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Theorem isfin2-2 9085
Description: FinII expressed in terms of minimal elements. (Contributed by Stefan O'Rear, 2-Nov-2014.) (Proof shortened by Mario Carneiro, 16-May-2015.)
Assertion
Ref Expression
isfin2-2 (𝐴𝑉 → (𝐴 ∈ FinII ↔ ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [] Or 𝑦) → 𝑦𝑦)))
Distinct variable group:   𝑦,𝐴
Allowed substitution hint:   𝑉(𝑦)

Proof of Theorem isfin2-2
Dummy variables 𝑏 𝑐 𝑚 𝑛 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elpwi 4140 . . . 4 (𝑦 ∈ 𝒫 𝒫 𝐴𝑦 ⊆ 𝒫 𝐴)
2 fin2i2 9084 . . . . 5 (((𝐴 ∈ FinII𝑦 ⊆ 𝒫 𝐴) ∧ (𝑦 ≠ ∅ ∧ [] Or 𝑦)) → 𝑦𝑦)
32ex 450 . . . 4 ((𝐴 ∈ FinII𝑦 ⊆ 𝒫 𝐴) → ((𝑦 ≠ ∅ ∧ [] Or 𝑦) → 𝑦𝑦))
41, 3sylan2 491 . . 3 ((𝐴 ∈ FinII𝑦 ∈ 𝒫 𝒫 𝐴) → ((𝑦 ≠ ∅ ∧ [] Or 𝑦) → 𝑦𝑦))
54ralrimiva 2960 . 2 (𝐴 ∈ FinII → ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [] Or 𝑦) → 𝑦𝑦))
6 elpwi 4140 . . . . 5 (𝑏 ∈ 𝒫 𝒫 𝐴𝑏 ⊆ 𝒫 𝐴)
7 simp1r 1084 . . . . . . . 8 (((𝐴𝑉𝑏 ⊆ 𝒫 𝐴) ∧ ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [] Or 𝑦) → 𝑦𝑦) ∧ (𝑏 ≠ ∅ ∧ [] Or 𝑏)) → 𝑏 ⊆ 𝒫 𝐴)
8 ssrab2 3666 . . . . . . . . . . 11 {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝑏} ⊆ 𝒫 𝐴
9 simp1l 1083 . . . . . . . . . . . 12 (((𝐴𝑉𝑏 ⊆ 𝒫 𝐴) ∧ ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [] Or 𝑦) → 𝑦𝑦) ∧ (𝑏 ≠ ∅ ∧ [] Or 𝑏)) → 𝐴𝑉)
10 pwexg 4810 . . . . . . . . . . . 12 (𝐴𝑉 → 𝒫 𝐴 ∈ V)
11 elpw2g 4787 . . . . . . . . . . . 12 (𝒫 𝐴 ∈ V → ({𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝑏} ∈ 𝒫 𝒫 𝐴 ↔ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝑏} ⊆ 𝒫 𝐴))
129, 10, 113syl 18 . . . . . . . . . . 11 (((𝐴𝑉𝑏 ⊆ 𝒫 𝐴) ∧ ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [] Or 𝑦) → 𝑦𝑦) ∧ (𝑏 ≠ ∅ ∧ [] Or 𝑏)) → ({𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝑏} ∈ 𝒫 𝒫 𝐴 ↔ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝑏} ⊆ 𝒫 𝐴))
138, 12mpbiri 248 . . . . . . . . . 10 (((𝐴𝑉𝑏 ⊆ 𝒫 𝐴) ∧ ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [] Or 𝑦) → 𝑦𝑦) ∧ (𝑏 ≠ ∅ ∧ [] Or 𝑏)) → {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝑏} ∈ 𝒫 𝒫 𝐴)
14 simp2 1060 . . . . . . . . . 10 (((𝐴𝑉𝑏 ⊆ 𝒫 𝐴) ∧ ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [] Or 𝑦) → 𝑦𝑦) ∧ (𝑏 ≠ ∅ ∧ [] Or 𝑏)) → ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [] Or 𝑦) → 𝑦𝑦))
15 simp3l 1087 . . . . . . . . . . . 12 (((𝐴𝑉𝑏 ⊆ 𝒫 𝐴) ∧ ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [] Or 𝑦) → 𝑦𝑦) ∧ (𝑏 ≠ ∅ ∧ [] Or 𝑏)) → 𝑏 ≠ ∅)
16 fin23lem7 9082 . . . . . . . . . . . 12 ((𝐴𝑉𝑏 ⊆ 𝒫 𝐴𝑏 ≠ ∅) → {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝑏} ≠ ∅)
179, 7, 15, 16syl3anc 1323 . . . . . . . . . . 11 (((𝐴𝑉𝑏 ⊆ 𝒫 𝐴) ∧ ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [] Or 𝑦) → 𝑦𝑦) ∧ (𝑏 ≠ ∅ ∧ [] Or 𝑏)) → {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝑏} ≠ ∅)
18 sorpsscmpl 6901 . . . . . . . . . . . . 13 ( [] Or 𝑏 → [] Or {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝑏})
1918adantl 482 . . . . . . . . . . . 12 ((𝑏 ≠ ∅ ∧ [] Or 𝑏) → [] Or {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝑏})
20193ad2ant3 1082 . . . . . . . . . . 11 (((𝐴𝑉𝑏 ⊆ 𝒫 𝐴) ∧ ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [] Or 𝑦) → 𝑦𝑦) ∧ (𝑏 ≠ ∅ ∧ [] Or 𝑏)) → [] Or {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝑏})
2117, 20jca 554 . . . . . . . . . 10 (((𝐴𝑉𝑏 ⊆ 𝒫 𝐴) ∧ ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [] Or 𝑦) → 𝑦𝑦) ∧ (𝑏 ≠ ∅ ∧ [] Or 𝑏)) → ({𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝑏} ≠ ∅ ∧ [] Or {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝑏}))
22 neeq1 2852 . . . . . . . . . . . . 13 (𝑦 = {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝑏} → (𝑦 ≠ ∅ ↔ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝑏} ≠ ∅))
23 soeq2 5015 . . . . . . . . . . . . 13 (𝑦 = {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝑏} → ( [] Or 𝑦 ↔ [] Or {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝑏}))
2422, 23anbi12d 746 . . . . . . . . . . . 12 (𝑦 = {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝑏} → ((𝑦 ≠ ∅ ∧ [] Or 𝑦) ↔ ({𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝑏} ≠ ∅ ∧ [] Or {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝑏})))
25 inteq 4443 . . . . . . . . . . . . 13 (𝑦 = {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝑏} → 𝑦 = {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝑏})
26 id 22 . . . . . . . . . . . . 13 (𝑦 = {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝑏} → 𝑦 = {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝑏})
2725, 26eleq12d 2692 . . . . . . . . . . . 12 (𝑦 = {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝑏} → ( 𝑦𝑦 {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝑏} ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝑏}))
2824, 27imbi12d 334 . . . . . . . . . . 11 (𝑦 = {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝑏} → (((𝑦 ≠ ∅ ∧ [] Or 𝑦) → 𝑦𝑦) ↔ (({𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝑏} ≠ ∅ ∧ [] Or {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝑏}) → {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝑏} ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝑏})))
2928rspcv 3291 . . . . . . . . . 10 ({𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝑏} ∈ 𝒫 𝒫 𝐴 → (∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [] Or 𝑦) → 𝑦𝑦) → (({𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝑏} ≠ ∅ ∧ [] Or {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝑏}) → {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝑏} ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝑏})))
3013, 14, 21, 29syl3c 66 . . . . . . . . 9 (((𝐴𝑉𝑏 ⊆ 𝒫 𝐴) ∧ ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [] Or 𝑦) → 𝑦𝑦) ∧ (𝑏 ≠ ∅ ∧ [] Or 𝑏)) → {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝑏} ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝑏})
31 sorpssint 6900 . . . . . . . . . 10 ( [] Or {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝑏} → (∃𝑧 ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝑏}∀𝑤 ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝑏} ¬ 𝑤𝑧 {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝑏} ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝑏}))
3220, 31syl 17 . . . . . . . . 9 (((𝐴𝑉𝑏 ⊆ 𝒫 𝐴) ∧ ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [] Or 𝑦) → 𝑦𝑦) ∧ (𝑏 ≠ ∅ ∧ [] Or 𝑏)) → (∃𝑧 ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝑏}∀𝑤 ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝑏} ¬ 𝑤𝑧 {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝑏} ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝑏}))
3330, 32mpbird 247 . . . . . . . 8 (((𝐴𝑉𝑏 ⊆ 𝒫 𝐴) ∧ ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [] Or 𝑦) → 𝑦𝑦) ∧ (𝑏 ≠ ∅ ∧ [] Or 𝑏)) → ∃𝑧 ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝑏}∀𝑤 ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝑏} ¬ 𝑤𝑧)
34 psseq1 3672 . . . . . . . . 9 (𝑚 = (𝐴𝑧) → (𝑚𝑛 ↔ (𝐴𝑧) ⊊ 𝑛))
35 psseq1 3672 . . . . . . . . 9 (𝑤 = (𝐴𝑛) → (𝑤𝑧 ↔ (𝐴𝑛) ⊊ 𝑧))
36 pssdifcom1 4026 . . . . . . . . 9 ((𝑧𝐴𝑛𝐴) → ((𝐴𝑧) ⊊ 𝑛 ↔ (𝐴𝑛) ⊊ 𝑧))
3734, 35, 36fin23lem11 9083 . . . . . . . 8 (𝑏 ⊆ 𝒫 𝐴 → (∃𝑧 ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝑏}∀𝑤 ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝑏} ¬ 𝑤𝑧 → ∃𝑚𝑏𝑛𝑏 ¬ 𝑚𝑛))
387, 33, 37sylc 65 . . . . . . 7 (((𝐴𝑉𝑏 ⊆ 𝒫 𝐴) ∧ ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [] Or 𝑦) → 𝑦𝑦) ∧ (𝑏 ≠ ∅ ∧ [] Or 𝑏)) → ∃𝑚𝑏𝑛𝑏 ¬ 𝑚𝑛)
39 simp3r 1088 . . . . . . . 8 (((𝐴𝑉𝑏 ⊆ 𝒫 𝐴) ∧ ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [] Or 𝑦) → 𝑦𝑦) ∧ (𝑏 ≠ ∅ ∧ [] Or 𝑏)) → [] Or 𝑏)
40 sorpssuni 6899 . . . . . . . 8 ( [] Or 𝑏 → (∃𝑚𝑏𝑛𝑏 ¬ 𝑚𝑛 𝑏𝑏))
4139, 40syl 17 . . . . . . 7 (((𝐴𝑉𝑏 ⊆ 𝒫 𝐴) ∧ ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [] Or 𝑦) → 𝑦𝑦) ∧ (𝑏 ≠ ∅ ∧ [] Or 𝑏)) → (∃𝑚𝑏𝑛𝑏 ¬ 𝑚𝑛 𝑏𝑏))
4238, 41mpbid 222 . . . . . 6 (((𝐴𝑉𝑏 ⊆ 𝒫 𝐴) ∧ ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [] Or 𝑦) → 𝑦𝑦) ∧ (𝑏 ≠ ∅ ∧ [] Or 𝑏)) → 𝑏𝑏)
43423exp 1261 . . . . 5 ((𝐴𝑉𝑏 ⊆ 𝒫 𝐴) → (∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [] Or 𝑦) → 𝑦𝑦) → ((𝑏 ≠ ∅ ∧ [] Or 𝑏) → 𝑏𝑏)))
446, 43sylan2 491 . . . 4 ((𝐴𝑉𝑏 ∈ 𝒫 𝒫 𝐴) → (∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [] Or 𝑦) → 𝑦𝑦) → ((𝑏 ≠ ∅ ∧ [] Or 𝑏) → 𝑏𝑏)))
4544ralrimdva 2963 . . 3 (𝐴𝑉 → (∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [] Or 𝑦) → 𝑦𝑦) → ∀𝑏 ∈ 𝒫 𝒫 𝐴((𝑏 ≠ ∅ ∧ [] Or 𝑏) → 𝑏𝑏)))
46 isfin2 9060 . . 3 (𝐴𝑉 → (𝐴 ∈ FinII ↔ ∀𝑏 ∈ 𝒫 𝒫 𝐴((𝑏 ≠ ∅ ∧ [] Or 𝑏) → 𝑏𝑏)))
4745, 46sylibrd 249 . 2 (𝐴𝑉 → (∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [] Or 𝑦) → 𝑦𝑦) → 𝐴 ∈ FinII))
485, 47impbid2 216 1 (𝐴𝑉 → (𝐴 ∈ FinII ↔ ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [] Or 𝑦) → 𝑦𝑦)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790  wral 2907  wrex 2908  {crab 2911  Vcvv 3186  cdif 3552  wss 3555  wpss 3556  c0 3891  𝒫 cpw 4130   cuni 4402   cint 4440   Or wor 4994   [] crpss 6889  FinIIcfin2 9045
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-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-rab 2916  df-v 3188  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-op 4155  df-uni 4403  df-int 4441  df-br 4614  df-opab 4674  df-po 4995  df-so 4996  df-xp 5080  df-rel 5081  df-rpss 6890  df-fin2 9052
This theorem is referenced by: (None)
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