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Theorem isfin2-2 9394
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 4325 . . . 4 (𝑦 ∈ 𝒫 𝒫 𝐴𝑦 ⊆ 𝒫 𝐴)
2 fin2i2 9393 . . . . 5 (((𝐴 ∈ FinII𝑦 ⊆ 𝒫 𝐴) ∧ (𝑦 ≠ ∅ ∧ [] Or 𝑦)) → 𝑦𝑦)
32ex 401 . . . 4 ((𝐴 ∈ FinII𝑦 ⊆ 𝒫 𝐴) → ((𝑦 ≠ ∅ ∧ [] Or 𝑦) → 𝑦𝑦))
41, 3sylan2 586 . . 3 ((𝐴 ∈ FinII𝑦 ∈ 𝒫 𝒫 𝐴) → ((𝑦 ≠ ∅ ∧ [] Or 𝑦) → 𝑦𝑦))
54ralrimiva 3113 . 2 (𝐴 ∈ FinII → ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [] Or 𝑦) → 𝑦𝑦))
6 elpwi 4325 . . . . 5 (𝑏 ∈ 𝒫 𝒫 𝐴𝑏 ⊆ 𝒫 𝐴)
7 simp1r 1255 . . . . . . . 8 (((𝐴𝑉𝑏 ⊆ 𝒫 𝐴) ∧ ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [] Or 𝑦) → 𝑦𝑦) ∧ (𝑏 ≠ ∅ ∧ [] Or 𝑏)) → 𝑏 ⊆ 𝒫 𝐴)
8 simp1l 1254 . . . . . . . . . . . 12 (((𝐴𝑉𝑏 ⊆ 𝒫 𝐴) ∧ ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [] Or 𝑦) → 𝑦𝑦) ∧ (𝑏 ≠ ∅ ∧ [] Or 𝑏)) → 𝐴𝑉)
9 simp3l 1258 . . . . . . . . . . . 12 (((𝐴𝑉𝑏 ⊆ 𝒫 𝐴) ∧ ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [] Or 𝑦) → 𝑦𝑦) ∧ (𝑏 ≠ ∅ ∧ [] Or 𝑏)) → 𝑏 ≠ ∅)
10 fin23lem7 9391 . . . . . . . . . . . 12 ((𝐴𝑉𝑏 ⊆ 𝒫 𝐴𝑏 ≠ ∅) → {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝑏} ≠ ∅)
118, 7, 9, 10syl3anc 1490 . . . . . . . . . . 11 (((𝐴𝑉𝑏 ⊆ 𝒫 𝐴) ∧ ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [] Or 𝑦) → 𝑦𝑦) ∧ (𝑏 ≠ ∅ ∧ [] Or 𝑏)) → {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝑏} ≠ ∅)
12 sorpsscmpl 7146 . . . . . . . . . . . . 13 ( [] Or 𝑏 → [] Or {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝑏})
1312adantl 473 . . . . . . . . . . . 12 ((𝑏 ≠ ∅ ∧ [] Or 𝑏) → [] Or {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝑏})
14133ad2ant3 1165 . . . . . . . . . . 11 (((𝐴𝑉𝑏 ⊆ 𝒫 𝐴) ∧ ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [] Or 𝑦) → 𝑦𝑦) ∧ (𝑏 ≠ ∅ ∧ [] Or 𝑏)) → [] Or {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝑏})
1511, 14jca 507 . . . . . . . . . 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}))
16 neeq1 2999 . . . . . . . . . . . . 13 (𝑦 = {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝑏} → (𝑦 ≠ ∅ ↔ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝑏} ≠ ∅))
17 soeq2 5218 . . . . . . . . . . . . 13 (𝑦 = {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝑏} → ( [] Or 𝑦 ↔ [] Or {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝑏}))
1816, 17anbi12d 624 . . . . . . . . . . . 12 (𝑦 = {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝑏} → ((𝑦 ≠ ∅ ∧ [] Or 𝑦) ↔ ({𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝑏} ≠ ∅ ∧ [] Or {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝑏})))
19 inteq 4636 . . . . . . . . . . . . 13 (𝑦 = {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝑏} → 𝑦 = {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝑏})
20 id 22 . . . . . . . . . . . . 13 (𝑦 = {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝑏} → 𝑦 = {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝑏})
2119, 20eleq12d 2838 . . . . . . . . . . . 12 (𝑦 = {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝑏} → ( 𝑦𝑦 {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝑏} ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝑏}))
2218, 21imbi12d 335 . . . . . . . . . . 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})))
23 simp2 1167 . . . . . . . . . . 11 (((𝐴𝑉𝑏 ⊆ 𝒫 𝐴) ∧ ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [] Or 𝑦) → 𝑦𝑦) ∧ (𝑏 ≠ ∅ ∧ [] Or 𝑏)) → ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [] Or 𝑦) → 𝑦𝑦))
24 ssrab2 3847 . . . . . . . . . . . 12 {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝑏} ⊆ 𝒫 𝐴
25 pwexg 5014 . . . . . . . . . . . . 13 (𝐴𝑉 → 𝒫 𝐴 ∈ V)
26 elpw2g 4985 . . . . . . . . . . . . 13 (𝒫 𝐴 ∈ V → ({𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝑏} ∈ 𝒫 𝒫 𝐴 ↔ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝑏} ⊆ 𝒫 𝐴))
278, 25, 263syl 18 . . . . . . . . . . . 12 (((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))
2824, 27mpbiri 249 . . . . . . . . . . 11 (((𝐴𝑉𝑏 ⊆ 𝒫 𝐴) ∧ ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [] Or 𝑦) → 𝑦𝑦) ∧ (𝑏 ≠ ∅ ∧ [] Or 𝑏)) → {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝑏} ∈ 𝒫 𝒫 𝐴)
2922, 23, 28rspcdva 3467 . . . . . . . . . 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}) → {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝑏} ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝑏}))
3015, 29mpd 15 . . . . . . . . 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 7145 . . . . . . . . . 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}))
3214, 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 248 . . . . . . . 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 3855 . . . . . . . . 9 (𝑚 = (𝐴𝑧) → (𝑚𝑛 ↔ (𝐴𝑧) ⊊ 𝑛))
35 psseq1 3855 . . . . . . . . 9 (𝑤 = (𝐴𝑛) → (𝑤𝑧 ↔ (𝐴𝑛) ⊊ 𝑧))
36 pssdifcom1 4214 . . . . . . . . 9 ((𝑧𝐴𝑛𝐴) → ((𝐴𝑧) ⊊ 𝑛 ↔ (𝐴𝑛) ⊊ 𝑧))
3734, 35, 36fin23lem11 9392 . . . . . . . 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 1259 . . . . . . . 8 (((𝐴𝑉𝑏 ⊆ 𝒫 𝐴) ∧ ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [] Or 𝑦) → 𝑦𝑦) ∧ (𝑏 ≠ ∅ ∧ [] Or 𝑏)) → [] Or 𝑏)
40 sorpssuni 7144 . . . . . . . 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 223 . . . . . 6 (((𝐴𝑉𝑏 ⊆ 𝒫 𝐴) ∧ ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [] Or 𝑦) → 𝑦𝑦) ∧ (𝑏 ≠ ∅ ∧ [] Or 𝑏)) → 𝑏𝑏)
43423exp 1148 . . . . 5 ((𝐴𝑉𝑏 ⊆ 𝒫 𝐴) → (∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [] Or 𝑦) → 𝑦𝑦) → ((𝑏 ≠ ∅ ∧ [] Or 𝑏) → 𝑏𝑏)))
446, 43sylan2 586 . . . 4 ((𝐴𝑉𝑏 ∈ 𝒫 𝒫 𝐴) → (∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [] Or 𝑦) → 𝑦𝑦) → ((𝑏 ≠ ∅ ∧ [] Or 𝑏) → 𝑏𝑏)))
4544ralrimdva 3116 . . 3 (𝐴𝑉 → (∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [] Or 𝑦) → 𝑦𝑦) → ∀𝑏 ∈ 𝒫 𝒫 𝐴((𝑏 ≠ ∅ ∧ [] Or 𝑏) → 𝑏𝑏)))
46 isfin2 9369 . . 3 (𝐴𝑉 → (𝐴 ∈ FinII ↔ ∀𝑏 ∈ 𝒫 𝒫 𝐴((𝑏 ≠ ∅ ∧ [] Or 𝑏) → 𝑏𝑏)))
4745, 46sylibrd 250 . 2 (𝐴𝑉 → (∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [] Or 𝑦) → 𝑦𝑦) → 𝐴 ∈ FinII))
485, 47impbid2 217 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 197  wa 384  w3a 1107   = wceq 1652  wcel 2155  wne 2937  wral 3055  wrex 3056  {crab 3059  Vcvv 3350  cdif 3729  wss 3732  wpss 3733  c0 4079  𝒫 cpw 4315   cuni 4594   cint 4633   Or wor 5197   [] crpss 7134  FinIIcfin2 9354
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-int 4634  df-br 4810  df-opab 4872  df-po 5198  df-so 5199  df-xp 5283  df-rel 5284  df-rpss 7135  df-fin2 9361
This theorem is referenced by: (None)
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