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Theorem fin1a2lem12 9515
Description: Lemma for fin1a2 9519. (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 473 . . 3 ((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) ∧ 𝐵 ∈ FinIII) → 𝐵 ∈ FinIII)
2 simpll1 1262 . . . . . . 7 ((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) ∧ 𝐵 ∈ FinIII) → 𝐴 ⊆ 𝒫 𝐵)
32adantr 468 . . . . . 6 (((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) ∧ 𝐵 ∈ FinIII) ∧ 𝑒 ∈ ω) → 𝐴 ⊆ 𝒫 𝐵)
4 ssrab2 3881 . . . . . . . 8 {𝑓𝐴𝑓𝑒} ⊆ 𝐴
54unissi 4651 . . . . . . 7 {𝑓𝐴𝑓𝑒} ⊆ 𝐴
6 sspwuni 4799 . . . . . . . 8 (𝐴 ⊆ 𝒫 𝐵 𝐴𝐵)
76biimpi 207 . . . . . . 7 (𝐴 ⊆ 𝒫 𝐵 𝐴𝐵)
85, 7syl5ss 3806 . . . . . 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 5016 . . . . . 6 (𝐵 ∈ FinIII → ( {𝑓𝐴𝑓𝑒} ∈ 𝒫 𝐵 {𝑓𝐴𝑓𝑒} ⊆ 𝐵))
1110ad2antlr 709 . . . . 5 (((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) ∧ 𝐵 ∈ FinIII) ∧ 𝑒 ∈ ω) → ( {𝑓𝐴𝑓𝑒} ∈ 𝒫 𝐵 {𝑓𝐴𝑓𝑒} ⊆ 𝐵))
129, 11mpbird 248 . . . 4 (((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) ∧ 𝐵 ∈ FinIII) ∧ 𝑒 ∈ ω) → {𝑓𝐴𝑓𝑒} ∈ 𝒫 𝐵)
1312fmpttd 6604 . . 3 ((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) ∧ 𝐵 ∈ FinIII) → (𝑒 ∈ ω ↦ {𝑓𝐴𝑓𝑒}):ω⟶𝒫 𝐵)
14 vex 3393 . . . . . . . . . . 11 𝑑 ∈ V
1514sucex 7238 . . . . . . . . . 10 suc 𝑑 ∈ V
16 sssucid 6012 . . . . . . . . . 10 𝑑 ⊆ suc 𝑑
17 ssdomg 8235 . . . . . . . . . 10 (suc 𝑑 ∈ V → (𝑑 ⊆ suc 𝑑𝑑 ≼ suc 𝑑))
1815, 16, 17mp2 9 . . . . . . . . 9 𝑑 ≼ suc 𝑑
19 domtr 8242 . . . . . . . . 9 ((𝑓𝑑𝑑 ≼ suc 𝑑) → 𝑓 ≼ suc 𝑑)
2018, 19mpan2 674 . . . . . . . 8 (𝑓𝑑𝑓 ≼ suc 𝑑)
2120a1i 11 . . . . . . 7 (𝑓𝐴 → (𝑓𝑑𝑓 ≼ suc 𝑑))
2221ss2rabi 3878 . . . . . 6 {𝑓𝐴𝑓𝑑} ⊆ {𝑓𝐴𝑓 ≼ suc 𝑑}
23 uniss 4649 . . . . . 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 5045 . . . . . . . . 9 (𝐵 ∈ FinIII → 𝒫 𝐵 ∈ V)
2726adantl 469 . . . . . . . 8 ((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) ∧ 𝐵 ∈ FinIII) → 𝒫 𝐵 ∈ V)
2827, 2ssexd 4997 . . . . . . 7 ((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) ∧ 𝐵 ∈ FinIII) → 𝐴 ∈ V)
29 rabexg 5003 . . . . . . 7 (𝐴 ∈ V → {𝑓𝐴𝑓𝑑} ∈ V)
30 uniexg 7182 . . . . . . 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 4844 . . . . . . . . 9 (𝑒 = 𝑑 → (𝑓𝑒𝑓𝑑))
3332rabbidv 3378 . . . . . . . 8 (𝑒 = 𝑑 → {𝑓𝐴𝑓𝑒} = {𝑓𝐴𝑓𝑑})
3433unieqd 4636 . . . . . . 7 (𝑒 = 𝑑 {𝑓𝐴𝑓𝑒} = {𝑓𝐴𝑓𝑑})
35 eqid 2805 . . . . . . 7 (𝑒 ∈ ω ↦ {𝑓𝐴𝑓𝑒}) = (𝑒 ∈ ω ↦ {𝑓𝐴𝑓𝑒})
3634, 35fvmptg 6498 . . . . . 6 ((𝑑 ∈ ω ∧ {𝑓𝐴𝑓𝑑} ∈ V) → ((𝑒 ∈ ω ↦ {𝑓𝐴𝑓𝑒})‘𝑑) = {𝑓𝐴𝑓𝑑})
3725, 31, 36syl2anr 586 . . . . 5 (((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) ∧ 𝐵 ∈ FinIII) ∧ 𝑑 ∈ ω) → ((𝑒 ∈ ω ↦ {𝑓𝐴𝑓𝑒})‘𝑑) = {𝑓𝐴𝑓𝑑})
38 peano2 7313 . . . . . 6 (𝑑 ∈ ω → suc 𝑑 ∈ ω)
39 rabexg 5003 . . . . . . 7 (𝐴 ∈ V → {𝑓𝐴𝑓 ≼ suc 𝑑} ∈ V)
40 uniexg 7182 . . . . . . 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 4844 . . . . . . . . 9 (𝑒 = suc 𝑑 → (𝑓𝑒𝑓 ≼ suc 𝑑))
4342rabbidv 3378 . . . . . . . 8 (𝑒 = suc 𝑑 → {𝑓𝐴𝑓𝑒} = {𝑓𝐴𝑓 ≼ suc 𝑑})
4443unieqd 4636 . . . . . . 7 (𝑒 = suc 𝑑 {𝑓𝐴𝑓𝑒} = {𝑓𝐴𝑓 ≼ suc 𝑑})
4544, 35fvmptg 6498 . . . . . 6 ((suc 𝑑 ∈ ω ∧ {𝑓𝐴𝑓 ≼ suc 𝑑} ∈ V) → ((𝑒 ∈ ω ↦ {𝑓𝐴𝑓𝑒})‘suc 𝑑) = {𝑓𝐴𝑓 ≼ suc 𝑑})
4638, 41, 45syl2anr 586 . . . . 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 3842 . . . 4 (((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) ∧ 𝐵 ∈ FinIII) ∧ 𝑑 ∈ ω) → ((𝑒 ∈ ω ↦ {𝑓𝐴𝑓𝑒})‘𝑑) ⊆ ((𝑒 ∈ ω ↦ {𝑓𝐴𝑓𝑒})‘suc 𝑑))
4847ralrimiva 3153 . . 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 9485 . . 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 1483 . 2 ((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) ∧ 𝐵 ∈ FinIII) → ran (𝑒 ∈ ω ↦ {𝑓𝐴𝑓𝑒}) ∈ ran (𝑒 ∈ ω ↦ {𝑓𝐴𝑓𝑒}))
51 fin1a2lem11 9514 . . . . . 6 (( [] Or 𝐴𝐴 ⊆ Fin) → ran (𝑒 ∈ ω ↦ {𝑓𝐴𝑓𝑒}) = (𝐴 ∪ {∅}))
5251adantrr 699 . . . . 5 (( [] Or 𝐴 ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) → ran (𝑒 ∈ ω ↦ {𝑓𝐴𝑓𝑒}) = (𝐴 ∪ {∅}))
53523ad2antl2 1230 . . . 4 (((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) → ran (𝑒 ∈ ω ↦ {𝑓𝐴𝑓𝑒}) = (𝐴 ∪ {∅}))
5453adantr 468 . . 3 ((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) ∧ 𝐵 ∈ FinIII) → ran (𝑒 ∈ ω ↦ {𝑓𝐴𝑓𝑒}) = (𝐴 ∪ {∅}))
55 simpll3 1266 . . . . . 6 ((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) ∧ 𝐵 ∈ FinIII) → ¬ 𝐴𝐴)
56 simplrr 787 . . . . . . 7 ((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) ∧ 𝐵 ∈ FinIII) → 𝐴 ≠ ∅)
57 sspwuni 4799 . . . . . . . . . . 11 (𝐴 ⊆ 𝒫 ∅ ↔ 𝐴 ⊆ ∅)
58 ss0b 4168 . . . . . . . . . . 11 ( 𝐴 ⊆ ∅ ↔ 𝐴 = ∅)
5957, 58bitri 266 . . . . . . . . . 10 (𝐴 ⊆ 𝒫 ∅ ↔ 𝐴 = ∅)
60 pw0 4530 . . . . . . . . . . . . 13 𝒫 ∅ = {∅}
6160sseq2i 3824 . . . . . . . . . . . 12 (𝐴 ⊆ 𝒫 ∅ ↔ 𝐴 ⊆ {∅})
62 sssn 4544 . . . . . . . . . . . 12 (𝐴 ⊆ {∅} ↔ (𝐴 = ∅ ∨ 𝐴 = {∅}))
6361, 62bitri 266 . . . . . . . . . . 11 (𝐴 ⊆ 𝒫 ∅ ↔ (𝐴 = ∅ ∨ 𝐴 = {∅}))
64 df-ne 2978 . . . . . . . . . . . 12 (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅)
65 0ex 4981 . . . . . . . . . . . . . . . . 17 ∅ ∈ V
6665unisn 4642 . . . . . . . . . . . . . . . 16 {∅} = ∅
6765snid 4399 . . . . . . . . . . . . . . . 16 ∅ ∈ {∅}
6866, 67eqeltri 2880 . . . . . . . . . . . . . . 15 {∅} ∈ {∅}
69 unieq 4634 . . . . . . . . . . . . . . . 16 (𝐴 = {∅} → 𝐴 = {∅})
70 id 22 . . . . . . . . . . . . . . . 16 (𝐴 = {∅} → 𝐴 = {∅})
7169, 70eleq12d 2878 . . . . . . . . . . . . . . 15 (𝐴 = {∅} → ( 𝐴𝐴 {∅} ∈ {∅}))
7268, 71mpbiri 249 . . . . . . . . . . . . . 14 (𝐴 = {∅} → 𝐴𝐴)
7372orim2i 925 . . . . . . . . . . . . 13 ((𝐴 = ∅ ∨ 𝐴 = {∅}) → (𝐴 = ∅ ∨ 𝐴𝐴))
7473ord 882 . . . . . . . . . . . 12 ((𝐴 = ∅ ∨ 𝐴 = {∅}) → (¬ 𝐴 = ∅ → 𝐴𝐴))
7564, 74syl5bi 233 . . . . . . . . . . 11 ((𝐴 = ∅ ∨ 𝐴 = {∅}) → (𝐴 ≠ ∅ → 𝐴𝐴))
7663, 75sylbi 208 . . . . . . . . . 10 (𝐴 ⊆ 𝒫 ∅ → (𝐴 ≠ ∅ → 𝐴𝐴))
7759, 76sylbir 226 . . . . . . . . 9 ( 𝐴 = ∅ → (𝐴 ≠ ∅ → 𝐴𝐴))
7877com12 32 . . . . . . . 8 (𝐴 ≠ ∅ → ( 𝐴 = ∅ → 𝐴𝐴))
7978con3d 149 . . . . . . 7 (𝐴 ≠ ∅ → (¬ 𝐴𝐴 → ¬ 𝐴 = ∅))
8056, 55, 79sylc 65 . . . . . 6 ((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) ∧ 𝐵 ∈ FinIII) → ¬ 𝐴 = ∅)
81 ioran 997 . . . . . 6 (¬ ( 𝐴𝐴 𝐴 = ∅) ↔ (¬ 𝐴𝐴 ∧ ¬ 𝐴 = ∅))
8255, 80, 81sylanbrc 574 . . . . 5 ((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) ∧ 𝐵 ∈ FinIII) → ¬ ( 𝐴𝐴 𝐴 = ∅))
83 uniun 4647 . . . . . . . 8 (𝐴 ∪ {∅}) = ( 𝐴 {∅})
8466uneq2i 3960 . . . . . . . 8 ( 𝐴 {∅}) = ( 𝐴 ∪ ∅)
85 un0 4162 . . . . . . . 8 ( 𝐴 ∪ ∅) = 𝐴
8683, 84, 853eqtri 2831 . . . . . . 7 (𝐴 ∪ {∅}) = 𝐴
8786eleq1i 2875 . . . . . 6 ( (𝐴 ∪ {∅}) ∈ (𝐴 ∪ {∅}) ↔ 𝐴 ∈ (𝐴 ∪ {∅}))
88 elun 3949 . . . . . 6 ( 𝐴 ∈ (𝐴 ∪ {∅}) ↔ ( 𝐴𝐴 𝐴 ∈ {∅}))
8965elsn2 4402 . . . . . . 7 ( 𝐴 ∈ {∅} ↔ 𝐴 = ∅)
9089orbi2i 927 . . . . . 6 (( 𝐴𝐴 𝐴 ∈ {∅}) ↔ ( 𝐴𝐴 𝐴 = ∅))
9187, 88, 903bitri 288 . . . . 5 ( (𝐴 ∪ {∅}) ∈ (𝐴 ∪ {∅}) ↔ ( 𝐴𝐴 𝐴 = ∅))
9282, 91sylnibr 320 . . . 4 ((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) ∧ 𝐵 ∈ FinIII) → ¬ (𝐴 ∪ {∅}) ∈ (𝐴 ∪ {∅}))
93 unieq 4634 . . . . . 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 2878 . . . . 5 (ran (𝑒 ∈ ω ↦ {𝑓𝐴𝑓𝑒}) = (𝐴 ∪ {∅}) → ( ran (𝑒 ∈ ω ↦ {𝑓𝐴𝑓𝑒}) ∈ ran (𝑒 ∈ ω ↦ {𝑓𝐴𝑓𝑒}) ↔ (𝐴 ∪ {∅}) ∈ (𝐴 ∪ {∅})))
9695notbid 309 . . . 4 (ran (𝑒 ∈ ω ↦ {𝑓𝐴𝑓𝑒}) = (𝐴 ∪ {∅}) → (¬ ran (𝑒 ∈ ω ↦ {𝑓𝐴𝑓𝑒}) ∈ ran (𝑒 ∈ ω ↦ {𝑓𝐴𝑓𝑒}) ↔ ¬ (𝐴 ∪ {∅}) ∈ (𝐴 ∪ {∅})))
9792, 96syl5ibrcom 238 . . 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 842 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 197  wa 384  wo 865  w3a 1100   = wceq 1637  wcel 2158  wne 2977  wral 3095  {crab 3099  Vcvv 3390  cun 3764  wss 3766  c0 4113  𝒫 cpw 4348  {csn 4367   cuni 4626   class class class wbr 4840  cmpt 4919   Or wor 5228  ran crn 5309  suc csuc 5935  wf 6094  cfv 6098   [] crpss 7163  ωcom 7292  cdom 8187  Fincfn 8189  FinIIIcfin3 9385
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1880  ax-4 1897  ax-5 2004  ax-6 2070  ax-7 2106  ax-8 2160  ax-9 2167  ax-10 2187  ax-11 2203  ax-12 2216  ax-13 2422  ax-ext 2784  ax-rep 4960  ax-sep 4971  ax-nul 4980  ax-pow 5032  ax-pr 5093  ax-un 7176
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1865  df-sb 2063  df-eu 2636  df-mo 2637  df-clab 2792  df-cleq 2798  df-clel 2801  df-nfc 2936  df-ne 2978  df-ral 3100  df-rex 3101  df-reu 3102  df-rmo 3103  df-rab 3104  df-v 3392  df-sbc 3631  df-csb 3726  df-dif 3769  df-un 3771  df-in 3773  df-ss 3780  df-pss 3782  df-nul 4114  df-if 4277  df-pw 4350  df-sn 4368  df-pr 4370  df-tp 4372  df-op 4374  df-uni 4627  df-int 4666  df-iun 4710  df-br 4841  df-opab 4903  df-mpt 4920  df-tr 4943  df-id 5216  df-eprel 5221  df-po 5229  df-so 5230  df-fr 5267  df-se 5268  df-we 5269  df-xp 5314  df-rel 5315  df-cnv 5316  df-co 5317  df-dm 5318  df-rn 5319  df-res 5320  df-ima 5321  df-pred 5890  df-ord 5936  df-on 5937  df-lim 5938  df-suc 5939  df-iota 6061  df-fun 6100  df-fn 6101  df-f 6102  df-f1 6103  df-fo 6104  df-f1o 6105  df-fv 6106  df-isom 6107  df-riota 6832  df-rpss 7164  df-om 7293  df-wrecs 7639  df-recs 7701  df-rdg 7739  df-1o 7793  df-er 7976  df-en 8190  df-dom 8191  df-sdom 8192  df-fin 8193  df-wdom 8700  df-card 9045  df-fin4 9391  df-fin3 9392
This theorem is referenced by:  fin1a2s  9518
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