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Theorem isfin2-2 9587
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 4463 . . . 4 (𝑦 ∈ 𝒫 𝒫 𝐴𝑦 ⊆ 𝒫 𝐴)
2 fin2i2 9586 . . . . 5 (((𝐴 ∈ FinII𝑦 ⊆ 𝒫 𝐴) ∧ (𝑦 ≠ ∅ ∧ [] Or 𝑦)) → 𝑦𝑦)
32ex 413 . . . 4 ((𝐴 ∈ FinII𝑦 ⊆ 𝒫 𝐴) → ((𝑦 ≠ ∅ ∧ [] Or 𝑦) → 𝑦𝑦))
41, 3sylan2 592 . . 3 ((𝐴 ∈ FinII𝑦 ∈ 𝒫 𝒫 𝐴) → ((𝑦 ≠ ∅ ∧ [] Or 𝑦) → 𝑦𝑦))
54ralrimiva 3149 . 2 (𝐴 ∈ FinII → ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [] Or 𝑦) → 𝑦𝑦))
6 elpwi 4463 . . . . 5 (𝑏 ∈ 𝒫 𝒫 𝐴𝑏 ⊆ 𝒫 𝐴)
7 simp1r 1191 . . . . . . . 8 (((𝐴𝑉𝑏 ⊆ 𝒫 𝐴) ∧ ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [] Or 𝑦) → 𝑦𝑦) ∧ (𝑏 ≠ ∅ ∧ [] Or 𝑏)) → 𝑏 ⊆ 𝒫 𝐴)
8 simp1l 1190 . . . . . . . . . . 11 (((𝐴𝑉𝑏 ⊆ 𝒫 𝐴) ∧ ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [] Or 𝑦) → 𝑦𝑦) ∧ (𝑏 ≠ ∅ ∧ [] Or 𝑏)) → 𝐴𝑉)
9 simp3l 1194 . . . . . . . . . . 11 (((𝐴𝑉𝑏 ⊆ 𝒫 𝐴) ∧ ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [] Or 𝑦) → 𝑦𝑦) ∧ (𝑏 ≠ ∅ ∧ [] Or 𝑏)) → 𝑏 ≠ ∅)
10 fin23lem7 9584 . . . . . . . . . . 11 ((𝐴𝑉𝑏 ⊆ 𝒫 𝐴𝑏 ≠ ∅) → {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝑏} ≠ ∅)
118, 7, 9, 10syl3anc 1364 . . . . . . . . . 10 (((𝐴𝑉𝑏 ⊆ 𝒫 𝐴) ∧ ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [] Or 𝑦) → 𝑦𝑦) ∧ (𝑏 ≠ ∅ ∧ [] Or 𝑏)) → {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝑏} ≠ ∅)
12 sorpsscmpl 7318 . . . . . . . . . . . 12 ( [] Or 𝑏 → [] Or {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝑏})
1312adantl 482 . . . . . . . . . . 11 ((𝑏 ≠ ∅ ∧ [] Or 𝑏) → [] Or {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝑏})
14133ad2ant3 1128 . . . . . . . . . 10 (((𝐴𝑉𝑏 ⊆ 𝒫 𝐴) ∧ ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [] Or 𝑦) → 𝑦𝑦) ∧ (𝑏 ≠ ∅ ∧ [] Or 𝑏)) → [] Or {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝑏})
15 neeq1 3046 . . . . . . . . . . . . 13 (𝑦 = {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝑏} → (𝑦 ≠ ∅ ↔ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝑏} ≠ ∅))
16 soeq2 5383 . . . . . . . . . . . . 13 (𝑦 = {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝑏} → ( [] Or 𝑦 ↔ [] Or {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝑏}))
1715, 16anbi12d 630 . . . . . . . . . . . 12 (𝑦 = {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝑏} → ((𝑦 ≠ ∅ ∧ [] Or 𝑦) ↔ ({𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝑏} ≠ ∅ ∧ [] Or {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝑏})))
18 inteq 4785 . . . . . . . . . . . . 13 (𝑦 = {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝑏} → 𝑦 = {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝑏})
19 id 22 . . . . . . . . . . . . 13 (𝑦 = {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝑏} → 𝑦 = {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝑏})
2018, 19eleq12d 2877 . . . . . . . . . . . 12 (𝑦 = {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝑏} → ( 𝑦𝑦 {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝑏} ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝑏}))
2117, 20imbi12d 346 . . . . . . . . . . 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})))
22 simp2 1130 . . . . . . . . . . 11 (((𝐴𝑉𝑏 ⊆ 𝒫 𝐴) ∧ ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [] Or 𝑦) → 𝑦𝑦) ∧ (𝑏 ≠ ∅ ∧ [] Or 𝑏)) → ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [] Or 𝑦) → 𝑦𝑦))
23 ssrab2 3977 . . . . . . . . . . . 12 {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝑏} ⊆ 𝒫 𝐴
24 pwexg 5170 . . . . . . . . . . . . 13 (𝐴𝑉 → 𝒫 𝐴 ∈ V)
25 elpw2g 5138 . . . . . . . . . . . . 13 (𝒫 𝐴 ∈ V → ({𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝑏} ∈ 𝒫 𝒫 𝐴 ↔ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝑏} ⊆ 𝒫 𝐴))
268, 24, 253syl 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))
2723, 26mpbiri 259 . . . . . . . . . . 11 (((𝐴𝑉𝑏 ⊆ 𝒫 𝐴) ∧ ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [] Or 𝑦) → 𝑦𝑦) ∧ (𝑏 ≠ ∅ ∧ [] Or 𝑏)) → {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝑏} ∈ 𝒫 𝒫 𝐴)
2821, 22, 27rspcdva 3565 . . . . . . . . . 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}))
2911, 14, 28mp2and 695 . . . . . . . . 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})
30 sorpssint 7317 . . . . . . . . . 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}))
3114, 30syl 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}))
3229, 31mpbird 258 . . . . . . . 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)
33 psseq1 3985 . . . . . . . . 9 (𝑚 = (𝐴𝑧) → (𝑚𝑛 ↔ (𝐴𝑧) ⊊ 𝑛))
34 psseq1 3985 . . . . . . . . 9 (𝑤 = (𝐴𝑛) → (𝑤𝑧 ↔ (𝐴𝑛) ⊊ 𝑧))
35 pssdifcom1 4349 . . . . . . . . 9 ((𝑧𝐴𝑛𝐴) → ((𝐴𝑧) ⊊ 𝑛 ↔ (𝐴𝑛) ⊊ 𝑧))
3633, 34, 35fin23lem11 9585 . . . . . . . 8 (𝑏 ⊆ 𝒫 𝐴 → (∃𝑧 ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝑏}∀𝑤 ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝑏} ¬ 𝑤𝑧 → ∃𝑚𝑏𝑛𝑏 ¬ 𝑚𝑛))
377, 32, 36sylc 65 . . . . . . 7 (((𝐴𝑉𝑏 ⊆ 𝒫 𝐴) ∧ ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [] Or 𝑦) → 𝑦𝑦) ∧ (𝑏 ≠ ∅ ∧ [] Or 𝑏)) → ∃𝑚𝑏𝑛𝑏 ¬ 𝑚𝑛)
38 simp3r 1195 . . . . . . . 8 (((𝐴𝑉𝑏 ⊆ 𝒫 𝐴) ∧ ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [] Or 𝑦) → 𝑦𝑦) ∧ (𝑏 ≠ ∅ ∧ [] Or 𝑏)) → [] Or 𝑏)
39 sorpssuni 7316 . . . . . . . 8 ( [] Or 𝑏 → (∃𝑚𝑏𝑛𝑏 ¬ 𝑚𝑛 𝑏𝑏))
4038, 39syl 17 . . . . . . 7 (((𝐴𝑉𝑏 ⊆ 𝒫 𝐴) ∧ ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [] Or 𝑦) → 𝑦𝑦) ∧ (𝑏 ≠ ∅ ∧ [] Or 𝑏)) → (∃𝑚𝑏𝑛𝑏 ¬ 𝑚𝑛 𝑏𝑏))
4137, 40mpbid 233 . . . . . 6 (((𝐴𝑉𝑏 ⊆ 𝒫 𝐴) ∧ ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [] Or 𝑦) → 𝑦𝑦) ∧ (𝑏 ≠ ∅ ∧ [] Or 𝑏)) → 𝑏𝑏)
42413exp 1112 . . . . 5 ((𝐴𝑉𝑏 ⊆ 𝒫 𝐴) → (∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [] Or 𝑦) → 𝑦𝑦) → ((𝑏 ≠ ∅ ∧ [] Or 𝑏) → 𝑏𝑏)))
436, 42sylan2 592 . . . 4 ((𝐴𝑉𝑏 ∈ 𝒫 𝒫 𝐴) → (∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [] Or 𝑦) → 𝑦𝑦) → ((𝑏 ≠ ∅ ∧ [] Or 𝑏) → 𝑏𝑏)))
4443ralrimdva 3156 . . 3 (𝐴𝑉 → (∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [] Or 𝑦) → 𝑦𝑦) → ∀𝑏 ∈ 𝒫 𝒫 𝐴((𝑏 ≠ ∅ ∧ [] Or 𝑏) → 𝑏𝑏)))
45 isfin2 9562 . . 3 (𝐴𝑉 → (𝐴 ∈ FinII ↔ ∀𝑏 ∈ 𝒫 𝒫 𝐴((𝑏 ≠ ∅ ∧ [] Or 𝑏) → 𝑏𝑏)))
4644, 45sylibrd 260 . 2 (𝐴𝑉 → (∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [] Or 𝑦) → 𝑦𝑦) → 𝐴 ∈ FinII))
475, 46impbid2 227 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 207  wa 396  w3a 1080   = wceq 1522  wcel 2081  wne 2984  wral 3105  wrex 3106  {crab 3109  Vcvv 3437  cdif 3856  wss 3859  wpss 3860  c0 4211  𝒫 cpw 4453   cuni 4745   cint 4782   Or wor 5361   [] crpss 7306  FinIIcfin2 9547
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-rab 3114  df-v 3439  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-pss 3876  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-op 4479  df-uni 4746  df-int 4783  df-br 4963  df-opab 5025  df-po 5362  df-so 5363  df-xp 5449  df-rel 5450  df-rpss 7307  df-fin2 9554
This theorem is referenced by: (None)
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