Theorem fin2i2 9742

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 4058	. . . . . 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 9740	. . . . . 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 7462	. . . . . 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 9719	. . . . 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 7460	. . . . 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 4067	. . . 4 ⊢ (𝑧 = (𝐴 ∖ 𝑚) → (𝑤 ⊊ 𝑧 ↔ 𝑤 ⊊ (𝐴 ∖ 𝑚)))
16		psseq2 4067	. . . 4 ⊢ (𝑛 = (𝐴 ∖ 𝑤) → (𝑚 ⊊ 𝑛 ↔ 𝑚 ⊊ (𝐴 ∖ 𝑤)))
17		pssdifcom2 4438	. . . 4 ⊢ ((𝑚 ⊆ 𝐴 ∧ 𝑤 ⊆ 𝐴) → (𝑤 ⊊ (𝐴 ∖ 𝑚) ↔ 𝑚 ⊊ (𝐴 ∖ 𝑤)))
18	15, 16, 17	fin23lem11 9741	. . 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 7461	. . 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 2114   ≠ wne 3018  ∀wral 3140  ∃wrex 3141  {crab 3144   ∖ cdif 3935   ⊆ wss 3938   ⊊ wpss 3939  ∅c0 4293  𝒫 cpw 4541  ∪ cuni 4840  ∩ cint 4878   Or wor 5475   [⊊] crpss 7450  Fin^IIcfin2 9703

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-int 4879  df-br 5069  df-opab 5131  df-po 5476  df-so 5477  df-xp 5563  df-rel 5564  df-rpss 7451  df-fin2 9710

This theorem is referenced by:  isfin2-2  9743  fin23lem40  9775  fin2so  34881