Theorem ordcmp 37327

Description: An ordinal topology is compact iff the underlying set is its supremum (union) only when the ordinal is 1_o. (Contributed by Chen-Pang He, 1-Nov-2015.)

Assertion

Ref	Expression
ordcmp	⊢ (Ord 𝐴 → (𝐴 ∈ Comp ↔ (∪ 𝐴 = ∪ ∪ 𝐴 → 𝐴 = 1o)))

Proof of Theorem ordcmp

Step	Hyp	Ref	Expression
1		orduni 7803	. . . 4 ⊢ (Ord 𝐴 → Ord ∪ 𝐴)
2		unizlim 6487	. . . . . 6 ⊢ (Ord ∪ 𝐴 → (∪ 𝐴 = ∪ ∪ 𝐴 ↔ (∪ 𝐴 = ∅ ∨ Lim ∪ 𝐴)))
3		uni0b 4893	. . . . . . 7 ⊢ (∪ 𝐴 = ∅ ↔ 𝐴 ⊆ {∅})
4	3	orbi1i 927	. . . . . 6 ⊢ ((∪ 𝐴 = ∅ ∨ Lim ∪ 𝐴) ↔ (𝐴 ⊆ {∅} ∨ Lim ∪ 𝐴))
5	2, 4	bitrdi 290	. . . . 5 ⊢ (Ord ∪ 𝐴 → (∪ 𝐴 = ∪ ∪ 𝐴 ↔ (𝐴 ⊆ {∅} ∨ Lim ∪ 𝐴)))
6	5	biimpd 232	. . . 4 ⊢ (Ord ∪ 𝐴 → (∪ 𝐴 = ∪ ∪ 𝐴 → (𝐴 ⊆ {∅} ∨ Lim ∪ 𝐴)))
7	1, 6	syl 18	. . 3 ⊢ (Ord 𝐴 → (∪ 𝐴 = ∪ ∪ 𝐴 → (𝐴 ⊆ {∅} ∨ Lim ∪ 𝐴)))
8		sssn 4781	. . . . . . 7 ⊢ (𝐴 ⊆ {∅} ↔ (𝐴 = ∅ ∨ 𝐴 = {∅}))
9		0ntop 23228	. . . . . . . . . . 11 ⊢ ¬ ∅ ∈ Top
10		cmptop 23718	. . . . . . . . . . 11 ⊢ (∅ ∈ Comp → ∅ ∈ Top)
11	9, 10	mto 200	. . . . . . . . . 10 ⊢ ¬ ∅ ∈ Comp
12		eleq1 2848	. . . . . . . . . 10 ⊢ (𝐴 = ∅ → (𝐴 ∈ Comp ↔ ∅ ∈ Comp))
13	11, 12	mtbiri 330	. . . . . . . . 9 ⊢ (𝐴 = ∅ → ¬ 𝐴 ∈ Comp)
14	13	pm2.21d 122	. . . . . . . 8 ⊢ (𝐴 = ∅ → (𝐴 ∈ Comp → 𝐴 = 1o))
15		id 23	. . . . . . . . . 10 ⊢ (𝐴 = {∅} → 𝐴 = {∅})
16		df1o2 8486	. . . . . . . . . 10 ⊢ 1o = {∅}
17	15, 16	eqtr4di 2813	. . . . . . . . 9 ⊢ (𝐴 = {∅} → 𝐴 = 1o)
18	17	a1d 26	. . . . . . . 8 ⊢ (𝐴 = {∅} → (𝐴 ∈ Comp → 𝐴 = 1o))
19	14, 18	jaoi 871	. . . . . . 7 ⊢ ((𝐴 = ∅ ∨ 𝐴 = {∅}) → (𝐴 ∈ Comp → 𝐴 = 1o))
20	8, 19	sylbi 220	. . . . . 6 ⊢ (𝐴 ⊆ {∅} → (𝐴 ∈ Comp → 𝐴 = 1o))
21	20	a1i 11	. . . . 5 ⊢ (Ord 𝐴 → (𝐴 ⊆ {∅} → (𝐴 ∈ Comp → 𝐴 = 1o)))
22		ordtop 37316	. . . . . . . . . . 11 ⊢ (Ord 𝐴 → (𝐴 ∈ Top ↔ 𝐴 ≠ ∪ 𝐴))
23	22	biimpd 232	. . . . . . . . . 10 ⊢ (Ord 𝐴 → (𝐴 ∈ Top → 𝐴 ≠ ∪ 𝐴))
24	23	necon2bd 2971	. . . . . . . . 9 ⊢ (Ord 𝐴 → (𝐴 = ∪ 𝐴 → ¬ 𝐴 ∈ Top))
25		cmptop 23718	. . . . . . . . . 10 ⊢ (𝐴 ∈ Comp → 𝐴 ∈ Top)
26	25	con3i 155	. . . . . . . . 9 ⊢ (¬ 𝐴 ∈ Top → ¬ 𝐴 ∈ Comp)
27	24, 26	syl6 36	. . . . . . . 8 ⊢ (Ord 𝐴 → (𝐴 = ∪ 𝐴 → ¬ 𝐴 ∈ Comp))
28	27	a1dd 51	. . . . . . 7 ⊢ (Ord 𝐴 → (𝐴 = ∪ 𝐴 → (Lim ∪ 𝐴 → ¬ 𝐴 ∈ Comp)))
29		limsucncmp 37326	. . . . . . . . 9 ⊢ (Lim ∪ 𝐴 → ¬ suc ∪ 𝐴 ∈ Comp)
30		eleq1 2848	. . . . . . . . . 10 ⊢ (𝐴 = suc ∪ 𝐴 → (𝐴 ∈ Comp ↔ suc ∪ 𝐴 ∈ Comp))
31	30	notbid 321	. . . . . . . . 9 ⊢ (𝐴 = suc ∪ 𝐴 → (¬ 𝐴 ∈ Comp ↔ ¬ suc ∪ 𝐴 ∈ Comp))
32	29, 31	imbitrrid 249	. . . . . . . 8 ⊢ (𝐴 = suc ∪ 𝐴 → (Lim ∪ 𝐴 → ¬ 𝐴 ∈ Comp))
33	32	a1i 11	. . . . . . 7 ⊢ (Ord 𝐴 → (𝐴 = suc ∪ 𝐴 → (Lim ∪ 𝐴 → ¬ 𝐴 ∈ Comp)))
34		orduniorsuc 7841	. . . . . . 7 ⊢ (Ord 𝐴 → (𝐴 = ∪ 𝐴 ∨ 𝐴 = suc ∪ 𝐴))
35	28, 33, 34	mpjaod 874	. . . . . 6 ⊢ (Ord 𝐴 → (Lim ∪ 𝐴 → ¬ 𝐴 ∈ Comp))
36		pm2.21 124	. . . . . 6 ⊢ (¬ 𝐴 ∈ Comp → (𝐴 ∈ Comp → 𝐴 = 1o))
37	35, 36	syl6 36	. . . . 5 ⊢ (Ord 𝐴 → (Lim ∪ 𝐴 → (𝐴 ∈ Comp → 𝐴 = 1o)))
38	21, 37	jaod 873	. . . 4 ⊢ (Ord 𝐴 → ((𝐴 ⊆ {∅} ∨ Lim ∪ 𝐴) → (𝐴 ∈ Comp → 𝐴 = 1o)))
39	38	com23 87	. . 3 ⊢ (Ord 𝐴 → (𝐴 ∈ Comp → ((𝐴 ⊆ {∅} ∨ Lim ∪ 𝐴) → 𝐴 = 1o)))
40	7, 39	syl5d 74	. 2 ⊢ (Ord 𝐴 → (𝐴 ∈ Comp → (∪ 𝐴 = ∪ ∪ 𝐴 → 𝐴 = 1o)))
41		ordeleqon 7796	. . . . . . 7 ⊢ (Ord 𝐴 ↔ (𝐴 ∈ On ∨ 𝐴 = On))
42		unon 7842	. . . . . . . . . . 11 ⊢ ∪ On = On
43	42	eqcomi 2769	. . . . . . . . . 10 ⊢ On = ∪ On
44	43	unieqi 4878	. . . . . . . . 9 ⊢ ∪ On = ∪ ∪ On
45		unieq 4877	. . . . . . . . 9 ⊢ (𝐴 = On → ∪ 𝐴 = ∪ On)
46	45	unieqd 4879	. . . . . . . . 9 ⊢ (𝐴 = On → ∪ ∪ 𝐴 = ∪ ∪ On)
47	44, 45, 46	3eqtr4a 2821	. . . . . . . 8 ⊢ (𝐴 = On → ∪ 𝐴 = ∪ ∪ 𝐴)
48	47	orim2i 924	. . . . . . 7 ⊢ ((𝐴 ∈ On ∨ 𝐴 = On) → (𝐴 ∈ On ∨ ∪ 𝐴 = ∪ ∪ 𝐴))
49	41, 48	sylbi 220	. . . . . 6 ⊢ (Ord 𝐴 → (𝐴 ∈ On ∨ ∪ 𝐴 = ∪ ∪ 𝐴))
50	49	orcomd 885	. . . . 5 ⊢ (Ord 𝐴 → (∪ 𝐴 = ∪ ∪ 𝐴 ∨ 𝐴 ∈ On))
51	50	ord 878	. . . 4 ⊢ (Ord 𝐴 → (¬ ∪ 𝐴 = ∪ ∪ 𝐴 → 𝐴 ∈ On))
52		unieq 4877	. . . . . . 7 ⊢ (𝐴 = ∪ 𝐴 → ∪ 𝐴 = ∪ ∪ 𝐴)
53	52	con3i 155	. . . . . 6 ⊢ (¬ ∪ 𝐴 = ∪ ∪ 𝐴 → ¬ 𝐴 = ∪ 𝐴)
54	34	ord 878	. . . . . 6 ⊢ (Ord 𝐴 → (¬ 𝐴 = ∪ 𝐴 → 𝐴 = suc ∪ 𝐴))
55	53, 54	syl5 35	. . . . 5 ⊢ (Ord 𝐴 → (¬ ∪ 𝐴 = ∪ ∪ 𝐴 → 𝐴 = suc ∪ 𝐴))
56		orduniorsuc 7841	. . . . . . . 8 ⊢ (Ord ∪ 𝐴 → (∪ 𝐴 = ∪ ∪ 𝐴 ∨ ∪ 𝐴 = suc ∪ ∪ 𝐴))
57	1, 56	syl 18	. . . . . . 7 ⊢ (Ord 𝐴 → (∪ 𝐴 = ∪ ∪ 𝐴 ∨ ∪ 𝐴 = suc ∪ ∪ 𝐴))
58	57	ord 878	. . . . . 6 ⊢ (Ord 𝐴 → (¬ ∪ 𝐴 = ∪ ∪ 𝐴 → ∪ 𝐴 = suc ∪ ∪ 𝐴))
59		suceq 6431	. . . . . 6 ⊢ (∪ 𝐴 = suc ∪ ∪ 𝐴 → suc ∪ 𝐴 = suc suc ∪ ∪ 𝐴)
60	58, 59	syl6 36	. . . . 5 ⊢ (Ord 𝐴 → (¬ ∪ 𝐴 = ∪ ∪ 𝐴 → suc ∪ 𝐴 = suc suc ∪ ∪ 𝐴))
61		eqtr 2780	. . . . . 6 ⊢ ((𝐴 = suc ∪ 𝐴 ∧ suc ∪ 𝐴 = suc suc ∪ ∪ 𝐴) → 𝐴 = suc suc ∪ ∪ 𝐴)
62	61	ex 418	. . . . 5 ⊢ (𝐴 = suc ∪ 𝐴 → (suc ∪ 𝐴 = suc suc ∪ ∪ 𝐴 → 𝐴 = suc suc ∪ ∪ 𝐴))
63	55, 60, 62	syl6c 71	. . . 4 ⊢ (Ord 𝐴 → (¬ ∪ 𝐴 = ∪ ∪ 𝐴 → 𝐴 = suc suc ∪ ∪ 𝐴))
64		onuni 7802	. . . . 5 ⊢ (𝐴 ∈ On → ∪ 𝐴 ∈ On)
65		onuni 7802	. . . . 5 ⊢ (∪ 𝐴 ∈ On → ∪ ∪ 𝐴 ∈ On)
66		onsucsuccmp 37324	. . . . 5 ⊢ (∪ ∪ 𝐴 ∈ On → suc suc ∪ ∪ 𝐴 ∈ Comp)
67		eleq1a 2855	. . . . 5 ⊢ (suc suc ∪ ∪ 𝐴 ∈ Comp → (𝐴 = suc suc ∪ ∪ 𝐴 → 𝐴 ∈ Comp))
68	64, 65, 66, 67	4syl 20	. . . 4 ⊢ (𝐴 ∈ On → (𝐴 = suc suc ∪ ∪ 𝐴 → 𝐴 ∈ Comp))
69	51, 63, 68	syl6c 71	. . 3 ⊢ (Ord 𝐴 → (¬ ∪ 𝐴 = ∪ ∪ 𝐴 → 𝐴 ∈ Comp))
70		id 23	. . . . . 6 ⊢ (𝐴 = 1o → 𝐴 = 1o)
71	70, 16	eqtrdi 2811	. . . . 5 ⊢ (𝐴 = 1o → 𝐴 = {∅})
72		0cmp 23717	. . . . 5 ⊢ {∅} ∈ Comp
73	71, 72	eqeltrdi 2868	. . . 4 ⊢ (𝐴 = 1o → 𝐴 ∈ Comp)
74	73	a1i 11	. . 3 ⊢ (Ord 𝐴 → (𝐴 = 1o → 𝐴 ∈ Comp))
75	69, 74	jad 189	. 2 ⊢ (Ord 𝐴 → ((∪ 𝐴 = ∪ ∪ 𝐴 → 𝐴 = 1o) → 𝐴 ∈ Comp))
76	40, 75	impbid 215	1 ⊢ (Ord 𝐴 → (𝐴 ∈ Comp ↔ (∪ 𝐴 = ∪ ∪ 𝐴 → 𝐴 = 1o)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∨ wo 861   = wceq 1569   ∈ wcel 2144   ≠ wne 2955   ⊆ wss 3895  ∅c0 4278  {csn 4584  ∪ cuni 4866  Ord word 6361  Oncon0 6362  Lim wlim 6363  suc csuc 6364  1_oc1o 8472  Topctop 23216  Compccmp 23709

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  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-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-om 7878  df-1o 8479  df-en 8977  df-fin 8980  df-topgen 17619  df-top 23217  df-topon 23234  df-bases 23269  df-cmp 23710

This theorem is used by: (None)