Theorem limsucncmpi 37155

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 3471	. . . . 5 ⊢ (suc 𝐴 ∈ Top → suc 𝐴 ∈ V)
2		sucexb 7801	. . . . 5 ⊢ (𝐴 ∈ V ↔ suc 𝐴 ∈ V)
3	1, 2	sylibr 237	. . . 4 ⊢ (suc 𝐴 ∈ Top → 𝐴 ∈ V)
4		sssucid 6434	. . . . 5 ⊢ 𝐴 ⊆ suc 𝐴
5		elpwg 4559	. . . . 5 ⊢ (𝐴 ∈ V → (𝐴 ∈ 𝒫 suc 𝐴 ↔ 𝐴 ⊆ suc 𝐴))
6	4, 5	mpbiri 261	. . . 4 ⊢ (𝐴 ∈ V → 𝐴 ∈ 𝒫 suc 𝐴)
7		limsucncmpi.1	. . . . . . 7 ⊢ Lim 𝐴
8		limuni 6414	. . . . . . 7 ⊢ (Lim 𝐴 → 𝐴 = ∪ 𝐴)
9	7, 8	ax-mp 5	. . . . . 6 ⊢ 𝐴 = ∪ 𝐴
10		elin 3914	. . . . . . . . . 10 ⊢ (𝑧 ∈ (𝒫 𝐴 ∩ Fin) ↔ (𝑧 ∈ 𝒫 𝐴 ∧ 𝑧 ∈ Fin))
11		elpwi 4563	. . . . . . . . . . 11 ⊢ (𝑧 ∈ 𝒫 𝐴 → 𝑧 ⊆ 𝐴)
12	11	anim1i 627	. . . . . . . . . 10 ⊢ ((𝑧 ∈ 𝒫 𝐴 ∧ 𝑧 ∈ Fin) → (𝑧 ⊆ 𝐴 ∧ 𝑧 ∈ Fin))
13	10, 12	sylbi 220	. . . . . . . . 9 ⊢ (𝑧 ∈ (𝒫 𝐴 ∩ Fin) → (𝑧 ⊆ 𝐴 ∧ 𝑧 ∈ Fin))
14		nlim0 6412	. . . . . . . . . . . . . . . 16 ⊢ ¬ Lim ∅
15	7, 14	2th 267	. . . . . . . . . . . . . . 15 ⊢ (Lim 𝐴 ↔ ¬ Lim ∅)
16		xor3 385	. . . . . . . . . . . . . . 15 ⊢ (¬ (Lim 𝐴 ↔ Lim ∅) ↔ (Lim 𝐴 ↔ ¬ Lim ∅))
17	15, 16	mpbir 234	. . . . . . . . . . . . . 14 ⊢ ¬ (Lim 𝐴 ↔ Lim ∅)
18		limeq 6363	. . . . . . . . . . . . . . 15 ⊢ (𝐴 = ∅ → (Lim 𝐴 ↔ Lim ∅))
19	18	necon3bi 2981	. . . . . . . . . . . . . 14 ⊢ (¬ (Lim 𝐴 ↔ Lim ∅) → 𝐴 ≠ ∅)
20	17, 19	ax-mp 5	. . . . . . . . . . . . 13 ⊢ 𝐴 ≠ ∅
21		uni0 4895	. . . . . . . . . . . . 13 ⊢ ∪ ∅ = ∅
22	20, 21	neeqtrri 3028	. . . . . . . . . . . 12 ⊢ 𝐴 ≠ ∪ ∅
23		unieq 4877	. . . . . . . . . . . . 13 ⊢ (𝑧 = ∅ → ∪ 𝑧 = ∪ ∅)
24	23	neeq2d 3015	. . . . . . . . . . . 12 ⊢ (𝑧 = ∅ → (𝐴 ≠ ∪ 𝑧 ↔ 𝐴 ≠ ∪ ∅))
25	22, 24	mpbiri 261	. . . . . . . . . . 11 ⊢ (𝑧 = ∅ → 𝐴 ≠ ∪ 𝑧)
26	25	a1i 11	. . . . . . . . . 10 ⊢ ((𝑧 ⊆ 𝐴 ∧ 𝑧 ∈ Fin) → (𝑧 = ∅ → 𝐴 ≠ ∪ 𝑧))
27		limord 6413	. . . . . . . . . . . . . 14 ⊢ (Lim 𝐴 → Ord 𝐴)
28		ordsson 7780	. . . . . . . . . . . . . 14 ⊢ (Ord 𝐴 → 𝐴 ⊆ On)
29	7, 27, 28	mp2b 10	. . . . . . . . . . . . 13 ⊢ 𝐴 ⊆ On
30		sstr2 3937	. . . . . . . . . . . . 13 ⊢ (𝑧 ⊆ 𝐴 → (𝐴 ⊆ On → 𝑧 ⊆ On))
31	29, 30	mpi 21	. . . . . . . . . . . 12 ⊢ (𝑧 ⊆ 𝐴 → 𝑧 ⊆ On)
32		ordunifi 9259	. . . . . . . . . . . . 13 ⊢ ((𝑧 ⊆ On ∧ 𝑧 ∈ Fin ∧ 𝑧 ≠ ∅) → ∪ 𝑧 ∈ 𝑧)
33	32	3expia 1139	. . . . . . . . . . . 12 ⊢ ((𝑧 ⊆ On ∧ 𝑧 ∈ Fin) → (𝑧 ≠ ∅ → ∪ 𝑧 ∈ 𝑧))
34	31, 33	sylan 592	. . . . . . . . . . 11 ⊢ ((𝑧 ⊆ 𝐴 ∧ 𝑧 ∈ Fin) → (𝑧 ≠ ∅ → ∪ 𝑧 ∈ 𝑧))
35		ssel 3924	. . . . . . . . . . . . 13 ⊢ (𝑧 ⊆ 𝐴 → (∪ 𝑧 ∈ 𝑧 → ∪ 𝑧 ∈ 𝐴))
36	7, 27	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ Ord 𝐴
37		nordeq 6370	. . . . . . . . . . . . . 14 ⊢ ((Ord 𝐴 ∧ ∪ 𝑧 ∈ 𝐴) → 𝐴 ≠ ∪ 𝑧)
38	36, 37	mpan 703	. . . . . . . . . . . . 13 ⊢ (∪ 𝑧 ∈ 𝐴 → 𝐴 ≠ ∪ 𝑧)
39	35, 38	syl6 36	. . . . . . . . . . . 12 ⊢ (𝑧 ⊆ 𝐴 → (∪ 𝑧 ∈ 𝑧 → 𝐴 ≠ ∪ 𝑧))
40	39	adantr 486	. . . . . . . . . . 11 ⊢ ((𝑧 ⊆ 𝐴 ∧ 𝑧 ∈ Fin) → (∪ 𝑧 ∈ 𝑧 → 𝐴 ≠ ∪ 𝑧))
41	34, 40	syld 48	. . . . . . . . . 10 ⊢ ((𝑧 ⊆ 𝐴 ∧ 𝑧 ∈ Fin) → (𝑧 ≠ ∅ → 𝐴 ≠ ∪ 𝑧))
42	26, 41	pm2.61dne 3041	. . . . . . . . 9 ⊢ ((𝑧 ⊆ 𝐴 ∧ 𝑧 ∈ Fin) → 𝐴 ≠ ∪ 𝑧)
43	13, 42	syl 18	. . . . . . . 8 ⊢ (𝑧 ∈ (𝒫 𝐴 ∩ Fin) → 𝐴 ≠ ∪ 𝑧)
44	43	neneqd 2960	. . . . . . 7 ⊢ (𝑧 ∈ (𝒫 𝐴 ∩ Fin) → ¬ 𝐴 = ∪ 𝑧)
45	44	nrex 3090	. . . . . 6 ⊢ ¬ ∃𝑧 ∈ (𝒫 𝐴 ∩ Fin)𝐴 = ∪ 𝑧
46		unieq 4877	. . . . . . . . 9 ⊢ (𝑦 = 𝐴 → ∪ 𝑦 = ∪ 𝐴)
47	46	eqeq2d 2771	. . . . . . . 8 ⊢ (𝑦 = 𝐴 → (𝐴 = ∪ 𝑦 ↔ 𝐴 = ∪ 𝐴))
48		pweq 4570	. . . . . . . . . . 11 ⊢ (𝑦 = 𝐴 → 𝒫 𝑦 = 𝒫 𝐴)
49	48	ineq1d 4164	. . . . . . . . . 10 ⊢ (𝑦 = 𝐴 → (𝒫 𝑦 ∩ Fin) = (𝒫 𝐴 ∩ Fin))
50	49	rexeqdv 3320	. . . . . . . . 9 ⊢ (𝑦 = 𝐴 → (∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝐴 = ∪ 𝑧 ↔ ∃𝑧 ∈ (𝒫 𝐴 ∩ Fin)𝐴 = ∪ 𝑧))
51	50	notbid 321	. . . . . . . 8 ⊢ (𝑦 = 𝐴 → (¬ ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝐴 = ∪ 𝑧 ↔ ¬ ∃𝑧 ∈ (𝒫 𝐴 ∩ Fin)𝐴 = ∪ 𝑧))
52	47, 51	anbi12d 644	. . . . . . 7 ⊢ (𝑦 = 𝐴 → ((𝐴 = ∪ 𝑦 ∧ ¬ ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝐴 = ∪ 𝑧) ↔ (𝐴 = ∪ 𝐴 ∧ ¬ ∃𝑧 ∈ (𝒫 𝐴 ∩ Fin)𝐴 = ∪ 𝑧)))
53	52	rspcev 3576	. . . . . 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 718	. . . . 5 ⊢ (𝐴 ∈ 𝒫 suc 𝐴 → ∃𝑦 ∈ 𝒫 suc 𝐴(𝐴 = ∪ 𝑦 ∧ ¬ ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝐴 = ∪ 𝑧))
55		rexanali 3116	. . . . 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 221	. . . 4 ⊢ (𝐴 ∈ 𝒫 suc 𝐴 → ¬ ∀𝑦 ∈ 𝒫 suc 𝐴(𝐴 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝐴 = ∪ 𝑧))
57	3, 6, 56	3syl 19	. . 3 ⊢ (suc 𝐴 ∈ Top → ¬ ∀𝑦 ∈ 𝒫 suc 𝐴(𝐴 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝐴 = ∪ 𝑧))
58		imnan 405	. . 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 233	. 2 ⊢ ¬ (suc 𝐴 ∈ Top ∧ ∀𝑦 ∈ 𝒫 suc 𝐴(𝐴 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝐴 = ∪ 𝑧))
60		ordunisuc 7826	. . . . 5 ⊢ (Ord 𝐴 → ∪ suc 𝐴 = 𝐴)
61	7, 27, 60	mp2b 10	. . . 4 ⊢ ∪ suc 𝐴 = 𝐴
62	61	eqcomi 2769	. . 3 ⊢ 𝐴 = ∪ suc 𝐴
63	62	iscmp 23668	. 2 ⊢ (suc 𝐴 ∈ Comp ↔ (suc 𝐴 ∈ Top ∧ ∀𝑦 ∈ 𝒫 suc 𝐴(𝐴 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝐴 = ∪ 𝑧)))
64	59, 63	mtbir 326	1 ⊢ ¬ suc 𝐴 ∈ Comp

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145   ≠ wne 2955  ∀wral 3076  ∃wrex 3086  Vcvv 3450   ∩ cin 3897   ⊆ wss 3898  ∅c0 4278  𝒫 cpw 4556  ∪ cuni 4866  Ord word 6350  Oncon0 6351  Lim wlim 6352  suc csuc 6353  Fincfn 8951  Topctop 23173  Compccmp 23666

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-tr 5212  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-ord 6354  df-on 6355  df-lim 6356  df-suc 6357  df-iota 6483  df-fun 6529  df-fn 6530  df-f 6531  df-f1 6532  df-fo 6533  df-f1o 6534  df-fv 6535  df-om 7861  df-en 8952  df-fin 8955  df-cmp 23667

This theorem is used by:  limsucncmp  37156