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Theorem isfin2-2 10316
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 4604 . . . 4 (𝑦 ∈ 𝒫 𝒫 𝐴𝑦 ⊆ 𝒫 𝐴)
2 fin2i2 10315 . . . . 5 (((𝐴 ∈ FinII𝑦 ⊆ 𝒫 𝐴) ∧ (𝑦 ≠ ∅ ∧ [] Or 𝑦)) → 𝑦𝑦)
32ex 412 . . . 4 ((𝐴 ∈ FinII𝑦 ⊆ 𝒫 𝐴) → ((𝑦 ≠ ∅ ∧ [] Or 𝑦) → 𝑦𝑦))
41, 3sylan2 592 . . 3 ((𝐴 ∈ FinII𝑦 ∈ 𝒫 𝒫 𝐴) → ((𝑦 ≠ ∅ ∧ [] Or 𝑦) → 𝑦𝑦))
54ralrimiva 3140 . 2 (𝐴 ∈ FinII → ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [] Or 𝑦) → 𝑦𝑦))
6 elpwi 4604 . . . . 5 (𝑏 ∈ 𝒫 𝒫 𝐴𝑏 ⊆ 𝒫 𝐴)
7 simp1r 1195 . . . . . . . 8 (((𝐴𝑉𝑏 ⊆ 𝒫 𝐴) ∧ ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [] Or 𝑦) → 𝑦𝑦) ∧ (𝑏 ≠ ∅ ∧ [] Or 𝑏)) → 𝑏 ⊆ 𝒫 𝐴)
8 simp1l 1194 . . . . . . . . . . 11 (((𝐴𝑉𝑏 ⊆ 𝒫 𝐴) ∧ ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [] Or 𝑦) → 𝑦𝑦) ∧ (𝑏 ≠ ∅ ∧ [] Or 𝑏)) → 𝐴𝑉)
9 simp3l 1198 . . . . . . . . . . 11 (((𝐴𝑉𝑏 ⊆ 𝒫 𝐴) ∧ ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [] Or 𝑦) → 𝑦𝑦) ∧ (𝑏 ≠ ∅ ∧ [] Or 𝑏)) → 𝑏 ≠ ∅)
10 fin23lem7 10313 . . . . . . . . . . 11 ((𝐴𝑉𝑏 ⊆ 𝒫 𝐴𝑏 ≠ ∅) → {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝑏} ≠ ∅)
118, 7, 9, 10syl3anc 1368 . . . . . . . . . 10 (((𝐴𝑉𝑏 ⊆ 𝒫 𝐴) ∧ ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [] Or 𝑦) → 𝑦𝑦) ∧ (𝑏 ≠ ∅ ∧ [] Or 𝑏)) → {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝑏} ≠ ∅)
12 sorpsscmpl 7721 . . . . . . . . . . . 12 ( [] Or 𝑏 → [] Or {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝑏})
1312adantl 481 . . . . . . . . . . 11 ((𝑏 ≠ ∅ ∧ [] Or 𝑏) → [] Or {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝑏})
14133ad2ant3 1132 . . . . . . . . . 10 (((𝐴𝑉𝑏 ⊆ 𝒫 𝐴) ∧ ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [] Or 𝑦) → 𝑦𝑦) ∧ (𝑏 ≠ ∅ ∧ [] Or 𝑏)) → [] Or {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝑏})
15 neeq1 2997 . . . . . . . . . . . . 13 (𝑦 = {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝑏} → (𝑦 ≠ ∅ ↔ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝑏} ≠ ∅))
16 soeq2 5603 . . . . . . . . . . . . 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 4946 . . . . . . . . . . . . 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 2821 . . . . . . . . . . . 12 (𝑦 = {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝑏} → ( 𝑦𝑦 {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝑏} ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝑏}))
2117, 20imbi12d 344 . . . . . . . . . . 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 1134 . . . . . . . . . . 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 4072 . . . . . . . . . . . 12 {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝑏} ⊆ 𝒫 𝐴
24 pwexg 5369 . . . . . . . . . . . . 13 (𝐴𝑉 → 𝒫 𝐴 ∈ V)
25 elpw2g 5337 . . . . . . . . . . . . 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 258 . . . . . . . . . . 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 3607 . . . . . . . . . 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 696 . . . . . . . . 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 7720 . . . . . . . . . 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 257 . . . . . . . 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 4082 . . . . . . . . 9 (𝑚 = (𝐴𝑧) → (𝑚𝑛 ↔ (𝐴𝑧) ⊊ 𝑛))
34 psseq1 4082 . . . . . . . . 9 (𝑤 = (𝐴𝑛) → (𝑤𝑧 ↔ (𝐴𝑛) ⊊ 𝑧))
35 pssdifcom1 4484 . . . . . . . . 9 ((𝑧𝐴𝑛𝐴) → ((𝐴𝑧) ⊊ 𝑛 ↔ (𝐴𝑛) ⊊ 𝑧))
3633, 34, 35fin23lem11 10314 . . . . . . . 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 1199 . . . . . . . 8 (((𝐴𝑉𝑏 ⊆ 𝒫 𝐴) ∧ ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [] Or 𝑦) → 𝑦𝑦) ∧ (𝑏 ≠ ∅ ∧ [] Or 𝑏)) → [] Or 𝑏)
39 sorpssuni 7719 . . . . . . . 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 231 . . . . . 6 (((𝐴𝑉𝑏 ⊆ 𝒫 𝐴) ∧ ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [] Or 𝑦) → 𝑦𝑦) ∧ (𝑏 ≠ ∅ ∧ [] Or 𝑏)) → 𝑏𝑏)
42413exp 1116 . . . . 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 3148 . . 3 (𝐴𝑉 → (∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [] Or 𝑦) → 𝑦𝑦) → ∀𝑏 ∈ 𝒫 𝒫 𝐴((𝑏 ≠ ∅ ∧ [] Or 𝑏) → 𝑏𝑏)))
45 isfin2 10291 . . 3 (𝐴𝑉 → (𝐴 ∈ FinII ↔ ∀𝑏 ∈ 𝒫 𝒫 𝐴((𝑏 ≠ ∅ ∧ [] Or 𝑏) → 𝑏𝑏)))
4644, 45sylibrd 259 . 2 (𝐴𝑉 → (∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [] Or 𝑦) → 𝑦𝑦) → 𝐴 ∈ FinII))
475, 46impbid2 225 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 205  wa 395  w3a 1084   = wceq 1533  wcel 2098  wne 2934  wral 3055  wrex 3064  {crab 3426  Vcvv 3468  cdif 3940  wss 3943  wpss 3944  c0 4317  𝒫 cpw 4597   cuni 4902   cint 4943   Or wor 5580   [] crpss 7709  FinIIcfin2 10276
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-br 5142  df-opab 5204  df-po 5581  df-so 5582  df-xp 5675  df-rel 5676  df-rpss 7710  df-fin2 10283
This theorem is referenced by: (None)
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