Theorem limsucncmpi 36458

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 3455	. . . . 5 ⊢ (suc 𝐴 ∈ Top → suc 𝐴 ∈ V)
2		sucexb 7732	. . . . 5 ⊢ (𝐴 ∈ V ↔ suc 𝐴 ∈ V)
3	1, 2	sylibr 234	. . . 4 ⊢ (suc 𝐴 ∈ Top → 𝐴 ∈ V)
4		sssucid 6384	. . . . 5 ⊢ 𝐴 ⊆ suc 𝐴
5		elpwg 4551	. . . . 5 ⊢ (𝐴 ∈ V → (𝐴 ∈ 𝒫 suc 𝐴 ↔ 𝐴 ⊆ suc 𝐴))
6	4, 5	mpbiri 258	. . . 4 ⊢ (𝐴 ∈ V → 𝐴 ∈ 𝒫 suc 𝐴)
7		limsucncmpi.1	. . . . . . 7 ⊢ Lim 𝐴
8		limuni 6364	. . . . . . 7 ⊢ (Lim 𝐴 → 𝐴 = ∪ 𝐴)
9	7, 8	ax-mp 5	. . . . . 6 ⊢ 𝐴 = ∪ 𝐴
10		elin 3916	. . . . . . . . . 10 ⊢ (𝑧 ∈ (𝒫 𝐴 ∩ Fin) ↔ (𝑧 ∈ 𝒫 𝐴 ∧ 𝑧 ∈ Fin))
11		elpwi 4555	. . . . . . . . . . 11 ⊢ (𝑧 ∈ 𝒫 𝐴 → 𝑧 ⊆ 𝐴)
12	11	anim1i 615	. . . . . . . . . 10 ⊢ ((𝑧 ∈ 𝒫 𝐴 ∧ 𝑧 ∈ Fin) → (𝑧 ⊆ 𝐴 ∧ 𝑧 ∈ Fin))
13	10, 12	sylbi 217	. . . . . . . . 9 ⊢ (𝑧 ∈ (𝒫 𝐴 ∩ Fin) → (𝑧 ⊆ 𝐴 ∧ 𝑧 ∈ Fin))
14		nlim0 6362	. . . . . . . . . . . . . . . 16 ⊢ ¬ Lim ∅
15	7, 14	2th 264	. . . . . . . . . . . . . . 15 ⊢ (Lim 𝐴 ↔ ¬ Lim ∅)
16		xor3 382	. . . . . . . . . . . . . . 15 ⊢ (¬ (Lim 𝐴 ↔ Lim ∅) ↔ (Lim 𝐴 ↔ ¬ Lim ∅))
17	15, 16	mpbir 231	. . . . . . . . . . . . . 14 ⊢ ¬ (Lim 𝐴 ↔ Lim ∅)
18		limeq 6314	. . . . . . . . . . . . . . 15 ⊢ (𝐴 = ∅ → (Lim 𝐴 ↔ Lim ∅))
19	18	necon3bi 2952	. . . . . . . . . . . . . 14 ⊢ (¬ (Lim 𝐴 ↔ Lim ∅) → 𝐴 ≠ ∅)
20	17, 19	ax-mp 5	. . . . . . . . . . . . 13 ⊢ 𝐴 ≠ ∅
21		uni0 4885	. . . . . . . . . . . . 13 ⊢ ∪ ∅ = ∅
22	20, 21	neeqtrri 2999	. . . . . . . . . . . 12 ⊢ 𝐴 ≠ ∪ ∅
23		unieq 4868	. . . . . . . . . . . . 13 ⊢ (𝑧 = ∅ → ∪ 𝑧 = ∪ ∅)
24	23	neeq2d 2986	. . . . . . . . . . . 12 ⊢ (𝑧 = ∅ → (𝐴 ≠ ∪ 𝑧 ↔ 𝐴 ≠ ∪ ∅))
25	22, 24	mpbiri 258	. . . . . . . . . . 11 ⊢ (𝑧 = ∅ → 𝐴 ≠ ∪ 𝑧)
26	25	a1i 11	. . . . . . . . . 10 ⊢ ((𝑧 ⊆ 𝐴 ∧ 𝑧 ∈ Fin) → (𝑧 = ∅ → 𝐴 ≠ ∪ 𝑧))
27		limord 6363	. . . . . . . . . . . . . 14 ⊢ (Lim 𝐴 → Ord 𝐴)
28		ordsson 7711	. . . . . . . . . . . . . 14 ⊢ (Ord 𝐴 → 𝐴 ⊆ On)
29	7, 27, 28	mp2b 10	. . . . . . . . . . . . 13 ⊢ 𝐴 ⊆ On
30		sstr2 3939	. . . . . . . . . . . . 13 ⊢ (𝑧 ⊆ 𝐴 → (𝐴 ⊆ On → 𝑧 ⊆ On))
31	29, 30	mpi 20	. . . . . . . . . . . 12 ⊢ (𝑧 ⊆ 𝐴 → 𝑧 ⊆ On)
32		ordunifi 9169	. . . . . . . . . . . . 13 ⊢ ((𝑧 ⊆ On ∧ 𝑧 ∈ Fin ∧ 𝑧 ≠ ∅) → ∪ 𝑧 ∈ 𝑧)
33	32	3expia 1121	. . . . . . . . . . . 12 ⊢ ((𝑧 ⊆ On ∧ 𝑧 ∈ Fin) → (𝑧 ≠ ∅ → ∪ 𝑧 ∈ 𝑧))
34	31, 33	sylan 580	. . . . . . . . . . 11 ⊢ ((𝑧 ⊆ 𝐴 ∧ 𝑧 ∈ Fin) → (𝑧 ≠ ∅ → ∪ 𝑧 ∈ 𝑧))
35		ssel 3926	. . . . . . . . . . . . 13 ⊢ (𝑧 ⊆ 𝐴 → (∪ 𝑧 ∈ 𝑧 → ∪ 𝑧 ∈ 𝐴))
36	7, 27	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ Ord 𝐴
37		nordeq 6321	. . . . . . . . . . . . . 14 ⊢ ((Ord 𝐴 ∧ ∪ 𝑧 ∈ 𝐴) → 𝐴 ≠ ∪ 𝑧)
38	36, 37	mpan 690	. . . . . . . . . . . . 13 ⊢ (∪ 𝑧 ∈ 𝐴 → 𝐴 ≠ ∪ 𝑧)
39	35, 38	syl6 35	. . . . . . . . . . . 12 ⊢ (𝑧 ⊆ 𝐴 → (∪ 𝑧 ∈ 𝑧 → 𝐴 ≠ ∪ 𝑧))
40	39	adantr 480	. . . . . . . . . . 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 3012	. . . . . . . . 9 ⊢ ((𝑧 ⊆ 𝐴 ∧ 𝑧 ∈ Fin) → 𝐴 ≠ ∪ 𝑧)
43	13, 42	syl 17	. . . . . . . 8 ⊢ (𝑧 ∈ (𝒫 𝐴 ∩ Fin) → 𝐴 ≠ ∪ 𝑧)
44	43	neneqd 2931	. . . . . . 7 ⊢ (𝑧 ∈ (𝒫 𝐴 ∩ Fin) → ¬ 𝐴 = ∪ 𝑧)
45	44	nrex 3058	. . . . . 6 ⊢ ¬ ∃𝑧 ∈ (𝒫 𝐴 ∩ Fin)𝐴 = ∪ 𝑧
46		unieq 4868	. . . . . . . . 9 ⊢ (𝑦 = 𝐴 → ∪ 𝑦 = ∪ 𝐴)
47	46	eqeq2d 2741	. . . . . . . 8 ⊢ (𝑦 = 𝐴 → (𝐴 = ∪ 𝑦 ↔ 𝐴 = ∪ 𝐴))
48		pweq 4562	. . . . . . . . . . 11 ⊢ (𝑦 = 𝐴 → 𝒫 𝑦 = 𝒫 𝐴)
49	48	ineq1d 4167	. . . . . . . . . 10 ⊢ (𝑦 = 𝐴 → (𝒫 𝑦 ∩ Fin) = (𝒫 𝐴 ∩ Fin))
50	49	rexeqdv 3291	. . . . . . . . 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 632	. . . . . . 7 ⊢ (𝑦 = 𝐴 → ((𝐴 = ∪ 𝑦 ∧ ¬ ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝐴 = ∪ 𝑧) ↔ (𝐴 = ∪ 𝐴 ∧ ¬ ∃𝑧 ∈ (𝒫 𝐴 ∩ Fin)𝐴 = ∪ 𝑧)))
53	52	rspcev 3575	. . . . . 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 705	. . . . 5 ⊢ (𝐴 ∈ 𝒫 suc 𝐴 → ∃𝑦 ∈ 𝒫 suc 𝐴(𝐴 = ∪ 𝑦 ∧ ¬ ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝐴 = ∪ 𝑧))
55		rexanali 3084	. . . . 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 218	. . . 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 399	. . 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 230	. 2 ⊢ ¬ (suc 𝐴 ∈ Top ∧ ∀𝑦 ∈ 𝒫 suc 𝐴(𝐴 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝐴 = ∪ 𝑧))
60		ordunisuc 7757	. . . . 5 ⊢ (Ord 𝐴 → ∪ suc 𝐴 = 𝐴)
61	7, 27, 60	mp2b 10	. . . 4 ⊢ ∪ suc 𝐴 = 𝐴
62	61	eqcomi 2739	. . 3 ⊢ 𝐴 = ∪ suc 𝐴
63	62	iscmp 23296	. 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 206   ∧ wa 395   = wceq 1541   ∈ wcel 2110   ≠ wne 2926  ∀wral 3045  ∃wrex 3054  Vcvv 3434   ∩ cin 3899   ⊆ wss 3900  ∅c0 4281  𝒫 cpw 4548  ∪ cuni 4857  Ord word 6301  Oncon0 6302  Lim wlim 6303  suc csuc 6304  Fincfn 8864  Topctop 22801  Compccmp 23294

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 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-10 2143  ax-11 2159  ax-12 2179  ax-ext 2702  ax-sep 5232  ax-nul 5242  ax-pr 5368  ax-un 7663

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3345  df-rab 3394  df-v 3436  df-sbc 3740  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-pss 3920  df-nul 4282  df-if 4474  df-pw 4550  df-sn 4575  df-pr 4577  df-op 4581  df-uni 4858  df-br 5090  df-opab 5152  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-ord 6305  df-on 6306  df-lim 6307  df-suc 6308  df-iota 6433  df-fun 6479  df-fn 6480  df-f 6481  df-f1 6482  df-fo 6483  df-f1o 6484  df-fv 6485  df-om 7792  df-en 8865  df-fin 8868  df-cmp 23295

This theorem is referenced by:  limsucncmp  36459