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Theorem fin1a2lem7 9172
Description: Lemma for fin1a2 9181. 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 7032 . . . . . 6 ∅ ∈ ω
2 ne0i 3897 . . . . . 6 (∅ ∈ ω → ω ≠ ∅)
3 brwdomn0 8418 . . . . . 6 (ω ≠ ∅ → (ω ≼* 𝐴 ↔ ∃𝑓 𝑓:𝐴onto→ω))
41, 2, 3mp2b 10 . . . . 5 (ω ≼* 𝐴 ↔ ∃𝑓 𝑓:𝐴onto→ω)
5 vex 3189 . . . . . . . . . 10 𝑓 ∈ V
6 fof 6072 . . . . . . . . . 10 (𝑓:𝐴onto→ω → 𝑓:𝐴⟶ω)
7 dmfex 7071 . . . . . . . . . 10 ((𝑓 ∈ V ∧ 𝑓:𝐴⟶ω) → 𝐴 ∈ V)
85, 6, 7sylancr 694 . . . . . . . . 9 (𝑓:𝐴onto→ω → 𝐴 ∈ V)
9 cnvimass 5444 . . . . . . . . . 10 (𝑓 “ ran 𝐸) ⊆ dom 𝑓
10 fdm 6008 . . . . . . . . . . 11 (𝑓:𝐴⟶ω → dom 𝑓 = 𝐴)
116, 10syl 17 . . . . . . . . . 10 (𝑓:𝐴onto→ω → dom 𝑓 = 𝐴)
129, 11syl5sseq 3632 . . . . . . . . 9 (𝑓:𝐴onto→ω → (𝑓 “ ran 𝐸) ⊆ 𝐴)
138, 12sselpwd 4767 . . . . . . . 8 (𝑓:𝐴onto→ω → (𝑓 “ ran 𝐸) ∈ 𝒫 𝐴)
14 fin1a2lem.b . . . . . . . . . . . . . 14 𝐸 = (𝑥 ∈ ω ↦ (2𝑜 ·𝑜 𝑥))
1514fin1a2lem4 9169 . . . . . . . . . . . . 13 𝐸:ω–1-1→ω
16 f1cnv 6117 . . . . . . . . . . . . 13 (𝐸:ω–1-1→ω → 𝐸:ran 𝐸1-1-onto→ω)
17 f1ofo 6101 . . . . . . . . . . . . 13 (𝐸:ran 𝐸1-1-onto→ω → 𝐸:ran 𝐸onto→ω)
1815, 16, 17mp2b 10 . . . . . . . . . . . 12 𝐸:ran 𝐸onto→ω
19 fofun 6073 . . . . . . . . . . . 12 (𝐸:ran 𝐸onto→ω → Fun 𝐸)
2018, 19ax-mp 5 . . . . . . . . . . 11 Fun 𝐸
215resex 5402 . . . . . . . . . . 11 (𝑓 ↾ (𝑓 “ ran 𝐸)) ∈ V
22 cofunexg 7077 . . . . . . . . . . 11 ((Fun 𝐸 ∧ (𝑓 ↾ (𝑓 “ ran 𝐸)) ∈ V) → (𝐸 ∘ (𝑓 ↾ (𝑓 “ ran 𝐸))) ∈ V)
2320, 21, 22mp2an 707 . . . . . . . . . 10 (𝐸 ∘ (𝑓 ↾ (𝑓 “ ran 𝐸))) ∈ V
24 fofun 6073 . . . . . . . . . . . . 13 (𝑓:𝐴onto→ω → Fun 𝑓)
25 fores 6081 . . . . . . . . . . . . 13 ((Fun 𝑓 ∧ (𝑓 “ ran 𝐸) ⊆ dom 𝑓) → (𝑓 ↾ (𝑓 “ ran 𝐸)):(𝑓 “ ran 𝐸)–onto→(𝑓 “ (𝑓 “ ran 𝐸)))
2624, 9, 25sylancl 693 . . . . . . . . . . . 12 (𝑓:𝐴onto→ω → (𝑓 ↾ (𝑓 “ ran 𝐸)):(𝑓 “ ran 𝐸)–onto→(𝑓 “ (𝑓 “ ran 𝐸)))
27 f1f 6058 . . . . . . . . . . . . . . 15 (𝐸:ω–1-1→ω → 𝐸:ω⟶ω)
28 frn 6010 . . . . . . . . . . . . . . 15 (𝐸:ω⟶ω → ran 𝐸 ⊆ ω)
2915, 27, 28mp2b 10 . . . . . . . . . . . . . 14 ran 𝐸 ⊆ ω
30 foimacnv 6111 . . . . . . . . . . . . . 14 ((𝑓:𝐴onto→ω ∧ ran 𝐸 ⊆ ω) → (𝑓 “ (𝑓 “ ran 𝐸)) = ran 𝐸)
3129, 30mpan2 706 . . . . . . . . . . . . 13 (𝑓:𝐴onto→ω → (𝑓 “ (𝑓 “ ran 𝐸)) = ran 𝐸)
32 foeq3 6070 . . . . . . . . . . . . 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))
3331, 32syl 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))
3426, 33mpbid 222 . . . . . . . . . . 11 (𝑓:𝐴onto→ω → (𝑓 ↾ (𝑓 “ ran 𝐸)):(𝑓 “ ran 𝐸)–onto→ran 𝐸)
35 foco 6082 . . . . . . . . . . 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)
3618, 34, 35sylancr 694 . . . . . . . . . 10 (𝑓:𝐴onto→ω → (𝐸 ∘ (𝑓 ↾ (𝑓 “ ran 𝐸))):(𝑓 “ ran 𝐸)–onto→ω)
37 fowdom 8420 . . . . . . . . . 10 (((𝐸 ∘ (𝑓 ↾ (𝑓 “ ran 𝐸))) ∈ V ∧ (𝐸 ∘ (𝑓 ↾ (𝑓 “ ran 𝐸))):(𝑓 “ ran 𝐸)–onto→ω) → ω ≼* (𝑓 “ ran 𝐸))
3823, 36, 37sylancr 694 . . . . . . . . 9 (𝑓:𝐴onto→ω → ω ≼* (𝑓 “ ran 𝐸))
395cnvex 7060 . . . . . . . . . . . 12 𝑓 ∈ V
4039imaex 7051 . . . . . . . . . . 11 (𝑓 “ ran 𝐸) ∈ V
41 isfin3-2 9133 . . . . . . . . . . 11 ((𝑓 “ ran 𝐸) ∈ V → ((𝑓 “ ran 𝐸) ∈ FinIII ↔ ¬ ω ≼* (𝑓 “ ran 𝐸)))
4240, 41ax-mp 5 . . . . . . . . . 10 ((𝑓 “ ran 𝐸) ∈ FinIII ↔ ¬ ω ≼* (𝑓 “ ran 𝐸))
4342con2bii 347 . . . . . . . . 9 (ω ≼* (𝑓 “ ran 𝐸) ↔ ¬ (𝑓 “ ran 𝐸) ∈ FinIII)
4438, 43sylib 208 . . . . . . . 8 (𝑓:𝐴onto→ω → ¬ (𝑓 “ ran 𝐸) ∈ FinIII)
45 fin1a2lem.aa . . . . . . . . . . . . . . 15 𝑆 = (𝑥 ∈ On ↦ suc 𝑥)
4614, 45fin1a2lem6 9171 . . . . . . . . . . . . . 14 (𝑆 ↾ ran 𝐸):ran 𝐸1-1-onto→(ω ∖ ran 𝐸)
47 f1ocnv 6106 . . . . . . . . . . . . . 14 ((𝑆 ↾ ran 𝐸):ran 𝐸1-1-onto→(ω ∖ ran 𝐸) → (𝑆 ↾ ran 𝐸):(ω ∖ ran 𝐸)–1-1-onto→ran 𝐸)
48 f1ofo 6101 . . . . . . . . . . . . . 14 ((𝑆 ↾ ran 𝐸):(ω ∖ ran 𝐸)–1-1-onto→ran 𝐸(𝑆 ↾ ran 𝐸):(ω ∖ ran 𝐸)–onto→ran 𝐸)
4946, 47, 48mp2b 10 . . . . . . . . . . . . 13 (𝑆 ↾ ran 𝐸):(ω ∖ ran 𝐸)–onto→ran 𝐸
50 foco 6082 . . . . . . . . . . . . 13 ((𝐸:ran 𝐸onto→ω ∧ (𝑆 ↾ ran 𝐸):(ω ∖ ran 𝐸)–onto→ran 𝐸) → (𝐸(𝑆 ↾ ran 𝐸)):(ω ∖ ran 𝐸)–onto→ω)
5118, 49, 50mp2an 707 . . . . . . . . . . . 12 (𝐸(𝑆 ↾ ran 𝐸)):(ω ∖ ran 𝐸)–onto→ω
52 fofun 6073 . . . . . . . . . . . 12 ((𝐸(𝑆 ↾ ran 𝐸)):(ω ∖ ran 𝐸)–onto→ω → Fun (𝐸(𝑆 ↾ ran 𝐸)))
5351, 52ax-mp 5 . . . . . . . . . . 11 Fun (𝐸(𝑆 ↾ ran 𝐸))
545resex 5402 . . . . . . . . . . 11 (𝑓 ↾ (𝐴 ∖ (𝑓 “ ran 𝐸))) ∈ V
55 cofunexg 7077 . . . . . . . . . . 11 ((Fun (𝐸(𝑆 ↾ ran 𝐸)) ∧ (𝑓 ↾ (𝐴 ∖ (𝑓 “ ran 𝐸))) ∈ V) → ((𝐸(𝑆 ↾ ran 𝐸)) ∘ (𝑓 ↾ (𝐴 ∖ (𝑓 “ ran 𝐸)))) ∈ V)
5653, 54, 55mp2an 707 . . . . . . . . . 10 ((𝐸(𝑆 ↾ ran 𝐸)) ∘ (𝑓 ↾ (𝐴 ∖ (𝑓 “ ran 𝐸)))) ∈ V
57 difss 3715 . . . . . . . . . . . . . 14 (𝐴 ∖ (𝑓 “ ran 𝐸)) ⊆ 𝐴
5857, 11syl5sseqr 3633 . . . . . . . . . . . . 13 (𝑓:𝐴onto→ω → (𝐴 ∖ (𝑓 “ ran 𝐸)) ⊆ dom 𝑓)
59 fores 6081 . . . . . . . . . . . . 13 ((Fun 𝑓 ∧ (𝐴 ∖ (𝑓 “ ran 𝐸)) ⊆ dom 𝑓) → (𝑓 ↾ (𝐴 ∖ (𝑓 “ ran 𝐸))):(𝐴 ∖ (𝑓 “ ran 𝐸))–onto→(𝑓 “ (𝐴 ∖ (𝑓 “ ran 𝐸))))
6024, 58, 59syl2anc 692 . . . . . . . . . . . 12 (𝑓:𝐴onto→ω → (𝑓 ↾ (𝐴 ∖ (𝑓 “ ran 𝐸))):(𝐴 ∖ (𝑓 “ ran 𝐸))–onto→(𝑓 “ (𝐴 ∖ (𝑓 “ ran 𝐸))))
61 funcnvcnv 5914 . . . . . . . . . . . . . . . 16 (Fun 𝑓 → Fun 𝑓)
62 imadif 5931 . . . . . . . . . . . . . . . 16 (Fun 𝑓 → (𝑓 “ (ω ∖ ran 𝐸)) = ((𝑓 “ ω) ∖ (𝑓 “ ran 𝐸)))
6324, 61, 623syl 18 . . . . . . . . . . . . . . 15 (𝑓:𝐴onto→ω → (𝑓 “ (ω ∖ ran 𝐸)) = ((𝑓 “ ω) ∖ (𝑓 “ ran 𝐸)))
6463imaeq2d 5425 . . . . . . . . . . . . . 14 (𝑓:𝐴onto→ω → (𝑓 “ (𝑓 “ (ω ∖ ran 𝐸))) = (𝑓 “ ((𝑓 “ ω) ∖ (𝑓 “ ran 𝐸))))
65 difss 3715 . . . . . . . . . . . . . . 15 (ω ∖ ran 𝐸) ⊆ ω
66 foimacnv 6111 . . . . . . . . . . . . . . 15 ((𝑓:𝐴onto→ω ∧ (ω ∖ ran 𝐸) ⊆ ω) → (𝑓 “ (𝑓 “ (ω ∖ ran 𝐸))) = (ω ∖ ran 𝐸))
6765, 66mpan2 706 . . . . . . . . . . . . . 14 (𝑓:𝐴onto→ω → (𝑓 “ (𝑓 “ (ω ∖ ran 𝐸))) = (ω ∖ ran 𝐸))
68 fimacnv 6303 . . . . . . . . . . . . . . . . 17 (𝑓:𝐴⟶ω → (𝑓 “ ω) = 𝐴)
696, 68syl 17 . . . . . . . . . . . . . . . 16 (𝑓:𝐴onto→ω → (𝑓 “ ω) = 𝐴)
7069difeq1d 3705 . . . . . . . . . . . . . . 15 (𝑓:𝐴onto→ω → ((𝑓 “ ω) ∖ (𝑓 “ ran 𝐸)) = (𝐴 ∖ (𝑓 “ ran 𝐸)))
7170imaeq2d 5425 . . . . . . . . . . . . . 14 (𝑓:𝐴onto→ω → (𝑓 “ ((𝑓 “ ω) ∖ (𝑓 “ ran 𝐸))) = (𝑓 “ (𝐴 ∖ (𝑓 “ ran 𝐸))))
7264, 67, 713eqtr3rd 2664 . . . . . . . . . . . . 13 (𝑓:𝐴onto→ω → (𝑓 “ (𝐴 ∖ (𝑓 “ ran 𝐸))) = (ω ∖ ran 𝐸))
73 foeq3 6070 . . . . . . . . . . . . 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)))
7472, 73syl 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)))
7560, 74mpbid 222 . . . . . . . . . . 11 (𝑓:𝐴onto→ω → (𝑓 ↾ (𝐴 ∖ (𝑓 “ ran 𝐸))):(𝐴 ∖ (𝑓 “ ran 𝐸))–onto→(ω ∖ ran 𝐸))
76 foco 6082 . . . . . . . . . . 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)
7751, 75, 76sylancr 694 . . . . . . . . . 10 (𝑓:𝐴onto→ω → ((𝐸(𝑆 ↾ ran 𝐸)) ∘ (𝑓 ↾ (𝐴 ∖ (𝑓 “ ran 𝐸)))):(𝐴 ∖ (𝑓 “ ran 𝐸))–onto→ω)
78 fowdom 8420 . . . . . . . . . 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)))
7956, 77, 78sylancr 694 . . . . . . . . 9 (𝑓:𝐴onto→ω → ω ≼* (𝐴 ∖ (𝑓 “ ran 𝐸)))
80 difexg 4768 . . . . . . . . . . 11 (𝐴 ∈ V → (𝐴 ∖ (𝑓 “ ran 𝐸)) ∈ V)
81 isfin3-2 9133 . . . . . . . . . . 11 ((𝐴 ∖ (𝑓 “ ran 𝐸)) ∈ V → ((𝐴 ∖ (𝑓 “ ran 𝐸)) ∈ FinIII ↔ ¬ ω ≼* (𝐴 ∖ (𝑓 “ ran 𝐸))))
828, 80, 813syl 18 . . . . . . . . . 10 (𝑓:𝐴onto→ω → ((𝐴 ∖ (𝑓 “ ran 𝐸)) ∈ FinIII ↔ ¬ ω ≼* (𝐴 ∖ (𝑓 “ ran 𝐸))))
8382con2bid 344 . . . . . . . . 9 (𝑓:𝐴onto→ω → (ω ≼* (𝐴 ∖ (𝑓 “ ran 𝐸)) ↔ ¬ (𝐴 ∖ (𝑓 “ ran 𝐸)) ∈ FinIII))
8479, 83mpbid 222 . . . . . . . 8 (𝑓:𝐴onto→ω → ¬ (𝐴 ∖ (𝑓 “ ran 𝐸)) ∈ FinIII)
85 eleq1 2686 . . . . . . . . . . . 12 (𝑦 = (𝑓 “ ran 𝐸) → (𝑦 ∈ FinIII ↔ (𝑓 “ ran 𝐸) ∈ FinIII))
86 difeq2 3700 . . . . . . . . . . . . 13 (𝑦 = (𝑓 “ ran 𝐸) → (𝐴𝑦) = (𝐴 ∖ (𝑓 “ ran 𝐸)))
8786eleq1d 2683 . . . . . . . . . . . 12 (𝑦 = (𝑓 “ ran 𝐸) → ((𝐴𝑦) ∈ FinIII ↔ (𝐴 ∖ (𝑓 “ ran 𝐸)) ∈ FinIII))
8885, 87orbi12d 745 . . . . . . . . . . 11 (𝑦 = (𝑓 “ ran 𝐸) → ((𝑦 ∈ FinIII ∨ (𝐴𝑦) ∈ FinIII) ↔ ((𝑓 “ ran 𝐸) ∈ FinIII ∨ (𝐴 ∖ (𝑓 “ ran 𝐸)) ∈ FinIII)))
8988notbid 308 . . . . . . . . . 10 (𝑦 = (𝑓 “ ran 𝐸) → (¬ (𝑦 ∈ FinIII ∨ (𝐴𝑦) ∈ FinIII) ↔ ¬ ((𝑓 “ ran 𝐸) ∈ FinIII ∨ (𝐴 ∖ (𝑓 “ ran 𝐸)) ∈ FinIII)))
90 ioran 511 . . . . . . . . . 10 (¬ ((𝑓 “ ran 𝐸) ∈ FinIII ∨ (𝐴 ∖ (𝑓 “ ran 𝐸)) ∈ FinIII) ↔ (¬ (𝑓 “ ran 𝐸) ∈ FinIII ∧ ¬ (𝐴 ∖ (𝑓 “ ran 𝐸)) ∈ FinIII))
9189, 90syl6bb 276 . . . . . . . . 9 (𝑦 = (𝑓 “ ran 𝐸) → (¬ (𝑦 ∈ FinIII ∨ (𝐴𝑦) ∈ FinIII) ↔ (¬ (𝑓 “ ran 𝐸) ∈ FinIII ∧ ¬ (𝐴 ∖ (𝑓 “ ran 𝐸)) ∈ FinIII)))
9291rspcev 3295 . . . . . . . 8 (((𝑓 “ ran 𝐸) ∈ 𝒫 𝐴 ∧ (¬ (𝑓 “ ran 𝐸) ∈ FinIII ∧ ¬ (𝐴 ∖ (𝑓 “ ran 𝐸)) ∈ FinIII)) → ∃𝑦 ∈ 𝒫 𝐴 ¬ (𝑦 ∈ FinIII ∨ (𝐴𝑦) ∈ FinIII))
9313, 44, 84, 92syl12anc 1321 . . . . . . 7 (𝑓:𝐴onto→ω → ∃𝑦 ∈ 𝒫 𝐴 ¬ (𝑦 ∈ FinIII ∨ (𝐴𝑦) ∈ FinIII))
94 rexnal 2989 . . . . . . 7 (∃𝑦 ∈ 𝒫 𝐴 ¬ (𝑦 ∈ FinIII ∨ (𝐴𝑦) ∈ FinIII) ↔ ¬ ∀𝑦 ∈ 𝒫 𝐴(𝑦 ∈ FinIII ∨ (𝐴𝑦) ∈ FinIII))
9593, 94sylib 208 . . . . . 6 (𝑓:𝐴onto→ω → ¬ ∀𝑦 ∈ 𝒫 𝐴(𝑦 ∈ FinIII ∨ (𝐴𝑦) ∈ FinIII))
9695exlimiv 1855 . . . . 5 (∃𝑓 𝑓:𝐴onto→ω → ¬ ∀𝑦 ∈ 𝒫 𝐴(𝑦 ∈ FinIII ∨ (𝐴𝑦) ∈ FinIII))
974, 96sylbi 207 . . . 4 (ω ≼* 𝐴 → ¬ ∀𝑦 ∈ 𝒫 𝐴(𝑦 ∈ FinIII ∨ (𝐴𝑦) ∈ FinIII))
9897con2i 134 . . 3 (∀𝑦 ∈ 𝒫 𝐴(𝑦 ∈ FinIII ∨ (𝐴𝑦) ∈ FinIII) → ¬ ω ≼* 𝐴)
99 isfin3-2 9133 . . 3 (𝐴𝑉 → (𝐴 ∈ FinIII ↔ ¬ ω ≼* 𝐴))
10098, 99syl5ibr 236 . 2 (𝐴𝑉 → (∀𝑦 ∈ 𝒫 𝐴(𝑦 ∈ FinIII ∨ (𝐴𝑦) ∈ FinIII) → 𝐴 ∈ FinIII))
101100imp 445 1 ((𝐴𝑉 ∧ ∀𝑦 ∈ 𝒫 𝐴(𝑦 ∈ FinIII ∨ (𝐴𝑦) ∈ FinIII)) → 𝐴 ∈ FinIII)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 383  wa 384   = wceq 1480  wex 1701  wcel 1987  wne 2790  wral 2907  wrex 2908  Vcvv 3186  cdif 3552  wss 3555  c0 3891  𝒫 cpw 4130   class class class wbr 4613  cmpt 4673  ccnv 5073  dom cdm 5074  ran crn 5075  cres 5076  cima 5077  ccom 5078  Oncon0 5682  suc csuc 5684  Fun wfun 5841  wf 5843  1-1wf1 5844  ontowfo 5845  1-1-ontowf1o 5846  (class class class)co 6604  ωcom 7012  2𝑜c2o 7499   ·𝑜 comu 7503  * cwdom 8406  FinIIIcfin3 9047
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-se 5034  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-isom 5856  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-seqom 7488  df-1o 7505  df-2o 7506  df-oadd 7509  df-omul 7510  df-er 7687  df-map 7804  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-wdom 8408  df-card 8709  df-fin4 9053  df-fin3 9054
This theorem is referenced by:  fin1a2lem8  9173
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