Theorem ordcmp 36629

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 7743	. . . 4 ⊢ (Ord 𝐴 → Ord ∪ 𝐴)
2		unizlim 6447	. . . . . 6 ⊢ (Ord ∪ 𝐴 → (∪ 𝐴 = ∪ ∪ 𝐴 ↔ (∪ 𝐴 = ∅ ∨ Lim ∪ 𝐴)))
3		uni0b 4876	. . . . . . 7 ⊢ (∪ 𝐴 = ∅ ↔ 𝐴 ⊆ {∅})
4	3	orbi1i 914	. . . . . 6 ⊢ ((∪ 𝐴 = ∅ ∨ Lim ∪ 𝐴) ↔ (𝐴 ⊆ {∅} ∨ Lim ∪ 𝐴))
5	2, 4	bitrdi 287	. . . . 5 ⊢ (Ord ∪ 𝐴 → (∪ 𝐴 = ∪ ∪ 𝐴 ↔ (𝐴 ⊆ {∅} ∨ Lim ∪ 𝐴)))
6	5	biimpd 229	. . . 4 ⊢ (Ord ∪ 𝐴 → (∪ 𝐴 = ∪ ∪ 𝐴 → (𝐴 ⊆ {∅} ∨ Lim ∪ 𝐴)))
7	1, 6	syl 17	. . 3 ⊢ (Ord 𝐴 → (∪ 𝐴 = ∪ ∪ 𝐴 → (𝐴 ⊆ {∅} ∨ Lim ∪ 𝐴)))
8		sssn 4769	. . . . . . 7 ⊢ (𝐴 ⊆ {∅} ↔ (𝐴 = ∅ ∨ 𝐴 = {∅}))
9		0ntop 22870	. . . . . . . . . . 11 ⊢ ¬ ∅ ∈ Top
10		cmptop 23360	. . . . . . . . . . 11 ⊢ (∅ ∈ Comp → ∅ ∈ Top)
11	9, 10	mto 197	. . . . . . . . . 10 ⊢ ¬ ∅ ∈ Comp
12		eleq1 2824	. . . . . . . . . 10 ⊢ (𝐴 = ∅ → (𝐴 ∈ Comp ↔ ∅ ∈ Comp))
13	11, 12	mtbiri 327	. . . . . . . . 9 ⊢ (𝐴 = ∅ → ¬ 𝐴 ∈ Comp)
14	13	pm2.21d 121	. . . . . . . 8 ⊢ (𝐴 = ∅ → (𝐴 ∈ Comp → 𝐴 = 1o))
15		id 22	. . . . . . . . . 10 ⊢ (𝐴 = {∅} → 𝐴 = {∅})
16		df1o2 8412	. . . . . . . . . 10 ⊢ 1o = {∅}
17	15, 16	eqtr4di 2789	. . . . . . . . 9 ⊢ (𝐴 = {∅} → 𝐴 = 1o)
18	17	a1d 25	. . . . . . . 8 ⊢ (𝐴 = {∅} → (𝐴 ∈ Comp → 𝐴 = 1o))
19	14, 18	jaoi 858	. . . . . . 7 ⊢ ((𝐴 = ∅ ∨ 𝐴 = {∅}) → (𝐴 ∈ Comp → 𝐴 = 1o))
20	8, 19	sylbi 217	. . . . . 6 ⊢ (𝐴 ⊆ {∅} → (𝐴 ∈ Comp → 𝐴 = 1o))
21	20	a1i 11	. . . . 5 ⊢ (Ord 𝐴 → (𝐴 ⊆ {∅} → (𝐴 ∈ Comp → 𝐴 = 1o)))
22		ordtop 36618	. . . . . . . . . . 11 ⊢ (Ord 𝐴 → (𝐴 ∈ Top ↔ 𝐴 ≠ ∪ 𝐴))
23	22	biimpd 229	. . . . . . . . . 10 ⊢ (Ord 𝐴 → (𝐴 ∈ Top → 𝐴 ≠ ∪ 𝐴))
24	23	necon2bd 2948	. . . . . . . . 9 ⊢ (Ord 𝐴 → (𝐴 = ∪ 𝐴 → ¬ 𝐴 ∈ Top))
25		cmptop 23360	. . . . . . . . . 10 ⊢ (𝐴 ∈ Comp → 𝐴 ∈ Top)
26	25	con3i 154	. . . . . . . . 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 36628	. . . . . . . . 9 ⊢ (Lim ∪ 𝐴 → ¬ suc ∪ 𝐴 ∈ Comp)
30		eleq1 2824	. . . . . . . . . 10 ⊢ (𝐴 = suc ∪ 𝐴 → (𝐴 ∈ Comp ↔ suc ∪ 𝐴 ∈ Comp))
31	30	notbid 318	. . . . . . . . 9 ⊢ (𝐴 = suc ∪ 𝐴 → (¬ 𝐴 ∈ Comp ↔ ¬ suc ∪ 𝐴 ∈ Comp))
32	29, 31	imbitrrid 246	. . . . . . . 8 ⊢ (𝐴 = suc ∪ 𝐴 → (Lim ∪ 𝐴 → ¬ 𝐴 ∈ Comp))
33	32	a1i 11	. . . . . . 7 ⊢ (Ord 𝐴 → (𝐴 = suc ∪ 𝐴 → (Lim ∪ 𝐴 → ¬ 𝐴 ∈ Comp)))
34		orduniorsuc 7781	. . . . . . 7 ⊢ (Ord 𝐴 → (𝐴 = ∪ 𝐴 ∨ 𝐴 = suc ∪ 𝐴))
35	28, 33, 34	mpjaod 861	. . . . . 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 860	. . . 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 7736	. . . . . . 7 ⊢ (Ord 𝐴 ↔ (𝐴 ∈ On ∨ 𝐴 = On))
42		unon 7782	. . . . . . . . . . 11 ⊢ ∪ On = On
43	42	eqcomi 2745	. . . . . . . . . 10 ⊢ On = ∪ On
44	43	unieqi 4862	. . . . . . . . 9 ⊢ ∪ On = ∪ ∪ On
45		unieq 4861	. . . . . . . . 9 ⊢ (𝐴 = On → ∪ 𝐴 = ∪ On)
46	45	unieqd 4863	. . . . . . . . 9 ⊢ (𝐴 = On → ∪ ∪ 𝐴 = ∪ ∪ On)
47	44, 45, 46	3eqtr4a 2797	. . . . . . . 8 ⊢ (𝐴 = On → ∪ 𝐴 = ∪ ∪ 𝐴)
48	47	orim2i 911	. . . . . . 7 ⊢ ((𝐴 ∈ On ∨ 𝐴 = On) → (𝐴 ∈ On ∨ ∪ 𝐴 = ∪ ∪ 𝐴))
49	41, 48	sylbi 217	. . . . . 6 ⊢ (Ord 𝐴 → (𝐴 ∈ On ∨ ∪ 𝐴 = ∪ ∪ 𝐴))
50	49	orcomd 872	. . . . 5 ⊢ (Ord 𝐴 → (∪ 𝐴 = ∪ ∪ 𝐴 ∨ 𝐴 ∈ On))
51	50	ord 865	. . . 4 ⊢ (Ord 𝐴 → (¬ ∪ 𝐴 = ∪ ∪ 𝐴 → 𝐴 ∈ On))
52		unieq 4861	. . . . . . 7 ⊢ (𝐴 = ∪ 𝐴 → ∪ 𝐴 = ∪ ∪ 𝐴)
53	52	con3i 154	. . . . . 6 ⊢ (¬ ∪ 𝐴 = ∪ ∪ 𝐴 → ¬ 𝐴 = ∪ 𝐴)
54	34	ord 865	. . . . . 6 ⊢ (Ord 𝐴 → (¬ 𝐴 = ∪ 𝐴 → 𝐴 = suc ∪ 𝐴))
55	53, 54	syl5 34	. . . . 5 ⊢ (Ord 𝐴 → (¬ ∪ 𝐴 = ∪ ∪ 𝐴 → 𝐴 = suc ∪ 𝐴))
56		orduniorsuc 7781	. . . . . . . 8 ⊢ (Ord ∪ 𝐴 → (∪ 𝐴 = ∪ ∪ 𝐴 ∨ ∪ 𝐴 = suc ∪ ∪ 𝐴))
57	1, 56	syl 17	. . . . . . 7 ⊢ (Ord 𝐴 → (∪ 𝐴 = ∪ ∪ 𝐴 ∨ ∪ 𝐴 = suc ∪ ∪ 𝐴))
58	57	ord 865	. . . . . 6 ⊢ (Ord 𝐴 → (¬ ∪ 𝐴 = ∪ ∪ 𝐴 → ∪ 𝐴 = suc ∪ ∪ 𝐴))
59		suceq 6391	. . . . . 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 2756	. . . . . 6 ⊢ ((𝐴 = suc ∪ 𝐴 ∧ suc ∪ 𝐴 = suc suc ∪ ∪ 𝐴) → 𝐴 = suc suc ∪ ∪ 𝐴)
62	61	ex 412	. . . . 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 7742	. . . . 5 ⊢ (𝐴 ∈ On → ∪ 𝐴 ∈ On)
65		onuni 7742	. . . . 5 ⊢ (∪ 𝐴 ∈ On → ∪ ∪ 𝐴 ∈ On)
66		onsucsuccmp 36626	. . . . 5 ⊢ (∪ ∪ 𝐴 ∈ On → suc suc ∪ ∪ 𝐴 ∈ Comp)
67		eleq1a 2831	. . . . 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	eqtrdi 2787	. . . . 5 ⊢ (𝐴 = 1o → 𝐴 = {∅})
72		0cmp 23359	. . . . 5 ⊢ {∅} ∈ Comp
73	71, 72	eqeltrdi 2844	. . . 4 ⊢ (𝐴 = 1o → 𝐴 ∈ Comp)
74	73	a1i 11	. . 3 ⊢ (Ord 𝐴 → (𝐴 = 1o → 𝐴 ∈ Comp))
75	69, 74	jad 187	. 2 ⊢ (Ord 𝐴 → ((∪ 𝐴 = ∪ ∪ 𝐴 → 𝐴 = 1o) → 𝐴 ∈ Comp))
76	40, 75	impbid 212	1 ⊢ (Ord 𝐴 → (𝐴 ∈ Comp ↔ (∪ 𝐴 = ∪ ∪ 𝐴 → 𝐴 = 1o)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∨ wo 848   = wceq 1542   ∈ wcel 2114   ≠ wne 2932   ⊆ wss 3889  ∅c0 4273  {csn 4567  ∪ cuni 4850  Ord word 6322  Oncon0 6323  Lim wlim 6324  suc csuc 6325  1_oc1o 8398  Topctop 22858  Compccmp 23351

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-om 7818  df-1o 8405  df-en 8894  df-fin 8897  df-topgen 17406  df-top 22859  df-topon 22876  df-bases 22911  df-cmp 23352

This theorem is referenced by: (None)