Theorem fin2i2 9734

Description: A II-finite set contains minimal elements for every nonempty chain. (Contributed by Mario Carneiro, 16-May-2015.)

Assertion

Ref	Expression
fin2i2	⊢ (((𝐴 ∈ FinII ∧ 𝐵 ⊆ 𝒫 𝐴) ∧ (𝐵 ≠ ∅ ∧ [⊊] Or 𝐵)) → ∩ 𝐵 ∈ 𝐵)

Proof of Theorem fin2i2

Dummy variables 𝑐 𝑚 𝑛 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simplr 767	. . 3 ⊢ (((𝐴 ∈ FinII ∧ 𝐵 ⊆ 𝒫 𝐴) ∧ (𝐵 ≠ ∅ ∧ [⊊] Or 𝐵)) → 𝐵 ⊆ 𝒫 𝐴)
2		simpll 765	. . . . 5 ⊢ (((𝐴 ∈ FinII ∧ 𝐵 ⊆ 𝒫 𝐴) ∧ (𝐵 ≠ ∅ ∧ [⊊] Or 𝐵)) → 𝐴 ∈ FinII)
3		ssrab2 4055	. . . . . 6 ⊢ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝐵} ⊆ 𝒫 𝐴
4	3	a1i 11	. . . . 5 ⊢ (((𝐴 ∈ FinII ∧ 𝐵 ⊆ 𝒫 𝐴) ∧ (𝐵 ≠ ∅ ∧ [⊊] Or 𝐵)) → {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝐵} ⊆ 𝒫 𝐴)
5		simprl 769	. . . . . 6 ⊢ (((𝐴 ∈ FinII ∧ 𝐵 ⊆ 𝒫 𝐴) ∧ (𝐵 ≠ ∅ ∧ [⊊] Or 𝐵)) → 𝐵 ≠ ∅)
6		fin23lem7 9732	. . . . . 6 ⊢ ((𝐴 ∈ FinII ∧ 𝐵 ⊆ 𝒫 𝐴 ∧ 𝐵 ≠ ∅) → {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝐵} ≠ ∅)
7	2, 1, 5, 6	syl3anc 1367	. . . . 5 ⊢ (((𝐴 ∈ FinII ∧ 𝐵 ⊆ 𝒫 𝐴) ∧ (𝐵 ≠ ∅ ∧ [⊊] Or 𝐵)) → {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝐵} ≠ ∅)
8		sorpsscmpl 7454	. . . . . 6 ⊢ ( [⊊] Or 𝐵 → [⊊] Or {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝐵})
9	8	ad2antll 727	. . . . 5 ⊢ (((𝐴 ∈ FinII ∧ 𝐵 ⊆ 𝒫 𝐴) ∧ (𝐵 ≠ ∅ ∧ [⊊] Or 𝐵)) → [⊊] Or {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝐵})
10		fin2i 9711	. . . . 5 ⊢ (((𝐴 ∈ FinII ∧ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝐵} ⊆ 𝒫 𝐴) ∧ ({𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝐵} ≠ ∅ ∧ [⊊] Or {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝐵})) → ∪ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝐵} ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝐵})
11	2, 4, 7, 9, 10	syl22anc 836	. . . 4 ⊢ (((𝐴 ∈ FinII ∧ 𝐵 ⊆ 𝒫 𝐴) ∧ (𝐵 ≠ ∅ ∧ [⊊] Or 𝐵)) → ∪ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝐵} ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝐵})
12		sorpssuni 7452	. . . . 5 ⊢ ( [⊊] Or {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝐵} → (∃𝑚 ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝐵}∀𝑛 ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝐵} ¬ 𝑚 ⊊ 𝑛 ↔ ∪ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝐵} ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝐵}))
13	9, 12	syl 17	. . . 4 ⊢ (((𝐴 ∈ FinII ∧ 𝐵 ⊆ 𝒫 𝐴) ∧ (𝐵 ≠ ∅ ∧ [⊊] Or 𝐵)) → (∃𝑚 ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝐵}∀𝑛 ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝐵} ¬ 𝑚 ⊊ 𝑛 ↔ ∪ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝐵} ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝐵}))
14	11, 13	mpbird 259	. . 3 ⊢ (((𝐴 ∈ FinII ∧ 𝐵 ⊆ 𝒫 𝐴) ∧ (𝐵 ≠ ∅ ∧ [⊊] Or 𝐵)) → ∃𝑚 ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝐵}∀𝑛 ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝐵} ¬ 𝑚 ⊊ 𝑛)
15		psseq2 4064	. . . 4 ⊢ (𝑧 = (𝐴 ∖ 𝑚) → (𝑤 ⊊ 𝑧 ↔ 𝑤 ⊊ (𝐴 ∖ 𝑚)))
16		psseq2 4064	. . . 4 ⊢ (𝑛 = (𝐴 ∖ 𝑤) → (𝑚 ⊊ 𝑛 ↔ 𝑚 ⊊ (𝐴 ∖ 𝑤)))
17		pssdifcom2 4435	. . . 4 ⊢ ((𝑚 ⊆ 𝐴 ∧ 𝑤 ⊆ 𝐴) → (𝑤 ⊊ (𝐴 ∖ 𝑚) ↔ 𝑚 ⊊ (𝐴 ∖ 𝑤)))
18	15, 16, 17	fin23lem11 9733	. . 3 ⊢ (𝐵 ⊆ 𝒫 𝐴 → (∃𝑚 ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝐵}∀𝑛 ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝐵} ¬ 𝑚 ⊊ 𝑛 → ∃𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 ¬ 𝑤 ⊊ 𝑧))
19	1, 14, 18	sylc 65	. 2 ⊢ (((𝐴 ∈ FinII ∧ 𝐵 ⊆ 𝒫 𝐴) ∧ (𝐵 ≠ ∅ ∧ [⊊] Or 𝐵)) → ∃𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 ¬ 𝑤 ⊊ 𝑧)
20		sorpssint 7453	. . 3 ⊢ ( [⊊] Or 𝐵 → (∃𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 ¬ 𝑤 ⊊ 𝑧 ↔ ∩ 𝐵 ∈ 𝐵))
21	20	ad2antll 727	. 2 ⊢ (((𝐴 ∈ FinII ∧ 𝐵 ⊆ 𝒫 𝐴) ∧ (𝐵 ≠ ∅ ∧ [⊊] Or 𝐵)) → (∃𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 ¬ 𝑤 ⊊ 𝑧 ↔ ∩ 𝐵 ∈ 𝐵))
22	19, 21	mpbid 234	1 ⊢ (((𝐴 ∈ FinII ∧ 𝐵 ⊆ 𝒫 𝐴) ∧ (𝐵 ≠ ∅ ∧ [⊊] Or 𝐵)) → ∩ 𝐵 ∈ 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   ∈ wcel 2110   ≠ wne 3016  ∀wral 3138  ∃wrex 3139  {crab 3142   ∖ cdif 3932   ⊆ wss 3935   ⊊ wpss 3936  ∅c0 4290  𝒫 cpw 4538  ∪ cuni 4831  ∩ cint 4868   Or wor 5467   [⊊] crpss 7442  Fin^IIcfin2 9695

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-int 4869  df-br 5059  df-opab 5121  df-po 5468  df-so 5469  df-xp 5555  df-rel 5556  df-rpss 7443  df-fin2 9702

This theorem is referenced by:  isfin2-2  9735  fin23lem40  9767  fin2so  34873