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Theorem fin2i2 9125
Description: A II-finite set contains minimal elements for every nonempty chain. (Contributed by Mario Carneiro, 16-May-2015.)
Assertion
Ref Expression
fin2i2 (((𝐴 ∈ FinII𝐵 ⊆ 𝒫 𝐴) ∧ (𝐵 ≠ ∅ ∧ [] Or 𝐵)) → 𝐵𝐵)

Proof of Theorem fin2i2
Dummy variables 𝑐 𝑚 𝑛 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simplr 791 . . 3 (((𝐴 ∈ FinII𝐵 ⊆ 𝒫 𝐴) ∧ (𝐵 ≠ ∅ ∧ [] Or 𝐵)) → 𝐵 ⊆ 𝒫 𝐴)
2 simpll 789 . . . . 5 (((𝐴 ∈ FinII𝐵 ⊆ 𝒫 𝐴) ∧ (𝐵 ≠ ∅ ∧ [] Or 𝐵)) → 𝐴 ∈ FinII)
3 ssrab2 3679 . . . . . 6 {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝐵} ⊆ 𝒫 𝐴
43a1i 11 . . . . 5 (((𝐴 ∈ FinII𝐵 ⊆ 𝒫 𝐴) ∧ (𝐵 ≠ ∅ ∧ [] Or 𝐵)) → {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝐵} ⊆ 𝒫 𝐴)
5 simprl 793 . . . . . 6 (((𝐴 ∈ FinII𝐵 ⊆ 𝒫 𝐴) ∧ (𝐵 ≠ ∅ ∧ [] Or 𝐵)) → 𝐵 ≠ ∅)
6 fin23lem7 9123 . . . . . 6 ((𝐴 ∈ FinII𝐵 ⊆ 𝒫 𝐴𝐵 ≠ ∅) → {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝐵} ≠ ∅)
72, 1, 5, 6syl3anc 1324 . . . . 5 (((𝐴 ∈ FinII𝐵 ⊆ 𝒫 𝐴) ∧ (𝐵 ≠ ∅ ∧ [] Or 𝐵)) → {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝐵} ≠ ∅)
8 sorpsscmpl 6933 . . . . . 6 ( [] Or 𝐵 → [] Or {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝐵})
98ad2antll 764 . . . . 5 (((𝐴 ∈ FinII𝐵 ⊆ 𝒫 𝐴) ∧ (𝐵 ≠ ∅ ∧ [] Or 𝐵)) → [] Or {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝐵})
10 fin2i 9102 . . . . 5 (((𝐴 ∈ FinII ∧ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝐵} ⊆ 𝒫 𝐴) ∧ ({𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝐵} ≠ ∅ ∧ [] Or {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝐵})) → {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝐵} ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝐵})
112, 4, 7, 9, 10syl22anc 1325 . . . 4 (((𝐴 ∈ FinII𝐵 ⊆ 𝒫 𝐴) ∧ (𝐵 ≠ ∅ ∧ [] Or 𝐵)) → {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝐵} ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝐵})
12 sorpssuni 6931 . . . . 5 ( [] Or {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝐵} → (∃𝑚 ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝐵}∀𝑛 ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝐵} ¬ 𝑚𝑛 {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝐵} ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝐵}))
139, 12syl 17 . . . 4 (((𝐴 ∈ FinII𝐵 ⊆ 𝒫 𝐴) ∧ (𝐵 ≠ ∅ ∧ [] Or 𝐵)) → (∃𝑚 ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝐵}∀𝑛 ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝐵} ¬ 𝑚𝑛 {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝐵} ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝐵}))
1411, 13mpbird 247 . . 3 (((𝐴 ∈ FinII𝐵 ⊆ 𝒫 𝐴) ∧ (𝐵 ≠ ∅ ∧ [] Or 𝐵)) → ∃𝑚 ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝐵}∀𝑛 ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝐵} ¬ 𝑚𝑛)
15 psseq2 3687 . . . 4 (𝑧 = (𝐴𝑚) → (𝑤𝑧𝑤 ⊊ (𝐴𝑚)))
16 psseq2 3687 . . . 4 (𝑛 = (𝐴𝑤) → (𝑚𝑛𝑚 ⊊ (𝐴𝑤)))
17 pssdifcom2 4046 . . . 4 ((𝑚𝐴𝑤𝐴) → (𝑤 ⊊ (𝐴𝑚) ↔ 𝑚 ⊊ (𝐴𝑤)))
1815, 16, 17fin23lem11 9124 . . 3 (𝐵 ⊆ 𝒫 𝐴 → (∃𝑚 ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝐵}∀𝑛 ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝐵} ¬ 𝑚𝑛 → ∃𝑧𝐵𝑤𝐵 ¬ 𝑤𝑧))
191, 14, 18sylc 65 . 2 (((𝐴 ∈ FinII𝐵 ⊆ 𝒫 𝐴) ∧ (𝐵 ≠ ∅ ∧ [] Or 𝐵)) → ∃𝑧𝐵𝑤𝐵 ¬ 𝑤𝑧)
20 sorpssint 6932 . . 3 ( [] Or 𝐵 → (∃𝑧𝐵𝑤𝐵 ¬ 𝑤𝑧 𝐵𝐵))
2120ad2antll 764 . 2 (((𝐴 ∈ FinII𝐵 ⊆ 𝒫 𝐴) ∧ (𝐵 ≠ ∅ ∧ [] Or 𝐵)) → (∃𝑧𝐵𝑤𝐵 ¬ 𝑤𝑧 𝐵𝐵))
2219, 21mpbid 222 1 (((𝐴 ∈ FinII𝐵 ⊆ 𝒫 𝐴) ∧ (𝐵 ≠ ∅ ∧ [] Or 𝐵)) → 𝐵𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  wcel 1988  wne 2791  wral 2909  wrex 2910  {crab 2913  cdif 3564  wss 3567  wpss 3568  c0 3907  𝒫 cpw 4149   cuni 4427   cint 4466   Or wor 5024   [] crpss 6921  FinIIcfin2 9086
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-int 4467  df-br 4645  df-opab 4704  df-po 5025  df-so 5026  df-xp 5110  df-rel 5111  df-rpss 6922  df-fin2 9093
This theorem is referenced by:  isfin2-2  9126  fin23lem40  9158  fin2so  33367
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