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Theorem fin1a2lem7 9816
Description: Lemma for fin1a2 9825. 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 7590 . . . . . 6 ∅ ∈ ω
2 ne0i 4297 . . . . . 6 (∅ ∈ ω → ω ≠ ∅)
3 brwdomn0 9021 . . . . . 6 (ω ≠ ∅ → (ω ≼* 𝐴 ↔ ∃𝑓 𝑓:𝐴onto→ω))
41, 2, 3mp2b 10 . . . . 5 (ω ≼* 𝐴 ↔ ∃𝑓 𝑓:𝐴onto→ω)
5 vex 3495 . . . . . . . . . 10 𝑓 ∈ V
6 fof 6583 . . . . . . . . . 10 (𝑓:𝐴onto→ω → 𝑓:𝐴⟶ω)
7 dmfex 7630 . . . . . . . . . 10 ((𝑓 ∈ V ∧ 𝑓:𝐴⟶ω) → 𝐴 ∈ V)
85, 6, 7sylancr 587 . . . . . . . . 9 (𝑓:𝐴onto→ω → 𝐴 ∈ V)
9 cnvimass 5942 . . . . . . . . . 10 (𝑓 “ ran 𝐸) ⊆ dom 𝑓
109, 6fssdm 6523 . . . . . . . . 9 (𝑓:𝐴onto→ω → (𝑓 “ ran 𝐸) ⊆ 𝐴)
118, 10sselpwd 5221 . . . . . . . 8 (𝑓:𝐴onto→ω → (𝑓 “ ran 𝐸) ∈ 𝒫 𝐴)
12 fin1a2lem.b . . . . . . . . . . . . . 14 𝐸 = (𝑥 ∈ ω ↦ (2o ·o 𝑥))
1312fin1a2lem4 9813 . . . . . . . . . . . . 13 𝐸:ω–1-1→ω
14 f1cnv 6631 . . . . . . . . . . . . 13 (𝐸:ω–1-1→ω → 𝐸:ran 𝐸1-1-onto→ω)
15 f1ofo 6615 . . . . . . . . . . . . 13 (𝐸:ran 𝐸1-1-onto→ω → 𝐸:ran 𝐸onto→ω)
1613, 14, 15mp2b 10 . . . . . . . . . . . 12 𝐸:ran 𝐸onto→ω
17 fofun 6584 . . . . . . . . . . . 12 (𝐸:ran 𝐸onto→ω → Fun 𝐸)
1816, 17ax-mp 5 . . . . . . . . . . 11 Fun 𝐸
195resex 5892 . . . . . . . . . . 11 (𝑓 ↾ (𝑓 “ ran 𝐸)) ∈ V
20 cofunexg 7639 . . . . . . . . . . 11 ((Fun 𝐸 ∧ (𝑓 ↾ (𝑓 “ ran 𝐸)) ∈ V) → (𝐸 ∘ (𝑓 ↾ (𝑓 “ ran 𝐸))) ∈ V)
2118, 19, 20mp2an 688 . . . . . . . . . 10 (𝐸 ∘ (𝑓 ↾ (𝑓 “ ran 𝐸))) ∈ V
22 fofun 6584 . . . . . . . . . . . . 13 (𝑓:𝐴onto→ω → Fun 𝑓)
23 fores 6593 . . . . . . . . . . . . 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 6568 . . . . . . . . . . . . . . 15 (𝐸:ω–1-1→ω → 𝐸:ω⟶ω)
26 frn 6513 . . . . . . . . . . . . . . 15 (𝐸:ω⟶ω → ran 𝐸 ⊆ ω)
2713, 25, 26mp2b 10 . . . . . . . . . . . . . 14 ran 𝐸 ⊆ ω
28 foimacnv 6625 . . . . . . . . . . . . . 14 ((𝑓:𝐴onto→ω ∧ ran 𝐸 ⊆ ω) → (𝑓 “ (𝑓 “ ran 𝐸)) = ran 𝐸)
2927, 28mpan2 687 . . . . . . . . . . . . 13 (𝑓:𝐴onto→ω → (𝑓 “ (𝑓 “ ran 𝐸)) = ran 𝐸)
30 foeq3 6581 . . . . . . . . . . . . 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 233 . . . . . . . . . . 11 (𝑓:𝐴onto→ω → (𝑓 ↾ (𝑓 “ ran 𝐸)):(𝑓 “ ran 𝐸)–onto→ran 𝐸)
33 foco 6595 . . . . . . . . . . 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 9023 . . . . . . . . . 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 7619 . . . . . . . . . . . 12 𝑓 ∈ V
3837imaex 7610 . . . . . . . . . . 11 (𝑓 “ ran 𝐸) ∈ V
39 isfin3-2 9777 . . . . . . . . . . 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 359 . . . . . . . . 9 (ω ≼* (𝑓 “ ran 𝐸) ↔ ¬ (𝑓 “ ran 𝐸) ∈ FinIII)
4236, 41sylib 219 . . . . . . . 8 (𝑓:𝐴onto→ω → ¬ (𝑓 “ ran 𝐸) ∈ FinIII)
43 fin1a2lem.aa . . . . . . . . . . . . . . 15 𝑆 = (𝑥 ∈ On ↦ suc 𝑥)
4412, 43fin1a2lem6 9815 . . . . . . . . . . . . . 14 (𝑆 ↾ ran 𝐸):ran 𝐸1-1-onto→(ω ∖ ran 𝐸)
45 f1ocnv 6620 . . . . . . . . . . . . . 14 ((𝑆 ↾ ran 𝐸):ran 𝐸1-1-onto→(ω ∖ ran 𝐸) → (𝑆 ↾ ran 𝐸):(ω ∖ ran 𝐸)–1-1-onto→ran 𝐸)
46 f1ofo 6615 . . . . . . . . . . . . . 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 6595 . . . . . . . . . . . . 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 688 . . . . . . . . . . . 12 (𝐸(𝑆 ↾ ran 𝐸)):(ω ∖ ran 𝐸)–onto→ω
50 fofun 6584 . . . . . . . . . . . 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 5892 . . . . . . . . . . 11 (𝑓 ↾ (𝐴 ∖ (𝑓 “ ran 𝐸))) ∈ V
53 cofunexg 7639 . . . . . . . . . . 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 688 . . . . . . . . . 10 ((𝐸(𝑆 ↾ ran 𝐸)) ∘ (𝑓 ↾ (𝐴 ∖ (𝑓 “ ran 𝐸)))) ∈ V
55 difss 4105 . . . . . . . . . . . . . 14 (𝐴 ∖ (𝑓 “ ran 𝐸)) ⊆ 𝐴
566fdmd 6516 . . . . . . . . . . . . . 14 (𝑓:𝐴onto→ω → dom 𝑓 = 𝐴)
5755, 56sseqtrrid 4017 . . . . . . . . . . . . 13 (𝑓:𝐴onto→ω → (𝐴 ∖ (𝑓 “ ran 𝐸)) ⊆ dom 𝑓)
58 fores 6593 . . . . . . . . . . . . 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 6414 . . . . . . . . . . . . . . . 16 (Fun 𝑓 → Fun 𝑓)
61 imadif 6431 . . . . . . . . . . . . . . . 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 5922 . . . . . . . . . . . . . 14 (𝑓:𝐴onto→ω → (𝑓 “ (𝑓 “ (ω ∖ ran 𝐸))) = (𝑓 “ ((𝑓 “ ω) ∖ (𝑓 “ ran 𝐸))))
64 difss 4105 . . . . . . . . . . . . . . 15 (ω ∖ ran 𝐸) ⊆ ω
65 foimacnv 6625 . . . . . . . . . . . . . . 15 ((𝑓:𝐴onto→ω ∧ (ω ∖ ran 𝐸) ⊆ ω) → (𝑓 “ (𝑓 “ (ω ∖ ran 𝐸))) = (ω ∖ ran 𝐸))
6664, 65mpan2 687 . . . . . . . . . . . . . 14 (𝑓:𝐴onto→ω → (𝑓 “ (𝑓 “ (ω ∖ ran 𝐸))) = (ω ∖ ran 𝐸))
67 fimacnv 6831 . . . . . . . . . . . . . . . . 17 (𝑓:𝐴⟶ω → (𝑓 “ ω) = 𝐴)
686, 67syl 17 . . . . . . . . . . . . . . . 16 (𝑓:𝐴onto→ω → (𝑓 “ ω) = 𝐴)
6968difeq1d 4095 . . . . . . . . . . . . . . 15 (𝑓:𝐴onto→ω → ((𝑓 “ ω) ∖ (𝑓 “ ran 𝐸)) = (𝐴 ∖ (𝑓 “ ran 𝐸)))
7069imaeq2d 5922 . . . . . . . . . . . . . 14 (𝑓:𝐴onto→ω → (𝑓 “ ((𝑓 “ ω) ∖ (𝑓 “ ran 𝐸))) = (𝑓 “ (𝐴 ∖ (𝑓 “ ran 𝐸))))
7163, 66, 703eqtr3rd 2862 . . . . . . . . . . . . 13 (𝑓:𝐴onto→ω → (𝑓 “ (𝐴 ∖ (𝑓 “ ran 𝐸))) = (ω ∖ ran 𝐸))
72 foeq3 6581 . . . . . . . . . . . . 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 233 . . . . . . . . . . 11 (𝑓:𝐴onto→ω → (𝑓 ↾ (𝐴 ∖ (𝑓 “ ran 𝐸))):(𝐴 ∖ (𝑓 “ ran 𝐸))–onto→(ω ∖ ran 𝐸))
75 foco 6595 . . . . . . . . . . 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 9023 . . . . . . . . . 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 5222 . . . . . . . . . . 11 (𝐴 ∈ V → (𝐴 ∖ (𝑓 “ ran 𝐸)) ∈ V)
80 isfin3-2 9777 . . . . . . . . . . 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 356 . . . . . . . . 9 (𝑓:𝐴onto→ω → (ω ≼* (𝐴 ∖ (𝑓 “ ran 𝐸)) ↔ ¬ (𝐴 ∖ (𝑓 “ ran 𝐸)) ∈ FinIII))
8378, 82mpbid 233 . . . . . . . 8 (𝑓:𝐴onto→ω → ¬ (𝐴 ∖ (𝑓 “ ran 𝐸)) ∈ FinIII)
84 eleq1 2897 . . . . . . . . . . . 12 (𝑦 = (𝑓 “ ran 𝐸) → (𝑦 ∈ FinIII ↔ (𝑓 “ ran 𝐸) ∈ FinIII))
85 difeq2 4090 . . . . . . . . . . . . 13 (𝑦 = (𝑓 “ ran 𝐸) → (𝐴𝑦) = (𝐴 ∖ (𝑓 “ ran 𝐸)))
8685eleq1d 2894 . . . . . . . . . . . 12 (𝑦 = (𝑓 “ ran 𝐸) → ((𝐴𝑦) ∈ FinIII ↔ (𝐴 ∖ (𝑓 “ ran 𝐸)) ∈ FinIII))
8784, 86orbi12d 912 . . . . . . . . . . 11 (𝑦 = (𝑓 “ ran 𝐸) → ((𝑦 ∈ FinIII ∨ (𝐴𝑦) ∈ FinIII) ↔ ((𝑓 “ ran 𝐸) ∈ FinIII ∨ (𝐴 ∖ (𝑓 “ ran 𝐸)) ∈ FinIII)))
8887notbid 319 . . . . . . . . . 10 (𝑦 = (𝑓 “ ran 𝐸) → (¬ (𝑦 ∈ FinIII ∨ (𝐴𝑦) ∈ FinIII) ↔ ¬ ((𝑓 “ ran 𝐸) ∈ FinIII ∨ (𝐴 ∖ (𝑓 “ ran 𝐸)) ∈ FinIII)))
89 ioran 977 . . . . . . . . . 10 (¬ ((𝑓 “ ran 𝐸) ∈ FinIII ∨ (𝐴 ∖ (𝑓 “ ran 𝐸)) ∈ FinIII) ↔ (¬ (𝑓 “ ran 𝐸) ∈ FinIII ∧ ¬ (𝐴 ∖ (𝑓 “ ran 𝐸)) ∈ FinIII))
9088, 89syl6bb 288 . . . . . . . . 9 (𝑦 = (𝑓 “ ran 𝐸) → (¬ (𝑦 ∈ FinIII ∨ (𝐴𝑦) ∈ FinIII) ↔ (¬ (𝑓 “ ran 𝐸) ∈ FinIII ∧ ¬ (𝐴 ∖ (𝑓 “ ran 𝐸)) ∈ FinIII)))
9190rspcev 3620 . . . . . . . 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 832 . . . . . . 7 (𝑓:𝐴onto→ω → ∃𝑦 ∈ 𝒫 𝐴 ¬ (𝑦 ∈ FinIII ∨ (𝐴𝑦) ∈ FinIII))
93 rexnal 3235 . . . . . . 7 (∃𝑦 ∈ 𝒫 𝐴 ¬ (𝑦 ∈ FinIII ∨ (𝐴𝑦) ∈ FinIII) ↔ ¬ ∀𝑦 ∈ 𝒫 𝐴(𝑦 ∈ FinIII ∨ (𝐴𝑦) ∈ FinIII))
9492, 93sylib 219 . . . . . 6 (𝑓:𝐴onto→ω → ¬ ∀𝑦 ∈ 𝒫 𝐴(𝑦 ∈ FinIII ∨ (𝐴𝑦) ∈ FinIII))
9594exlimiv 1922 . . . . 5 (∃𝑓 𝑓:𝐴onto→ω → ¬ ∀𝑦 ∈ 𝒫 𝐴(𝑦 ∈ FinIII ∨ (𝐴𝑦) ∈ FinIII))
964, 95sylbi 218 . . . 4 (ω ≼* 𝐴 → ¬ ∀𝑦 ∈ 𝒫 𝐴(𝑦 ∈ FinIII ∨ (𝐴𝑦) ∈ FinIII))
9796con2i 141 . . 3 (∀𝑦 ∈ 𝒫 𝐴(𝑦 ∈ FinIII ∨ (𝐴𝑦) ∈ FinIII) → ¬ ω ≼* 𝐴)
98 isfin3-2 9777 . . 3 (𝐴𝑉 → (𝐴 ∈ FinIII ↔ ¬ ω ≼* 𝐴))
9997, 98syl5ibr 247 . 2 (𝐴𝑉 → (∀𝑦 ∈ 𝒫 𝐴(𝑦 ∈ FinIII ∨ (𝐴𝑦) ∈ FinIII) → 𝐴 ∈ FinIII))
10099imp 407 1 ((𝐴𝑉 ∧ ∀𝑦 ∈ 𝒫 𝐴(𝑦 ∈ FinIII ∨ (𝐴𝑦) ∈ FinIII)) → 𝐴 ∈ FinIII)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  wo 841   = wceq 1528  wex 1771  wcel 2105  wne 3013  wral 3135  wrex 3136  Vcvv 3492  cdif 3930  wss 3933  c0 4288  𝒫 cpw 4535   class class class wbr 5057  cmpt 5137  ccnv 5547  dom cdm 5548  ran crn 5549  cres 5550  cima 5551  ccom 5552  Oncon0 6184  suc csuc 6186  Fun wfun 6342  wf 6344  1-1wf1 6345  ontowfo 6346  1-1-ontowf1o 6347  (class class class)co 7145  ωcom 7569  2oc2o 8085   ·o comu 8089  * cwdom 9009  FinIIIcfin3 9691
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-se 5508  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-seqom 8073  df-1o 8091  df-2o 8092  df-oadd 8095  df-omul 8096  df-er 8278  df-map 8397  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-wdom 9011  df-card 9356  df-fin4 9697  df-fin3 9698
This theorem is referenced by:  fin1a2lem8  9817
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