Theorem ordcmp 33795

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 7509	. . . 4 ⊢ (Ord 𝐴 → Ord ∪ 𝐴)
2		unizlim 6307	. . . . . 6 ⊢ (Ord ∪ 𝐴 → (∪ 𝐴 = ∪ ∪ 𝐴 ↔ (∪ 𝐴 = ∅ ∨ Lim ∪ 𝐴)))
3		uni0b 4864	. . . . . . 7 ⊢ (∪ 𝐴 = ∅ ↔ 𝐴 ⊆ {∅})
4	3	orbi1i 910	. . . . . 6 ⊢ ((∪ 𝐴 = ∅ ∨ Lim ∪ 𝐴) ↔ (𝐴 ⊆ {∅} ∨ Lim ∪ 𝐴))
5	2, 4	syl6bb 289	. . . . 5 ⊢ (Ord ∪ 𝐴 → (∪ 𝐴 = ∪ ∪ 𝐴 ↔ (𝐴 ⊆ {∅} ∨ Lim ∪ 𝐴)))
6	5	biimpd 231	. . . 4 ⊢ (Ord ∪ 𝐴 → (∪ 𝐴 = ∪ ∪ 𝐴 → (𝐴 ⊆ {∅} ∨ Lim ∪ 𝐴)))
7	1, 6	syl 17	. . 3 ⊢ (Ord 𝐴 → (∪ 𝐴 = ∪ ∪ 𝐴 → (𝐴 ⊆ {∅} ∨ Lim ∪ 𝐴)))
8		sssn 4759	. . . . . . 7 ⊢ (𝐴 ⊆ {∅} ↔ (𝐴 = ∅ ∨ 𝐴 = {∅}))
9		0ntop 21513	. . . . . . . . . . 11 ⊢ ¬ ∅ ∈ Top
10		cmptop 22003	. . . . . . . . . . 11 ⊢ (∅ ∈ Comp → ∅ ∈ Top)
11	9, 10	mto 199	. . . . . . . . . 10 ⊢ ¬ ∅ ∈ Comp
12		eleq1 2900	. . . . . . . . . 10 ⊢ (𝐴 = ∅ → (𝐴 ∈ Comp ↔ ∅ ∈ Comp))
13	11, 12	mtbiri 329	. . . . . . . . 9 ⊢ (𝐴 = ∅ → ¬ 𝐴 ∈ Comp)
14	13	pm2.21d 121	. . . . . . . 8 ⊢ (𝐴 = ∅ → (𝐴 ∈ Comp → 𝐴 = 1o))
15		id 22	. . . . . . . . . 10 ⊢ (𝐴 = {∅} → 𝐴 = {∅})
16		df1o2 8116	. . . . . . . . . 10 ⊢ 1o = {∅}
17	15, 16	syl6eqr 2874	. . . . . . . . 9 ⊢ (𝐴 = {∅} → 𝐴 = 1o)
18	17	a1d 25	. . . . . . . 8 ⊢ (𝐴 = {∅} → (𝐴 ∈ Comp → 𝐴 = 1o))
19	14, 18	jaoi 853	. . . . . . 7 ⊢ ((𝐴 = ∅ ∨ 𝐴 = {∅}) → (𝐴 ∈ Comp → 𝐴 = 1o))
20	8, 19	sylbi 219	. . . . . 6 ⊢ (𝐴 ⊆ {∅} → (𝐴 ∈ Comp → 𝐴 = 1o))
21	20	a1i 11	. . . . 5 ⊢ (Ord 𝐴 → (𝐴 ⊆ {∅} → (𝐴 ∈ Comp → 𝐴 = 1o)))
22		ordtop 33784	. . . . . . . . . . 11 ⊢ (Ord 𝐴 → (𝐴 ∈ Top ↔ 𝐴 ≠ ∪ 𝐴))
23	22	biimpd 231	. . . . . . . . . 10 ⊢ (Ord 𝐴 → (𝐴 ∈ Top → 𝐴 ≠ ∪ 𝐴))
24	23	necon2bd 3032	. . . . . . . . 9 ⊢ (Ord 𝐴 → (𝐴 = ∪ 𝐴 → ¬ 𝐴 ∈ Top))
25		cmptop 22003	. . . . . . . . . 10 ⊢ (𝐴 ∈ Comp → 𝐴 ∈ Top)
26	25	con3i 157	. . . . . . . . 9 ⊢ (¬ 𝐴 ∈ Top → ¬ 𝐴 ∈ Comp)
27	24, 26	syl6 35	. . . . . . . 8 ⊢ (Ord 𝐴 → (𝐴 = ∪ 𝐴 → ¬ 𝐴 ∈ Comp))
28	27	a1dd 50	. . . . . . 7 ⊢ (Ord 𝐴 → (𝐴 = ∪ 𝐴 → (Lim ∪ 𝐴 → ¬ 𝐴 ∈ Comp)))
29		limsucncmp 33794	. . . . . . . . 9 ⊢ (Lim ∪ 𝐴 → ¬ suc ∪ 𝐴 ∈ Comp)
30		eleq1 2900	. . . . . . . . . 10 ⊢ (𝐴 = suc ∪ 𝐴 → (𝐴 ∈ Comp ↔ suc ∪ 𝐴 ∈ Comp))
31	30	notbid 320	. . . . . . . . 9 ⊢ (𝐴 = suc ∪ 𝐴 → (¬ 𝐴 ∈ Comp ↔ ¬ suc ∪ 𝐴 ∈ Comp))
32	29, 31	syl5ibr 248	. . . . . . . 8 ⊢ (𝐴 = suc ∪ 𝐴 → (Lim ∪ 𝐴 → ¬ 𝐴 ∈ Comp))
33	32	a1i 11	. . . . . . 7 ⊢ (Ord 𝐴 → (𝐴 = suc ∪ 𝐴 → (Lim ∪ 𝐴 → ¬ 𝐴 ∈ Comp)))
34		orduniorsuc 7545	. . . . . . 7 ⊢ (Ord 𝐴 → (𝐴 = ∪ 𝐴 ∨ 𝐴 = suc ∪ 𝐴))
35	28, 33, 34	mpjaod 856	. . . . . 6 ⊢ (Ord 𝐴 → (Lim ∪ 𝐴 → ¬ 𝐴 ∈ Comp))
36		pm2.21 123	. . . . . 6 ⊢ (¬ 𝐴 ∈ Comp → (𝐴 ∈ Comp → 𝐴 = 1o))
37	35, 36	syl6 35	. . . . 5 ⊢ (Ord 𝐴 → (Lim ∪ 𝐴 → (𝐴 ∈ Comp → 𝐴 = 1o)))
38	21, 37	jaod 855	. . . 4 ⊢ (Ord 𝐴 → ((𝐴 ⊆ {∅} ∨ Lim ∪ 𝐴) → (𝐴 ∈ Comp → 𝐴 = 1o)))
39	38	com23 86	. . 3 ⊢ (Ord 𝐴 → (𝐴 ∈ Comp → ((𝐴 ⊆ {∅} ∨ Lim ∪ 𝐴) → 𝐴 = 1o)))
40	7, 39	syl5d 73	. 2 ⊢ (Ord 𝐴 → (𝐴 ∈ Comp → (∪ 𝐴 = ∪ ∪ 𝐴 → 𝐴 = 1o)))
41		ordeleqon 7503	. . . . . . 7 ⊢ (Ord 𝐴 ↔ (𝐴 ∈ On ∨ 𝐴 = On))
42		unon 7546	. . . . . . . . . . 11 ⊢ ∪ On = On
43	42	eqcomi 2830	. . . . . . . . . 10 ⊢ On = ∪ On
44	43	unieqi 4851	. . . . . . . . 9 ⊢ ∪ On = ∪ ∪ On
45		unieq 4849	. . . . . . . . 9 ⊢ (𝐴 = On → ∪ 𝐴 = ∪ On)
46	45	unieqd 4852	. . . . . . . . 9 ⊢ (𝐴 = On → ∪ ∪ 𝐴 = ∪ ∪ On)
47	44, 45, 46	3eqtr4a 2882	. . . . . . . 8 ⊢ (𝐴 = On → ∪ 𝐴 = ∪ ∪ 𝐴)
48	47	orim2i 907	. . . . . . 7 ⊢ ((𝐴 ∈ On ∨ 𝐴 = On) → (𝐴 ∈ On ∨ ∪ 𝐴 = ∪ ∪ 𝐴))
49	41, 48	sylbi 219	. . . . . 6 ⊢ (Ord 𝐴 → (𝐴 ∈ On ∨ ∪ 𝐴 = ∪ ∪ 𝐴))
50	49	orcomd 867	. . . . 5 ⊢ (Ord 𝐴 → (∪ 𝐴 = ∪ ∪ 𝐴 ∨ 𝐴 ∈ On))
51	50	ord 860	. . . 4 ⊢ (Ord 𝐴 → (¬ ∪ 𝐴 = ∪ ∪ 𝐴 → 𝐴 ∈ On))
52		unieq 4849	. . . . . . 7 ⊢ (𝐴 = ∪ 𝐴 → ∪ 𝐴 = ∪ ∪ 𝐴)
53	52	con3i 157	. . . . . 6 ⊢ (¬ ∪ 𝐴 = ∪ ∪ 𝐴 → ¬ 𝐴 = ∪ 𝐴)
54	34	ord 860	. . . . . 6 ⊢ (Ord 𝐴 → (¬ 𝐴 = ∪ 𝐴 → 𝐴 = suc ∪ 𝐴))
55	53, 54	syl5 34	. . . . 5 ⊢ (Ord 𝐴 → (¬ ∪ 𝐴 = ∪ ∪ 𝐴 → 𝐴 = suc ∪ 𝐴))
56		orduniorsuc 7545	. . . . . . . 8 ⊢ (Ord ∪ 𝐴 → (∪ 𝐴 = ∪ ∪ 𝐴 ∨ ∪ 𝐴 = suc ∪ ∪ 𝐴))
57	1, 56	syl 17	. . . . . . 7 ⊢ (Ord 𝐴 → (∪ 𝐴 = ∪ ∪ 𝐴 ∨ ∪ 𝐴 = suc ∪ ∪ 𝐴))
58	57	ord 860	. . . . . 6 ⊢ (Ord 𝐴 → (¬ ∪ 𝐴 = ∪ ∪ 𝐴 → ∪ 𝐴 = suc ∪ ∪ 𝐴))
59		suceq 6256	. . . . . 6 ⊢ (∪ 𝐴 = suc ∪ ∪ 𝐴 → suc ∪ 𝐴 = suc suc ∪ ∪ 𝐴)
60	58, 59	syl6 35	. . . . 5 ⊢ (Ord 𝐴 → (¬ ∪ 𝐴 = ∪ ∪ 𝐴 → suc ∪ 𝐴 = suc suc ∪ ∪ 𝐴))
61		eqtr 2841	. . . . . 6 ⊢ ((𝐴 = suc ∪ 𝐴 ∧ suc ∪ 𝐴 = suc suc ∪ ∪ 𝐴) → 𝐴 = suc suc ∪ ∪ 𝐴)
62	61	ex 415	. . . . 5 ⊢ (𝐴 = suc ∪ 𝐴 → (suc ∪ 𝐴 = suc suc ∪ ∪ 𝐴 → 𝐴 = suc suc ∪ ∪ 𝐴))
63	55, 60, 62	syl6c 70	. . . 4 ⊢ (Ord 𝐴 → (¬ ∪ 𝐴 = ∪ ∪ 𝐴 → 𝐴 = suc suc ∪ ∪ 𝐴))
64		onuni 7508	. . . . 5 ⊢ (𝐴 ∈ On → ∪ 𝐴 ∈ On)
65		onuni 7508	. . . . 5 ⊢ (∪ 𝐴 ∈ On → ∪ ∪ 𝐴 ∈ On)
66		onsucsuccmp 33792	. . . . 5 ⊢ (∪ ∪ 𝐴 ∈ On → suc suc ∪ ∪ 𝐴 ∈ Comp)
67		eleq1a 2908	. . . . 5 ⊢ (suc suc ∪ ∪ 𝐴 ∈ Comp → (𝐴 = suc suc ∪ ∪ 𝐴 → 𝐴 ∈ Comp))
68	64, 65, 66, 67	4syl 19	. . . 4 ⊢ (𝐴 ∈ On → (𝐴 = suc suc ∪ ∪ 𝐴 → 𝐴 ∈ Comp))
69	51, 63, 68	syl6c 70	. . 3 ⊢ (Ord 𝐴 → (¬ ∪ 𝐴 = ∪ ∪ 𝐴 → 𝐴 ∈ Comp))
70		id 22	. . . . . 6 ⊢ (𝐴 = 1o → 𝐴 = 1o)
71	70, 16	syl6eq 2872	. . . . 5 ⊢ (𝐴 = 1o → 𝐴 = {∅})
72		0cmp 22002	. . . . 5 ⊢ {∅} ∈ Comp
73	71, 72	eqeltrdi 2921	. . . 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 214	1 ⊢ (Ord 𝐴 → (𝐴 ∈ Comp ↔ (∪ 𝐴 = ∪ ∪ 𝐴 → 𝐴 = 1o)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∨ wo 843   = wceq 1537   ∈ wcel 2114   ≠ wne 3016   ⊆ wss 3936  ∅c0 4291  {csn 4567  ∪ cuni 4838  Ord word 6190  Oncon0 6191  Lim wlim 6192  suc csuc 6193  1_oc1o 8095  Topctop 21501  Compccmp 21994

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 2793  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461

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 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-om 7581  df-1o 8102  df-er 8289  df-en 8510  df-fin 8513  df-topgen 16717  df-top 21502  df-topon 21519  df-bases 21554  df-cmp 21995

This theorem is referenced by: (None)