Theorem limsucncmpi 34634

Description: The successor of a limit ordinal is not compact. (Contributed by Chen-Pang He, 20-Oct-2015.)

Hypothesis

Ref	Expression
limsucncmpi.1	⊢ Lim 𝐴

Assertion

Ref	Expression
limsucncmpi	⊢ ¬ suc 𝐴 ∈ Comp

Proof of Theorem limsucncmpi

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3450	. . . . 5 ⊢ (suc 𝐴 ∈ Top → suc 𝐴 ∈ V)
2		sucexb 7654	. . . . 5 ⊢ (𝐴 ∈ V ↔ suc 𝐴 ∈ V)
3	1, 2	sylibr 233	. . . 4 ⊢ (suc 𝐴 ∈ Top → 𝐴 ∈ V)
4		sssucid 6343	. . . . 5 ⊢ 𝐴 ⊆ suc 𝐴
5		elpwg 4536	. . . . 5 ⊢ (𝐴 ∈ V → (𝐴 ∈ 𝒫 suc 𝐴 ↔ 𝐴 ⊆ suc 𝐴))
6	4, 5	mpbiri 257	. . . 4 ⊢ (𝐴 ∈ V → 𝐴 ∈ 𝒫 suc 𝐴)
7		limsucncmpi.1	. . . . . . 7 ⊢ Lim 𝐴
8		limuni 6326	. . . . . . 7 ⊢ (Lim 𝐴 → 𝐴 = ∪ 𝐴)
9	7, 8	ax-mp 5	. . . . . 6 ⊢ 𝐴 = ∪ 𝐴
10		elin 3903	. . . . . . . . . 10 ⊢ (𝑧 ∈ (𝒫 𝐴 ∩ Fin) ↔ (𝑧 ∈ 𝒫 𝐴 ∧ 𝑧 ∈ Fin))
11		elpwi 4542	. . . . . . . . . . 11 ⊢ (𝑧 ∈ 𝒫 𝐴 → 𝑧 ⊆ 𝐴)
12	11	anim1i 615	. . . . . . . . . 10 ⊢ ((𝑧 ∈ 𝒫 𝐴 ∧ 𝑧 ∈ Fin) → (𝑧 ⊆ 𝐴 ∧ 𝑧 ∈ Fin))
13	10, 12	sylbi 216	. . . . . . . . 9 ⊢ (𝑧 ∈ (𝒫 𝐴 ∩ Fin) → (𝑧 ⊆ 𝐴 ∧ 𝑧 ∈ Fin))
14		nlim0 6324	. . . . . . . . . . . . . . . 16 ⊢ ¬ Lim ∅
15	7, 14	2th 263	. . . . . . . . . . . . . . 15 ⊢ (Lim 𝐴 ↔ ¬ Lim ∅)
16		xor3 384	. . . . . . . . . . . . . . 15 ⊢ (¬ (Lim 𝐴 ↔ Lim ∅) ↔ (Lim 𝐴 ↔ ¬ Lim ∅))
17	15, 16	mpbir 230	. . . . . . . . . . . . . 14 ⊢ ¬ (Lim 𝐴 ↔ Lim ∅)
18		limeq 6278	. . . . . . . . . . . . . . 15 ⊢ (𝐴 = ∅ → (Lim 𝐴 ↔ Lim ∅))
19	18	necon3bi 2970	. . . . . . . . . . . . . 14 ⊢ (¬ (Lim 𝐴 ↔ Lim ∅) → 𝐴 ≠ ∅)
20	17, 19	ax-mp 5	. . . . . . . . . . . . 13 ⊢ 𝐴 ≠ ∅
21		uni0 4869	. . . . . . . . . . . . 13 ⊢ ∪ ∅ = ∅
22	20, 21	neeqtrri 3017	. . . . . . . . . . . 12 ⊢ 𝐴 ≠ ∪ ∅
23		unieq 4850	. . . . . . . . . . . . 13 ⊢ (𝑧 = ∅ → ∪ 𝑧 = ∪ ∅)
24	23	neeq2d 3004	. . . . . . . . . . . 12 ⊢ (𝑧 = ∅ → (𝐴 ≠ ∪ 𝑧 ↔ 𝐴 ≠ ∪ ∅))
25	22, 24	mpbiri 257	. . . . . . . . . . 11 ⊢ (𝑧 = ∅ → 𝐴 ≠ ∪ 𝑧)
26	25	a1i 11	. . . . . . . . . 10 ⊢ ((𝑧 ⊆ 𝐴 ∧ 𝑧 ∈ Fin) → (𝑧 = ∅ → 𝐴 ≠ ∪ 𝑧))
27		limord 6325	. . . . . . . . . . . . . 14 ⊢ (Lim 𝐴 → Ord 𝐴)
28		ordsson 7633	. . . . . . . . . . . . . 14 ⊢ (Ord 𝐴 → 𝐴 ⊆ On)
29	7, 27, 28	mp2b 10	. . . . . . . . . . . . 13 ⊢ 𝐴 ⊆ On
30		sstr2 3928	. . . . . . . . . . . . 13 ⊢ (𝑧 ⊆ 𝐴 → (𝐴 ⊆ On → 𝑧 ⊆ On))
31	29, 30	mpi 20	. . . . . . . . . . . 12 ⊢ (𝑧 ⊆ 𝐴 → 𝑧 ⊆ On)
32		ordunifi 9064	. . . . . . . . . . . . 13 ⊢ ((𝑧 ⊆ On ∧ 𝑧 ∈ Fin ∧ 𝑧 ≠ ∅) → ∪ 𝑧 ∈ 𝑧)
33	32	3expia 1120	. . . . . . . . . . . 12 ⊢ ((𝑧 ⊆ On ∧ 𝑧 ∈ Fin) → (𝑧 ≠ ∅ → ∪ 𝑧 ∈ 𝑧))
34	31, 33	sylan 580	. . . . . . . . . . 11 ⊢ ((𝑧 ⊆ 𝐴 ∧ 𝑧 ∈ Fin) → (𝑧 ≠ ∅ → ∪ 𝑧 ∈ 𝑧))
35		ssel 3914	. . . . . . . . . . . . 13 ⊢ (𝑧 ⊆ 𝐴 → (∪ 𝑧 ∈ 𝑧 → ∪ 𝑧 ∈ 𝐴))
36	7, 27	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ Ord 𝐴
37		nordeq 6285	. . . . . . . . . . . . . 14 ⊢ ((Ord 𝐴 ∧ ∪ 𝑧 ∈ 𝐴) → 𝐴 ≠ ∪ 𝑧)
38	36, 37	mpan 687	. . . . . . . . . . . . 13 ⊢ (∪ 𝑧 ∈ 𝐴 → 𝐴 ≠ ∪ 𝑧)
39	35, 38	syl6 35	. . . . . . . . . . . 12 ⊢ (𝑧 ⊆ 𝐴 → (∪ 𝑧 ∈ 𝑧 → 𝐴 ≠ ∪ 𝑧))
40	39	adantr 481	. . . . . . . . . . 11 ⊢ ((𝑧 ⊆ 𝐴 ∧ 𝑧 ∈ Fin) → (∪ 𝑧 ∈ 𝑧 → 𝐴 ≠ ∪ 𝑧))
41	34, 40	syld 47	. . . . . . . . . 10 ⊢ ((𝑧 ⊆ 𝐴 ∧ 𝑧 ∈ Fin) → (𝑧 ≠ ∅ → 𝐴 ≠ ∪ 𝑧))
42	26, 41	pm2.61dne 3031	. . . . . . . . 9 ⊢ ((𝑧 ⊆ 𝐴 ∧ 𝑧 ∈ Fin) → 𝐴 ≠ ∪ 𝑧)
43	13, 42	syl 17	. . . . . . . 8 ⊢ (𝑧 ∈ (𝒫 𝐴 ∩ Fin) → 𝐴 ≠ ∪ 𝑧)
44	43	neneqd 2948	. . . . . . 7 ⊢ (𝑧 ∈ (𝒫 𝐴 ∩ Fin) → ¬ 𝐴 = ∪ 𝑧)
45	44	nrex 3197	. . . . . 6 ⊢ ¬ ∃𝑧 ∈ (𝒫 𝐴 ∩ Fin)𝐴 = ∪ 𝑧
46		unieq 4850	. . . . . . . . 9 ⊢ (𝑦 = 𝐴 → ∪ 𝑦 = ∪ 𝐴)
47	46	eqeq2d 2749	. . . . . . . 8 ⊢ (𝑦 = 𝐴 → (𝐴 = ∪ 𝑦 ↔ 𝐴 = ∪ 𝐴))
48		pweq 4549	. . . . . . . . . . 11 ⊢ (𝑦 = 𝐴 → 𝒫 𝑦 = 𝒫 𝐴)
49	48	ineq1d 4145	. . . . . . . . . 10 ⊢ (𝑦 = 𝐴 → (𝒫 𝑦 ∩ Fin) = (𝒫 𝐴 ∩ Fin))
50	49	rexeqdv 3349	. . . . . . . . 9 ⊢ (𝑦 = 𝐴 → (∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝐴 = ∪ 𝑧 ↔ ∃𝑧 ∈ (𝒫 𝐴 ∩ Fin)𝐴 = ∪ 𝑧))
51	50	notbid 318	. . . . . . . 8 ⊢ (𝑦 = 𝐴 → (¬ ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝐴 = ∪ 𝑧 ↔ ¬ ∃𝑧 ∈ (𝒫 𝐴 ∩ Fin)𝐴 = ∪ 𝑧))
52	47, 51	anbi12d 631	. . . . . . 7 ⊢ (𝑦 = 𝐴 → ((𝐴 = ∪ 𝑦 ∧ ¬ ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝐴 = ∪ 𝑧) ↔ (𝐴 = ∪ 𝐴 ∧ ¬ ∃𝑧 ∈ (𝒫 𝐴 ∩ Fin)𝐴 = ∪ 𝑧)))
53	52	rspcev 3561	. . . . . 6 ⊢ ((𝐴 ∈ 𝒫 suc 𝐴 ∧ (𝐴 = ∪ 𝐴 ∧ ¬ ∃𝑧 ∈ (𝒫 𝐴 ∩ Fin)𝐴 = ∪ 𝑧)) → ∃𝑦 ∈ 𝒫 suc 𝐴(𝐴 = ∪ 𝑦 ∧ ¬ ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝐴 = ∪ 𝑧))
54	9, 45, 53	mpanr12 702	. . . . 5 ⊢ (𝐴 ∈ 𝒫 suc 𝐴 → ∃𝑦 ∈ 𝒫 suc 𝐴(𝐴 = ∪ 𝑦 ∧ ¬ ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝐴 = ∪ 𝑧))
55		rexanali 3192	. . . . 5 ⊢ (∃𝑦 ∈ 𝒫 suc 𝐴(𝐴 = ∪ 𝑦 ∧ ¬ ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝐴 = ∪ 𝑧) ↔ ¬ ∀𝑦 ∈ 𝒫 suc 𝐴(𝐴 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝐴 = ∪ 𝑧))
56	54, 55	sylib 217	. . . 4 ⊢ (𝐴 ∈ 𝒫 suc 𝐴 → ¬ ∀𝑦 ∈ 𝒫 suc 𝐴(𝐴 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝐴 = ∪ 𝑧))
57	3, 6, 56	3syl 18	. . 3 ⊢ (suc 𝐴 ∈ Top → ¬ ∀𝑦 ∈ 𝒫 suc 𝐴(𝐴 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝐴 = ∪ 𝑧))
58		imnan 400	. . 3 ⊢ ((suc 𝐴 ∈ Top → ¬ ∀𝑦 ∈ 𝒫 suc 𝐴(𝐴 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝐴 = ∪ 𝑧)) ↔ ¬ (suc 𝐴 ∈ Top ∧ ∀𝑦 ∈ 𝒫 suc 𝐴(𝐴 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝐴 = ∪ 𝑧)))
59	57, 58	mpbi 229	. 2 ⊢ ¬ (suc 𝐴 ∈ Top ∧ ∀𝑦 ∈ 𝒫 suc 𝐴(𝐴 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝐴 = ∪ 𝑧))
60		ordunisuc 7679	. . . . 5 ⊢ (Ord 𝐴 → ∪ suc 𝐴 = 𝐴)
61	7, 27, 60	mp2b 10	. . . 4 ⊢ ∪ suc 𝐴 = 𝐴
62	61	eqcomi 2747	. . 3 ⊢ 𝐴 = ∪ suc 𝐴
63	62	iscmp 22539	. 2 ⊢ (suc 𝐴 ∈ Comp ↔ (suc 𝐴 ∈ Top ∧ ∀𝑦 ∈ 𝒫 suc 𝐴(𝐴 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝐴 = ∪ 𝑧)))
64	59, 63	mtbir 323	1 ⊢ ¬ suc 𝐴 ∈ Comp

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106   ≠ wne 2943  ∀wral 3064  ∃wrex 3065  Vcvv 3432   ∩ cin 3886   ⊆ wss 3887  ∅c0 4256  𝒫 cpw 4533  ∪ cuni 4839  Ord word 6265  Oncon0 6266  Lim wlim 6267  suc csuc 6268  Fincfn 8733  Topctop 22042  Compccmp 22537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-om 7713  df-en 8734  df-fin 8737  df-cmp 22538

This theorem is referenced by:  limsucncmp  34635