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Theorem fin1a2lem12 10347
Description: Lemma for fin1a2 10351. (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 1212 . . . . . . 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 4037 . . . . . . . 8 {𝑓𝐴𝑓𝑒} ⊆ 𝐴
54unissi 4874 . . . . . . 7 {𝑓𝐴𝑓𝑒} ⊆ 𝐴
6 sspwuni 5060 . . . . . . . 8 (𝐴 ⊆ 𝒫 𝐵 𝐴𝐵)
76biimpi 215 . . . . . . 7 (𝐴 ⊆ 𝒫 𝐵 𝐴𝐵)
85, 7sstrid 3955 . . . . . 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 5301 . . . . . 6 (𝐵 ∈ FinIII → ( {𝑓𝐴𝑓𝑒} ∈ 𝒫 𝐵 {𝑓𝐴𝑓𝑒} ⊆ 𝐵))
1110ad2antlr 725 . . . . 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 7063 . . 3 ((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) ∧ 𝐵 ∈ FinIII) → (𝑒 ∈ ω ↦ {𝑓𝐴𝑓𝑒}):ω⟶𝒫 𝐵)
14 vex 3449 . . . . . . . . . . 11 𝑑 ∈ V
1514sucex 7741 . . . . . . . . . 10 suc 𝑑 ∈ V
16 sssucid 6397 . . . . . . . . . 10 𝑑 ⊆ suc 𝑑
17 ssdomg 8940 . . . . . . . . . 10 (suc 𝑑 ∈ V → (𝑑 ⊆ suc 𝑑𝑑 ≼ suc 𝑑))
1815, 16, 17mp2 9 . . . . . . . . 9 𝑑 ≼ suc 𝑑
19 domtr 8947 . . . . . . . . 9 ((𝑓𝑑𝑑 ≼ suc 𝑑) → 𝑓 ≼ suc 𝑑)
2018, 19mpan2 689 . . . . . . . 8 (𝑓𝑑𝑓 ≼ suc 𝑑)
2120a1i 11 . . . . . . 7 (𝑓𝐴 → (𝑓𝑑𝑓 ≼ suc 𝑑))
2221ss2rabi 4034 . . . . . 6 {𝑓𝐴𝑓𝑑} ⊆ {𝑓𝐴𝑓 ≼ suc 𝑑}
23 uniss 4873 . . . . . 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 5333 . . . . . . . . 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 5281 . . . . . . 7 ((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) ∧ 𝐵 ∈ FinIII) → 𝐴 ∈ V)
29 rabexg 5288 . . . . . . 7 (𝐴 ∈ V → {𝑓𝐴𝑓𝑑} ∈ V)
30 uniexg 7677 . . . . . . 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 5109 . . . . . . . . 9 (𝑒 = 𝑑 → (𝑓𝑒𝑓𝑑))
3332rabbidv 3415 . . . . . . . 8 (𝑒 = 𝑑 → {𝑓𝐴𝑓𝑒} = {𝑓𝐴𝑓𝑑})
3433unieqd 4879 . . . . . . 7 (𝑒 = 𝑑 {𝑓𝐴𝑓𝑒} = {𝑓𝐴𝑓𝑑})
35 eqid 2736 . . . . . . 7 (𝑒 ∈ ω ↦ {𝑓𝐴𝑓𝑒}) = (𝑒 ∈ ω ↦ {𝑓𝐴𝑓𝑒})
3634, 35fvmptg 6946 . . . . . 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 7827 . . . . . 6 (𝑑 ∈ ω → suc 𝑑 ∈ ω)
39 rabexg 5288 . . . . . . 7 (𝐴 ∈ V → {𝑓𝐴𝑓 ≼ suc 𝑑} ∈ V)
40 uniexg 7677 . . . . . . 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 5109 . . . . . . . . 9 (𝑒 = suc 𝑑 → (𝑓𝑒𝑓 ≼ suc 𝑑))
4342rabbidv 3415 . . . . . . . 8 (𝑒 = suc 𝑑 → {𝑓𝐴𝑓𝑒} = {𝑓𝐴𝑓 ≼ suc 𝑑})
4443unieqd 4879 . . . . . . 7 (𝑒 = suc 𝑑 {𝑓𝐴𝑓𝑒} = {𝑓𝐴𝑓 ≼ suc 𝑑})
4544, 35fvmptg 6946 . . . . . 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 3991 . . . 4 (((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) ∧ 𝐵 ∈ FinIII) ∧ 𝑑 ∈ ω) → ((𝑒 ∈ ω ↦ {𝑓𝐴𝑓𝑒})‘𝑑) ⊆ ((𝑒 ∈ ω ↦ {𝑓𝐴𝑓𝑒})‘suc 𝑑))
4847ralrimiva 3143 . . 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 10317 . . 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 1371 . 2 ((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) ∧ 𝐵 ∈ FinIII) → ran (𝑒 ∈ ω ↦ {𝑓𝐴𝑓𝑒}) ∈ ran (𝑒 ∈ ω ↦ {𝑓𝐴𝑓𝑒}))
51 fin1a2lem11 10346 . . . . . 6 (( [] Or 𝐴𝐴 ⊆ Fin) → ran (𝑒 ∈ ω ↦ {𝑓𝐴𝑓𝑒}) = (𝐴 ∪ {∅}))
5251adantrr 715 . . . . 5 (( [] Or 𝐴 ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) → ran (𝑒 ∈ ω ↦ {𝑓𝐴𝑓𝑒}) = (𝐴 ∪ {∅}))
53523ad2antl2 1186 . . . 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 1214 . . . . . 6 ((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) ∧ 𝐵 ∈ FinIII) → ¬ 𝐴𝐴)
56 simplrr 776 . . . . . . 7 ((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) ∧ 𝐵 ∈ FinIII) → 𝐴 ≠ ∅)
57 sspwuni 5060 . . . . . . . . . . 11 (𝐴 ⊆ 𝒫 ∅ ↔ 𝐴 ⊆ ∅)
58 ss0b 4357 . . . . . . . . . . 11 ( 𝐴 ⊆ ∅ ↔ 𝐴 = ∅)
5957, 58bitri 274 . . . . . . . . . 10 (𝐴 ⊆ 𝒫 ∅ ↔ 𝐴 = ∅)
60 pw0 4772 . . . . . . . . . . . . 13 𝒫 ∅ = {∅}
6160sseq2i 3973 . . . . . . . . . . . 12 (𝐴 ⊆ 𝒫 ∅ ↔ 𝐴 ⊆ {∅})
62 sssn 4786 . . . . . . . . . . . 12 (𝐴 ⊆ {∅} ↔ (𝐴 = ∅ ∨ 𝐴 = {∅}))
6361, 62bitri 274 . . . . . . . . . . 11 (𝐴 ⊆ 𝒫 ∅ ↔ (𝐴 = ∅ ∨ 𝐴 = {∅}))
64 df-ne 2944 . . . . . . . . . . . 12 (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅)
65 0ex 5264 . . . . . . . . . . . . . . . . 17 ∅ ∈ V
6665unisn 4887 . . . . . . . . . . . . . . . 16 {∅} = ∅
6765snid 4622 . . . . . . . . . . . . . . . 16 ∅ ∈ {∅}
6866, 67eqeltri 2834 . . . . . . . . . . . . . . 15 {∅} ∈ {∅}
69 unieq 4876 . . . . . . . . . . . . . . . 16 (𝐴 = {∅} → 𝐴 = {∅})
70 id 22 . . . . . . . . . . . . . . . 16 (𝐴 = {∅} → 𝐴 = {∅})
7169, 70eleq12d 2832 . . . . . . . . . . . . . . 15 (𝐴 = {∅} → ( 𝐴𝐴 {∅} ∈ {∅}))
7268, 71mpbiri 257 . . . . . . . . . . . . . 14 (𝐴 = {∅} → 𝐴𝐴)
7372orim2i 909 . . . . . . . . . . . . 13 ((𝐴 = ∅ ∨ 𝐴 = {∅}) → (𝐴 = ∅ ∨ 𝐴𝐴))
7473ord 862 . . . . . . . . . . . 12 ((𝐴 = ∅ ∨ 𝐴 = {∅}) → (¬ 𝐴 = ∅ → 𝐴𝐴))
7564, 74biimtrid 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 982 . . . . . 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 4891 . . . . . . . 8 (𝐴 ∪ {∅}) = ( 𝐴 {∅})
8466uneq2i 4120 . . . . . . . 8 ( 𝐴 {∅}) = ( 𝐴 ∪ ∅)
85 un0 4350 . . . . . . . 8 ( 𝐴 ∪ ∅) = 𝐴
8683, 84, 853eqtri 2768 . . . . . . 7 (𝐴 ∪ {∅}) = 𝐴
8786eleq1i 2828 . . . . . 6 ( (𝐴 ∪ {∅}) ∈ (𝐴 ∪ {∅}) ↔ 𝐴 ∈ (𝐴 ∪ {∅}))
88 elun 4108 . . . . . 6 ( 𝐴 ∈ (𝐴 ∪ {∅}) ↔ ( 𝐴𝐴 𝐴 ∈ {∅}))
8965elsn2 4625 . . . . . . 7 ( 𝐴 ∈ {∅} ↔ 𝐴 = ∅)
9089orbi2i 911 . . . . . 6 (( 𝐴𝐴 𝐴 ∈ {∅}) ↔ ( 𝐴𝐴 𝐴 = ∅))
9187, 88, 903bitri 296 . . . . 5 ( (𝐴 ∪ {∅}) ∈ (𝐴 ∪ {∅}) ↔ ( 𝐴𝐴 𝐴 = ∅))
9282, 91sylnibr 328 . . . 4 ((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) ∧ 𝐵 ∈ FinIII) → ¬ (𝐴 ∪ {∅}) ∈ (𝐴 ∪ {∅}))
93 unieq 4876 . . . . . 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 2832 . . . . 5 (ran (𝑒 ∈ ω ↦ {𝑓𝐴𝑓𝑒}) = (𝐴 ∪ {∅}) → ( ran (𝑒 ∈ ω ↦ {𝑓𝐴𝑓𝑒}) ∈ ran (𝑒 ∈ ω ↦ {𝑓𝐴𝑓𝑒}) ↔ (𝐴 ∪ {∅}) ∈ (𝐴 ∪ {∅})))
9695notbid 317 . . . 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 815 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 845  w3a 1087   = wceq 1541  wcel 2106  wne 2943  wral 3064  {crab 3407  Vcvv 3445  cun 3908  wss 3910  c0 4282  𝒫 cpw 4560  {csn 4586   cuni 4865   class class class wbr 5105  cmpt 5188   Or wor 5544  ran crn 5634  suc csuc 6319  wf 6492  cfv 6496   [] crpss 7659  ωcom 7802  cdom 8881  Fincfn 8883  FinIIIcfin3 10217
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-rpss 7660  df-om 7803  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-wdom 9501  df-card 9875  df-fin4 10223  df-fin3 10224
This theorem is referenced by:  fin1a2s  10350
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