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Theorem fin1a2lem12 10167
Description: Lemma for fin1a2 10171. (Contributed by Stefan O'Rear, 8-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.)
Assertion
Ref Expression
fin1a2lem12 (((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) → ¬ 𝐵 ∈ FinIII)

Proof of Theorem fin1a2lem12
Dummy variables 𝑑 𝑒 𝑓 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr 485 . . 3 ((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) ∧ 𝐵 ∈ FinIII) → 𝐵 ∈ FinIII)
2 simpll1 1211 . . . . . . 7 ((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) ∧ 𝐵 ∈ FinIII) → 𝐴 ⊆ 𝒫 𝐵)
32adantr 481 . . . . . 6 (((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) ∧ 𝐵 ∈ FinIII) ∧ 𝑒 ∈ ω) → 𝐴 ⊆ 𝒫 𝐵)
4 ssrab2 4013 . . . . . . . 8 {𝑓𝐴𝑓𝑒} ⊆ 𝐴
54unissi 4848 . . . . . . 7 {𝑓𝐴𝑓𝑒} ⊆ 𝐴
6 sspwuni 5029 . . . . . . . 8 (𝐴 ⊆ 𝒫 𝐵 𝐴𝐵)
76biimpi 215 . . . . . . 7 (𝐴 ⊆ 𝒫 𝐵 𝐴𝐵)
85, 7sstrid 3932 . . . . . 6 (𝐴 ⊆ 𝒫 𝐵 {𝑓𝐴𝑓𝑒} ⊆ 𝐵)
93, 8syl 17 . . . . 5 (((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) ∧ 𝐵 ∈ FinIII) ∧ 𝑒 ∈ ω) → {𝑓𝐴𝑓𝑒} ⊆ 𝐵)
10 elpw2g 5268 . . . . . 6 (𝐵 ∈ FinIII → ( {𝑓𝐴𝑓𝑒} ∈ 𝒫 𝐵 {𝑓𝐴𝑓𝑒} ⊆ 𝐵))
1110ad2antlr 724 . . . . 5 (((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) ∧ 𝐵 ∈ FinIII) ∧ 𝑒 ∈ ω) → ( {𝑓𝐴𝑓𝑒} ∈ 𝒫 𝐵 {𝑓𝐴𝑓𝑒} ⊆ 𝐵))
129, 11mpbird 256 . . . 4 (((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) ∧ 𝐵 ∈ FinIII) ∧ 𝑒 ∈ ω) → {𝑓𝐴𝑓𝑒} ∈ 𝒫 𝐵)
1312fmpttd 6989 . . 3 ((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) ∧ 𝐵 ∈ FinIII) → (𝑒 ∈ ω ↦ {𝑓𝐴𝑓𝑒}):ω⟶𝒫 𝐵)
14 vex 3436 . . . . . . . . . . 11 𝑑 ∈ V
1514sucex 7656 . . . . . . . . . 10 suc 𝑑 ∈ V
16 sssucid 6343 . . . . . . . . . 10 𝑑 ⊆ suc 𝑑
17 ssdomg 8786 . . . . . . . . . 10 (suc 𝑑 ∈ V → (𝑑 ⊆ suc 𝑑𝑑 ≼ suc 𝑑))
1815, 16, 17mp2 9 . . . . . . . . 9 𝑑 ≼ suc 𝑑
19 domtr 8793 . . . . . . . . 9 ((𝑓𝑑𝑑 ≼ suc 𝑑) → 𝑓 ≼ suc 𝑑)
2018, 19mpan2 688 . . . . . . . 8 (𝑓𝑑𝑓 ≼ suc 𝑑)
2120a1i 11 . . . . . . 7 (𝑓𝐴 → (𝑓𝑑𝑓 ≼ suc 𝑑))
2221ss2rabi 4010 . . . . . 6 {𝑓𝐴𝑓𝑑} ⊆ {𝑓𝐴𝑓 ≼ suc 𝑑}
23 uniss 4847 . . . . . 6 ({𝑓𝐴𝑓𝑑} ⊆ {𝑓𝐴𝑓 ≼ suc 𝑑} → {𝑓𝐴𝑓𝑑} ⊆ {𝑓𝐴𝑓 ≼ suc 𝑑})
2422, 23mp1i 13 . . . . 5 (((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) ∧ 𝐵 ∈ FinIII) ∧ 𝑑 ∈ ω) → {𝑓𝐴𝑓𝑑} ⊆ {𝑓𝐴𝑓 ≼ suc 𝑑})
25 id 22 . . . . . 6 (𝑑 ∈ ω → 𝑑 ∈ ω)
26 pwexg 5301 . . . . . . . . 9 (𝐵 ∈ FinIII → 𝒫 𝐵 ∈ V)
2726adantl 482 . . . . . . . 8 ((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) ∧ 𝐵 ∈ FinIII) → 𝒫 𝐵 ∈ V)
2827, 2ssexd 5248 . . . . . . 7 ((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) ∧ 𝐵 ∈ FinIII) → 𝐴 ∈ V)
29 rabexg 5255 . . . . . . 7 (𝐴 ∈ V → {𝑓𝐴𝑓𝑑} ∈ V)
30 uniexg 7593 . . . . . . 7 ({𝑓𝐴𝑓𝑑} ∈ V → {𝑓𝐴𝑓𝑑} ∈ V)
3128, 29, 303syl 18 . . . . . 6 ((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) ∧ 𝐵 ∈ FinIII) → {𝑓𝐴𝑓𝑑} ∈ V)
32 breq2 5078 . . . . . . . . 9 (𝑒 = 𝑑 → (𝑓𝑒𝑓𝑑))
3332rabbidv 3414 . . . . . . . 8 (𝑒 = 𝑑 → {𝑓𝐴𝑓𝑒} = {𝑓𝐴𝑓𝑑})
3433unieqd 4853 . . . . . . 7 (𝑒 = 𝑑 {𝑓𝐴𝑓𝑒} = {𝑓𝐴𝑓𝑑})
35 eqid 2738 . . . . . . 7 (𝑒 ∈ ω ↦ {𝑓𝐴𝑓𝑒}) = (𝑒 ∈ ω ↦ {𝑓𝐴𝑓𝑒})
3634, 35fvmptg 6873 . . . . . 6 ((𝑑 ∈ ω ∧ {𝑓𝐴𝑓𝑑} ∈ V) → ((𝑒 ∈ ω ↦ {𝑓𝐴𝑓𝑒})‘𝑑) = {𝑓𝐴𝑓𝑑})
3725, 31, 36syl2anr 597 . . . . 5 (((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) ∧ 𝐵 ∈ FinIII) ∧ 𝑑 ∈ ω) → ((𝑒 ∈ ω ↦ {𝑓𝐴𝑓𝑒})‘𝑑) = {𝑓𝐴𝑓𝑑})
38 peano2 7737 . . . . . 6 (𝑑 ∈ ω → suc 𝑑 ∈ ω)
39 rabexg 5255 . . . . . . 7 (𝐴 ∈ V → {𝑓𝐴𝑓 ≼ suc 𝑑} ∈ V)
40 uniexg 7593 . . . . . . 7 ({𝑓𝐴𝑓 ≼ suc 𝑑} ∈ V → {𝑓𝐴𝑓 ≼ suc 𝑑} ∈ V)
4128, 39, 403syl 18 . . . . . 6 ((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) ∧ 𝐵 ∈ FinIII) → {𝑓𝐴𝑓 ≼ suc 𝑑} ∈ V)
42 breq2 5078 . . . . . . . . 9 (𝑒 = suc 𝑑 → (𝑓𝑒𝑓 ≼ suc 𝑑))
4342rabbidv 3414 . . . . . . . 8 (𝑒 = suc 𝑑 → {𝑓𝐴𝑓𝑒} = {𝑓𝐴𝑓 ≼ suc 𝑑})
4443unieqd 4853 . . . . . . 7 (𝑒 = suc 𝑑 {𝑓𝐴𝑓𝑒} = {𝑓𝐴𝑓 ≼ suc 𝑑})
4544, 35fvmptg 6873 . . . . . 6 ((suc 𝑑 ∈ ω ∧ {𝑓𝐴𝑓 ≼ suc 𝑑} ∈ V) → ((𝑒 ∈ ω ↦ {𝑓𝐴𝑓𝑒})‘suc 𝑑) = {𝑓𝐴𝑓 ≼ suc 𝑑})
4638, 41, 45syl2anr 597 . . . . 5 (((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) ∧ 𝐵 ∈ FinIII) ∧ 𝑑 ∈ ω) → ((𝑒 ∈ ω ↦ {𝑓𝐴𝑓𝑒})‘suc 𝑑) = {𝑓𝐴𝑓 ≼ suc 𝑑})
4724, 37, 463sstr4d 3968 . . . 4 (((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) ∧ 𝐵 ∈ FinIII) ∧ 𝑑 ∈ ω) → ((𝑒 ∈ ω ↦ {𝑓𝐴𝑓𝑒})‘𝑑) ⊆ ((𝑒 ∈ ω ↦ {𝑓𝐴𝑓𝑒})‘suc 𝑑))
4847ralrimiva 3103 . . 3 ((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) ∧ 𝐵 ∈ FinIII) → ∀𝑑 ∈ ω ((𝑒 ∈ ω ↦ {𝑓𝐴𝑓𝑒})‘𝑑) ⊆ ((𝑒 ∈ ω ↦ {𝑓𝐴𝑓𝑒})‘suc 𝑑))
49 fin34i 10137 . . 3 ((𝐵 ∈ FinIII ∧ (𝑒 ∈ ω ↦ {𝑓𝐴𝑓𝑒}):ω⟶𝒫 𝐵 ∧ ∀𝑑 ∈ ω ((𝑒 ∈ ω ↦ {𝑓𝐴𝑓𝑒})‘𝑑) ⊆ ((𝑒 ∈ ω ↦ {𝑓𝐴𝑓𝑒})‘suc 𝑑)) → ran (𝑒 ∈ ω ↦ {𝑓𝐴𝑓𝑒}) ∈ ran (𝑒 ∈ ω ↦ {𝑓𝐴𝑓𝑒}))
501, 13, 48, 49syl3anc 1370 . 2 ((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) ∧ 𝐵 ∈ FinIII) → ran (𝑒 ∈ ω ↦ {𝑓𝐴𝑓𝑒}) ∈ ran (𝑒 ∈ ω ↦ {𝑓𝐴𝑓𝑒}))
51 fin1a2lem11 10166 . . . . . 6 (( [] Or 𝐴𝐴 ⊆ Fin) → ran (𝑒 ∈ ω ↦ {𝑓𝐴𝑓𝑒}) = (𝐴 ∪ {∅}))
5251adantrr 714 . . . . 5 (( [] Or 𝐴 ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) → ran (𝑒 ∈ ω ↦ {𝑓𝐴𝑓𝑒}) = (𝐴 ∪ {∅}))
53523ad2antl2 1185 . . . 4 (((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) → ran (𝑒 ∈ ω ↦ {𝑓𝐴𝑓𝑒}) = (𝐴 ∪ {∅}))
5453adantr 481 . . 3 ((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) ∧ 𝐵 ∈ FinIII) → ran (𝑒 ∈ ω ↦ {𝑓𝐴𝑓𝑒}) = (𝐴 ∪ {∅}))
55 simpll3 1213 . . . . . 6 ((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) ∧ 𝐵 ∈ FinIII) → ¬ 𝐴𝐴)
56 simplrr 775 . . . . . . 7 ((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) ∧ 𝐵 ∈ FinIII) → 𝐴 ≠ ∅)
57 sspwuni 5029 . . . . . . . . . . 11 (𝐴 ⊆ 𝒫 ∅ ↔ 𝐴 ⊆ ∅)
58 ss0b 4331 . . . . . . . . . . 11 ( 𝐴 ⊆ ∅ ↔ 𝐴 = ∅)
5957, 58bitri 274 . . . . . . . . . 10 (𝐴 ⊆ 𝒫 ∅ ↔ 𝐴 = ∅)
60 pw0 4745 . . . . . . . . . . . . 13 𝒫 ∅ = {∅}
6160sseq2i 3950 . . . . . . . . . . . 12 (𝐴 ⊆ 𝒫 ∅ ↔ 𝐴 ⊆ {∅})
62 sssn 4759 . . . . . . . . . . . 12 (𝐴 ⊆ {∅} ↔ (𝐴 = ∅ ∨ 𝐴 = {∅}))
6361, 62bitri 274 . . . . . . . . . . 11 (𝐴 ⊆ 𝒫 ∅ ↔ (𝐴 = ∅ ∨ 𝐴 = {∅}))
64 df-ne 2944 . . . . . . . . . . . 12 (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅)
65 0ex 5231 . . . . . . . . . . . . . . . . 17 ∅ ∈ V
6665unisn 4861 . . . . . . . . . . . . . . . 16 {∅} = ∅
6765snid 4597 . . . . . . . . . . . . . . . 16 ∅ ∈ {∅}
6866, 67eqeltri 2835 . . . . . . . . . . . . . . 15 {∅} ∈ {∅}
69 unieq 4850 . . . . . . . . . . . . . . . 16 (𝐴 = {∅} → 𝐴 = {∅})
70 id 22 . . . . . . . . . . . . . . . 16 (𝐴 = {∅} → 𝐴 = {∅})
7169, 70eleq12d 2833 . . . . . . . . . . . . . . 15 (𝐴 = {∅} → ( 𝐴𝐴 {∅} ∈ {∅}))
7268, 71mpbiri 257 . . . . . . . . . . . . . 14 (𝐴 = {∅} → 𝐴𝐴)
7372orim2i 908 . . . . . . . . . . . . 13 ((𝐴 = ∅ ∨ 𝐴 = {∅}) → (𝐴 = ∅ ∨ 𝐴𝐴))
7473ord 861 . . . . . . . . . . . 12 ((𝐴 = ∅ ∨ 𝐴 = {∅}) → (¬ 𝐴 = ∅ → 𝐴𝐴))
7564, 74syl5bi 241 . . . . . . . . . . 11 ((𝐴 = ∅ ∨ 𝐴 = {∅}) → (𝐴 ≠ ∅ → 𝐴𝐴))
7663, 75sylbi 216 . . . . . . . . . 10 (𝐴 ⊆ 𝒫 ∅ → (𝐴 ≠ ∅ → 𝐴𝐴))
7759, 76sylbir 234 . . . . . . . . 9 ( 𝐴 = ∅ → (𝐴 ≠ ∅ → 𝐴𝐴))
7877com12 32 . . . . . . . 8 (𝐴 ≠ ∅ → ( 𝐴 = ∅ → 𝐴𝐴))
7978con3d 152 . . . . . . 7 (𝐴 ≠ ∅ → (¬ 𝐴𝐴 → ¬ 𝐴 = ∅))
8056, 55, 79sylc 65 . . . . . 6 ((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) ∧ 𝐵 ∈ FinIII) → ¬ 𝐴 = ∅)
81 ioran 981 . . . . . 6 (¬ ( 𝐴𝐴 𝐴 = ∅) ↔ (¬ 𝐴𝐴 ∧ ¬ 𝐴 = ∅))
8255, 80, 81sylanbrc 583 . . . . 5 ((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) ∧ 𝐵 ∈ FinIII) → ¬ ( 𝐴𝐴 𝐴 = ∅))
83 uniun 4864 . . . . . . . 8 (𝐴 ∪ {∅}) = ( 𝐴 {∅})
8466uneq2i 4094 . . . . . . . 8 ( 𝐴 {∅}) = ( 𝐴 ∪ ∅)
85 un0 4324 . . . . . . . 8 ( 𝐴 ∪ ∅) = 𝐴
8683, 84, 853eqtri 2770 . . . . . . 7 (𝐴 ∪ {∅}) = 𝐴
8786eleq1i 2829 . . . . . 6 ( (𝐴 ∪ {∅}) ∈ (𝐴 ∪ {∅}) ↔ 𝐴 ∈ (𝐴 ∪ {∅}))
88 elun 4083 . . . . . 6 ( 𝐴 ∈ (𝐴 ∪ {∅}) ↔ ( 𝐴𝐴 𝐴 ∈ {∅}))
8965elsn2 4600 . . . . . . 7 ( 𝐴 ∈ {∅} ↔ 𝐴 = ∅)
9089orbi2i 910 . . . . . 6 (( 𝐴𝐴 𝐴 ∈ {∅}) ↔ ( 𝐴𝐴 𝐴 = ∅))
9187, 88, 903bitri 297 . . . . 5 ( (𝐴 ∪ {∅}) ∈ (𝐴 ∪ {∅}) ↔ ( 𝐴𝐴 𝐴 = ∅))
9282, 91sylnibr 329 . . . 4 ((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) ∧ 𝐵 ∈ FinIII) → ¬ (𝐴 ∪ {∅}) ∈ (𝐴 ∪ {∅}))
93 unieq 4850 . . . . . 6 (ran (𝑒 ∈ ω ↦ {𝑓𝐴𝑓𝑒}) = (𝐴 ∪ {∅}) → ran (𝑒 ∈ ω ↦ {𝑓𝐴𝑓𝑒}) = (𝐴 ∪ {∅}))
94 id 22 . . . . . 6 (ran (𝑒 ∈ ω ↦ {𝑓𝐴𝑓𝑒}) = (𝐴 ∪ {∅}) → ran (𝑒 ∈ ω ↦ {𝑓𝐴𝑓𝑒}) = (𝐴 ∪ {∅}))
9593, 94eleq12d 2833 . . . . 5 (ran (𝑒 ∈ ω ↦ {𝑓𝐴𝑓𝑒}) = (𝐴 ∪ {∅}) → ( ran (𝑒 ∈ ω ↦ {𝑓𝐴𝑓𝑒}) ∈ ran (𝑒 ∈ ω ↦ {𝑓𝐴𝑓𝑒}) ↔ (𝐴 ∪ {∅}) ∈ (𝐴 ∪ {∅})))
9695notbid 318 . . . 4 (ran (𝑒 ∈ ω ↦ {𝑓𝐴𝑓𝑒}) = (𝐴 ∪ {∅}) → (¬ ran (𝑒 ∈ ω ↦ {𝑓𝐴𝑓𝑒}) ∈ ran (𝑒 ∈ ω ↦ {𝑓𝐴𝑓𝑒}) ↔ ¬ (𝐴 ∪ {∅}) ∈ (𝐴 ∪ {∅})))
9792, 96syl5ibrcom 246 . . 3 ((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) ∧ 𝐵 ∈ FinIII) → (ran (𝑒 ∈ ω ↦ {𝑓𝐴𝑓𝑒}) = (𝐴 ∪ {∅}) → ¬ ran (𝑒 ∈ ω ↦ {𝑓𝐴𝑓𝑒}) ∈ ran (𝑒 ∈ ω ↦ {𝑓𝐴𝑓𝑒})))
9854, 97mpd 15 . 2 ((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) ∧ 𝐵 ∈ FinIII) → ¬ ran (𝑒 ∈ ω ↦ {𝑓𝐴𝑓𝑒}) ∈ ran (𝑒 ∈ ω ↦ {𝑓𝐴𝑓𝑒}))
9950, 98pm2.65da 814 1 (((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) → ¬ 𝐵 ∈ FinIII)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  wo 844  w3a 1086   = wceq 1539  wcel 2106  wne 2943  wral 3064  {crab 3068  Vcvv 3432  cun 3885  wss 3887  c0 4256  𝒫 cpw 4533  {csn 4561   cuni 4839   class class class wbr 5074  cmpt 5157   Or wor 5502  ran crn 5590  suc csuc 6268  wf 6429  cfv 6433   [] crpss 7575  ωcom 7712  cdom 8731  Fincfn 8733  FinIIIcfin3 10037
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-se 5545  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-isom 6442  df-riota 7232  df-ov 7278  df-rpss 7576  df-om 7713  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-er 8498  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-wdom 9324  df-card 9697  df-fin4 10043  df-fin3 10044
This theorem is referenced by:  fin1a2s  10170
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