𝐸)–onto→(𝑓 “ (𝑓 “ ran 𝐸)))
2422, 9, 23sylancl 586 . . . . . . . . . . . 12 (𝑓:𝐴onto→ω → (𝑓 ↾ (𝑓 “ ran 𝐸)):(𝑓 “ ran 𝐸)–onto→(𝑓 “ (𝑓 “ ran 𝐸)))
25 f1f 6804 . . . . . . . . . . . . . . 15 (𝐸:ω–1-1→ω → 𝐸:ω⟶ω)
26 frn 6743 . . . . . . . . . . . . . . 15 (𝐸:ω⟶ω → ran 𝐸 ⊆ ω)
2713, 25, 26mp2b 10 . . . . . . . . . . . . . 14 ran 𝐸 ⊆ ω
28 foimacnv 6865 . . . . . . . . . . . . . 14 ((𝑓:𝐴onto→ω ∧ ran 𝐸 ⊆ ω) → (𝑓 “ (𝑓 “ ran 𝐸)) = ran 𝐸)
2927, 28mpan2 691 . . . . . . . . . . . . 13 (𝑓:𝐴onto→ω → (𝑓 “ (𝑓 “ ran 𝐸)) = ran 𝐸)
30 foeq3 6818 . . . . . . . . . . . . 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))
3129, 30syl 17 . . . . . . . . . . . 12 (𝑓:𝐴onto→ω → ((𝑓 ↾ (𝑓 “ ran 𝐸)):(𝑓 “ ran 𝐸)–onto→(𝑓 “ (𝑓 “ ran 𝐸)) ↔ (𝑓 ↾ (𝑓 “ ran 𝐸)):(𝑓 “ ran 𝐸)–onto→ran 𝐸))
3224, 31mpbid 232 . . . . . . . . . . 11 (𝑓:𝐴onto→ω → (𝑓 ↾ (𝑓 “ ran 𝐸)):(𝑓 “ ran 𝐸)–onto→ran 𝐸)
33 foco 6834 . . . . . . . . . . 11 ((𝐸:ran 𝐸onto→ω ∧ (𝑓 ↾ (𝑓 “ ran 𝐸)):(𝑓 “ ran 𝐸)–onto→ran 𝐸) → (𝐸 ∘ (𝑓 ↾ (𝑓 “ ran 𝐸))):(𝑓 “ ran 𝐸)–onto→ω)
3416, 32, 33sylancr 587 . . . . . . . . . 10 (𝑓:𝐴onto→ω → (𝐸 ∘ (𝑓 ↾ (𝑓 “ ran 𝐸))):(𝑓 “ ran 𝐸)–onto→ω)
35 fowdom 9608 . . . . . . . . . 10 (((𝐸 ∘ (𝑓 ↾ (𝑓 “ ran 𝐸))) ∈ V ∧ (𝐸 ∘ (𝑓 ↾ (𝑓 “ ran 𝐸))):(𝑓 “ ran 𝐸)–onto→ω) → ω ≼* (𝑓 “ ran 𝐸))
3621, 34, 35sylancr 587 . . . . . . . . 9 (𝑓:𝐴onto→ω → ω ≼* (𝑓 “ ran 𝐸))
375cnvex 7947 . . . . . . . . . . . 12 𝑓 ∈ V
3837imaex 7936 . . . . . . . . . . 11 (𝑓 “ ran 𝐸) ∈ V
39 isfin3-2 10404 . . . . . . . . . . 11 ((𝑓 “ ran 𝐸) ∈ V → ((𝑓 “ ran 𝐸) ∈ FinIII ↔ ¬ ω ≼* (𝑓 “ ran 𝐸)))
4038, 39ax-mp 5 . . . . . . . . . 10 ((𝑓 “ ran 𝐸) ∈ FinIII ↔ ¬ ω ≼* (𝑓 “ ran 𝐸))
4140con2bii 357 . . . . . . . . 9 (ω ≼* (𝑓 “ ran 𝐸) ↔ ¬ (𝑓 “ ran 𝐸) ∈ FinIII)
4236, 41sylib 218 . . . . . . . 8 (𝑓:𝐴onto→ω → ¬ (𝑓 “ ran 𝐸) ∈ FinIII)
43 fin1a2lem.aa . . . . . . . . . . . . . . 15 𝑆 = (𝑥 ∈ On ↦ suc 𝑥)
4412, 43fin1a2lem6 10442 . . . . . . . . . . . . . 14 (𝑆 ↾ ran 𝐸):ran 𝐸1-1-onto→(ω ∖ ran 𝐸)
45 f1ocnv 6860 . . . . . . . . . . . . . 14 ((𝑆 ↾ ran 𝐸):ran 𝐸1-1-onto→(ω ∖ ran 𝐸) → (𝑆 ↾ ran 𝐸):(ω ∖ ran 𝐸)–1-1-onto→ran 𝐸)
46 f1ofo 6855 . . . . . . . . . . . . . 14 ((𝑆 ↾ ran 𝐸):(ω ∖ ran 𝐸)–1-1-onto→ran 𝐸(𝑆 ↾ ran 𝐸):(ω ∖ ran 𝐸)–onto→ran 𝐸)
4744, 45, 46mp2b 10 . . . . . . . . . . . . 13 (𝑆 ↾ ran 𝐸):(ω ∖ ran 𝐸)–onto→ran 𝐸
48 foco 6834 . . . . . . . . . . . . 13 ((𝐸:ran 𝐸onto→ω ∧ (𝑆 ↾ ran 𝐸):(ω ∖ ran 𝐸)–onto→ran 𝐸) → (𝐸(𝑆 ↾ ran 𝐸)):(ω ∖ ran 𝐸)–onto→ω)
4916, 47, 48mp2an 692 . . . . . . . . . . . 12 (𝐸(𝑆 ↾ ran 𝐸)):(ω ∖ ran 𝐸)–onto→ω
50 fofun 6821 . . . . . . . . . . . 12 ((𝐸(𝑆 ↾ ran 𝐸)):(ω ∖ ran 𝐸)–onto→ω → Fun (𝐸(𝑆 ↾ ran 𝐸)))
5149, 50ax-mp 5 . . . . . . . . . . 11 Fun (𝐸(𝑆 ↾ ran 𝐸))
525resex 6048 . . . . . . . . . . 11 (𝑓 ↾ (𝐴 ∖ (𝑓 “ ran 𝐸))) ∈ V
53 cofunexg 7971 . . . . . . . . . . 11 ((Fun (𝐸(𝑆 ↾ ran 𝐸)) ∧ (𝑓 ↾ (𝐴 ∖ (𝑓 “ ran 𝐸))) ∈ V) → ((𝐸(𝑆 ↾ ran 𝐸)) ∘ (𝑓 ↾ (𝐴 ∖ (𝑓 “ ran 𝐸)))) ∈ V)
5451, 52, 53mp2an 692 . . . . . . . . . 10 ((𝐸(𝑆 ↾ ran 𝐸)) ∘ (𝑓 ↾ (𝐴 ∖ (𝑓 “ ran 𝐸)))) ∈ V
55 difss 4145 . . . . . . . . . . . . . 14 (𝐴 ∖ (𝑓 “ ran 𝐸)) ⊆ 𝐴
566fdmd 6746 . . . . . . . . . . . . . 14 (𝑓:𝐴onto→ω → dom 𝑓 = 𝐴)
5755, 56sseqtrrid 4048 . . . . . . . . . . . . 13 (𝑓:𝐴onto→ω → (𝐴 ∖ (𝑓 “ ran 𝐸)) ⊆ dom 𝑓)
58 fores 6830 . . . . . . . . . . . . 13 ((Fun 𝑓 ∧ (𝐴 ∖ (𝑓 “ ran 𝐸)) ⊆ dom 𝑓) → (𝑓 ↾ (𝐴 ∖ (𝑓 “ ran 𝐸))):(𝐴 ∖ (𝑓 “ ran 𝐸))–onto→(𝑓 “ (𝐴 ∖ (𝑓 “ ran 𝐸))))
5922, 57, 58syl2anc 584 . . . . . . . . . . . 12 (𝑓:𝐴onto→ω → (𝑓 ↾ (𝐴 ∖ (𝑓 “ ran 𝐸))):(𝐴 ∖ (𝑓 “ ran 𝐸))–onto→(𝑓 “ (𝐴 ∖ (𝑓 “ ran 𝐸))))
60 funcnvcnv 6634 . . . . . . . . . . . . . . . 16 (Fun 𝑓 → Fun 𝑓)
61 imadif 6651 . . . . . . . . . . . . . . . 16 (Fun 𝑓 → (𝑓 “ (ω ∖ ran 𝐸)) = ((𝑓 “ ω) ∖ (𝑓 “ ran 𝐸)))
6222, 60, 613syl 18 . . . . . . . . . . . . . . 15 (𝑓:𝐴onto→ω → (𝑓 “ (ω ∖ ran 𝐸)) = ((𝑓 “ ω) ∖ (𝑓 “ ran 𝐸)))
6362imaeq2d 6079 . . . . . . . . . . . . . 14 (𝑓:𝐴onto→ω → (𝑓 “ (𝑓 “ (ω ∖ ran 𝐸))) = (𝑓 “ ((𝑓 “ ω) ∖ (𝑓 “ ran 𝐸))))
64 difss 4145 . . . . . . . . . . . . . . 15 (ω ∖ ran 𝐸) ⊆ ω
65 foimacnv 6865 . . . . . . . . . . . . . . 15 ((𝑓:𝐴onto→ω ∧ (ω ∖ ran 𝐸) ⊆ ω) → (𝑓 “ (𝑓 “ (ω ∖ ran 𝐸))) = (ω ∖ ran 𝐸))
6664, 65mpan2 691 . . . . . . . . . . . . . 14 (𝑓:𝐴onto→ω → (𝑓 “ (𝑓 “ (ω ∖ ran 𝐸))) = (ω ∖ ran 𝐸))
67 fimacnv 6758 . . . . . . . . . . . . . . . . 17 (𝑓:𝐴⟶ω → (𝑓 “ ω) = 𝐴)
686, 67syl 17 . . . . . . . . . . . . . . . 16 (𝑓:𝐴onto→ω → (𝑓 “ ω) = 𝐴)
6968difeq1d 4134 . . . . . . . . . . . . . . 15 (𝑓:𝐴onto→ω → ((𝑓 “ ω) ∖ (𝑓 “ ran 𝐸)) = (𝐴 ∖ (𝑓 “ ran 𝐸)))
7069imaeq2d 6079 . . . . . . . . . . . . . 14 (𝑓:𝐴onto→ω → (𝑓 “ ((𝑓 “ ω) ∖ (𝑓 “ ran 𝐸))) = (𝑓 “ (𝐴 ∖ (𝑓 “ ran 𝐸))))
7163, 66, 703eqtr3rd 2783 . . . . . . . . . . . . 13 (𝑓:𝐴onto→ω → (𝑓 “ (𝐴 ∖ (𝑓 “ ran 𝐸))) = (ω ∖ ran 𝐸))
72 foeq3 6818 . . . . . . . . . . . . 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)))
7371, 72syl 17 . . . . . . . . . . . 12 (𝑓:𝐴onto→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)))
7459, 73mpbid 232 . . . . . . . . . . 11 (𝑓:𝐴onto→ω → (𝑓 ↾ (𝐴 ∖ (𝑓 “ ran 𝐸))):(𝐴 ∖ (𝑓 “ ran 𝐸))–onto→(ω ∖ ran 𝐸))
75 foco 6834 . . . . . . . . . . 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)
7649, 74, 75sylancr 587 . . . . . . . . . 10 (𝑓:𝐴onto→ω → ((𝐸(𝑆 ↾ ran 𝐸)) ∘ (𝑓 ↾ (𝐴 ∖ (𝑓 “ ran 𝐸)))):(𝐴 ∖ (𝑓 “ ran 𝐸))–onto→ω)
77 fowdom 9608 . . . . . . . . . 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)))
7854, 76, 77sylancr 587 . . . . . . . . 9 (𝑓:𝐴onto→ω → ω ≼* (𝐴 ∖ (𝑓 “ ran 𝐸)))
79 difexg 5334 . . . . . . . . . . 11 (𝐴 ∈ V → (𝐴 ∖ (𝑓 “ ran 𝐸)) ∈ V)
80 isfin3-2 10404 . . . . . . . . . . 11 ((𝐴 ∖ (𝑓 “ ran 𝐸)) ∈ V → ((𝐴 ∖ (𝑓 “ ran 𝐸)) ∈ FinIII ↔ ¬ ω ≼* (𝐴 ∖ (𝑓 “ ran 𝐸))))
818, 79, 803syl 18 . . . . . . . . . 10 (𝑓:𝐴onto→ω → ((𝐴 ∖ (𝑓 “ ran 𝐸)) ∈ FinIII ↔ ¬ ω ≼* (𝐴 ∖ (𝑓 “ ran 𝐸))))
8281con2bid 354 . . . . . . . . 9 (𝑓:𝐴onto→ω → (ω ≼* (𝐴 ∖ (𝑓 “ ran 𝐸)) ↔ ¬ (𝐴 ∖ (𝑓 “ ran 𝐸)) ∈ FinIII))
8378, 82mpbid 232 . . . . . . . 8 (𝑓:𝐴onto→ω → ¬ (𝐴 ∖ (𝑓 “ ran 𝐸)) ∈ FinIII)
84 eleq1 2826 . . . . . . . . . . . 12 (𝑦 = (𝑓 “ ran 𝐸) → (𝑦 ∈ FinIII ↔ (𝑓 “ ran 𝐸) ∈ FinIII))
85 difeq2 4129 . . . . . . . . . . . . 13 (𝑦 = (𝑓 “ ran 𝐸) → (𝐴𝑦) = (𝐴 ∖ (𝑓 “ ran 𝐸)))
8685eleq1d 2823 . . . . . . . . . . . 12 (𝑦 = (𝑓 “ ran 𝐸) → ((𝐴𝑦) ∈ FinIII ↔ (𝐴 ∖ (𝑓 “ ran 𝐸)) ∈ FinIII))
8784, 86orbi12d 918 . . . . . . . . . . 11 (𝑦 = (𝑓 “ ran 𝐸) → ((𝑦 ∈ FinIII ∨ (𝐴𝑦) ∈ FinIII) ↔ ((𝑓 “ ran 𝐸) ∈ FinIII ∨ (𝐴 ∖ (𝑓 “ ran 𝐸)) ∈ FinIII)))
8887notbid 318 . . . . . . . . . 10 (𝑦 = (𝑓 “ ran 𝐸) → (¬ (𝑦 ∈ FinIII ∨ (𝐴𝑦) ∈ FinIII) ↔ ¬ ((𝑓 “ ran 𝐸) ∈ FinIII ∨ (𝐴 ∖ (𝑓 “ ran 𝐸)) ∈ FinIII)))
89 ioran 985 . . . . . . . . . 10 (¬ ((𝑓 “ ran 𝐸) ∈ FinIII ∨ (𝐴 ∖ (𝑓 “ ran 𝐸)) ∈ FinIII) ↔ (¬ (𝑓 “ ran 𝐸) ∈ FinIII ∧ ¬ (𝐴 ∖ (𝑓 “ ran 𝐸)) ∈ FinIII))
9088, 89bitrdi 287 . . . . . . . . 9 (𝑦 = (𝑓 “ ran 𝐸) → (¬ (𝑦 ∈ FinIII ∨ (𝐴𝑦) ∈ FinIII) ↔ (¬ (𝑓 “ ran 𝐸) ∈ FinIII ∧ ¬ (𝐴 ∖ (𝑓 “ ran 𝐸)) ∈ FinIII)))
9190rspcev 3621 . . . . . . . 8 (((𝑓 “ ran 𝐸) ∈ 𝒫 𝐴 ∧ (¬ (𝑓 “ ran 𝐸) ∈ FinIII ∧ ¬ (𝐴 ∖ (𝑓 “ ran 𝐸)) ∈ FinIII)) → ∃𝑦 ∈ 𝒫 𝐴 ¬ (𝑦 ∈ FinIII ∨ (𝐴𝑦) ∈ FinIII))
9211, 42, 83, 91syl12anc 837 . . . . . . 7 (𝑓:𝐴onto→ω → ∃𝑦 ∈ 𝒫 𝐴 ¬ (𝑦 ∈ FinIII ∨ (𝐴𝑦) ∈ FinIII))
93 rexnal 3097 . . . . . . 7 (∃𝑦 ∈ 𝒫 𝐴 ¬ (𝑦 ∈ FinIII ∨ (𝐴𝑦) ∈ FinIII) ↔ ¬ ∀𝑦 ∈ 𝒫 𝐴(𝑦 ∈ FinIII ∨ (𝐴𝑦) ∈ FinIII))
9492, 93sylib 218 . . . . . 6 (𝑓:𝐴onto→ω → ¬ ∀𝑦 ∈ 𝒫 𝐴(𝑦 ∈ FinIII ∨ (𝐴𝑦) ∈ FinIII))
9594exlimiv 1927 . . . . 5 (∃𝑓 𝑓:𝐴onto→ω → ¬ ∀𝑦 ∈ 𝒫 𝐴(𝑦 ∈ FinIII ∨ (𝐴𝑦) ∈ FinIII))
964, 95sylbi 217 . . . 4 (ω ≼* 𝐴 → ¬ ∀𝑦 ∈ 𝒫 𝐴(𝑦 ∈ FinIII ∨ (𝐴𝑦) ∈ FinIII))
9796con2i 139 . . 3 (∀𝑦 ∈ 𝒫 𝐴(𝑦 ∈ FinIII ∨ (𝐴𝑦) ∈ FinIII) → ¬ ω ≼* 𝐴)
98 isfin3-2 10404 . . 3 (𝐴𝑉 → (𝐴 ∈ FinIII ↔ ¬ ω ≼* 𝐴))
9997, 98imbitrrid 246 . 2 (𝐴𝑉 → (∀𝑦 ∈ 𝒫 𝐴(𝑦 ∈ FinIII ∨ (𝐴𝑦) ∈ FinIII) → 𝐴 ∈ FinIII))
10099imp 406 1 ((𝐴𝑉 ∧ ∀𝑦 ∈ 𝒫 𝐴(𝑦 ∈ FinIII ∨ (𝐴𝑦) ∈ FinIII)) → 𝐴 ∈ FinIII)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wo 847   = wceq 1536  wex 1775  wcel 2105  wne 2937  wral 3058  wrex 3067  Vcvv 3477  cdif 3959  wss 3962  c0 4338  𝒫 cpw 4604   class class class wbr 5147  cmpt 5230  ccnv 5687  dom cdm 5688  ran crn 5689  cres 5690  cima 5691  ccom 5692  Oncon0 6385  suc csuc 6387  Fun wfun 6556  wf 6558  1-1wf1 6559  ontowfo 6560  1-1-ontowf1o 6561  (class class class)co 7430  ωcom 7886  2oc2o 8498   ·o comu 8502  * cwdom 9601  FinIIIcfin3 10318
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-se 5641  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-isom 6571  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-seqom 8486  df-1o 8504  df-2o 8505  df-oadd 8508  df-omul 8509  df-er 8743  df-map 8866  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-wdom 9602  df-card 9976  df-fin4 10324  df-fin3 10325
This theorem is referenced by:  fin1a2lem8  10444
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