Theorem onint1 34638

Description: The ordinal T₁ spaces are 1_o and 2_o, proven without the Axiom of Regularity. (Contributed by Chen-Pang He, 9-Nov-2015.)

Assertion

Ref	Expression
onint1	⊢ (On ∩ Fre) = {1o, 2o}

Proof of Theorem onint1

Dummy variables 𝑗 𝑎 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elin 3903	. . . . 5 ⊢ (𝑗 ∈ (On ∩ Fre) ↔ (𝑗 ∈ On ∧ 𝑗 ∈ Fre))
2		eqid 2738	. . . . . . . . . . 11 ⊢ ∪ 𝑗 = ∪ 𝑗
3	2	ist1 22472	. . . . . . . . . 10 ⊢ (𝑗 ∈ Fre ↔ (𝑗 ∈ Top ∧ ∀𝑎 ∈ ∪ 𝑗{𝑎} ∈ (Clsd‘𝑗)))
4	3	simprbi 497	. . . . . . . . 9 ⊢ (𝑗 ∈ Fre → ∀𝑎 ∈ ∪ 𝑗{𝑎} ∈ (Clsd‘𝑗))
5		onelon 6291	. . . . . . . . . . . . . . 15 ⊢ ((𝑗 ∈ On ∧ (∪ 𝑗 ∖ {∅}) ∈ 𝑗) → (∪ 𝑗 ∖ {∅}) ∈ On)
6	5	ex 413	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ On → ((∪ 𝑗 ∖ {∅}) ∈ 𝑗 → (∪ 𝑗 ∖ {∅}) ∈ On))
7		neldifsnd 4726	. . . . . . . . . . . . . . . . 17 ⊢ (2o ∈ 𝑗 → ¬ ∅ ∈ (∪ 𝑗 ∖ {∅}))
8		p0ex 5307	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ {∅} ∈ V
9	8	prid2 4699	. . . . . . . . . . . . . . . . . . . . 21 ⊢ {∅} ∈ {∅, {∅}}
10		df2o2 8306	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 2o = {∅, {∅}}
11	9, 10	eleqtrri 2838	. . . . . . . . . . . . . . . . . . . 20 ⊢ {∅} ∈ 2o
12		elunii 4844	. . . . . . . . . . . . . . . . . . . 20 ⊢ (({∅} ∈ 2o ∧ 2o ∈ 𝑗) → {∅} ∈ ∪ 𝑗)
13	11, 12	mpan 687	. . . . . . . . . . . . . . . . . . 19 ⊢ (2o ∈ 𝑗 → {∅} ∈ ∪ 𝑗)
14		df1o2 8304	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 1o = {∅}
15		1on 8309	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 1o ∈ On
16	14, 15	eqeltrri 2836	. . . . . . . . . . . . . . . . . . . . 21 ⊢ {∅} ∈ On
17	16	onirri 6373	. . . . . . . . . . . . . . . . . . . 20 ⊢ ¬ {∅} ∈ {∅}
18	17	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ (2o ∈ 𝑗 → ¬ {∅} ∈ {∅})
19	13, 18	eldifd 3898	. . . . . . . . . . . . . . . . . 18 ⊢ (2o ∈ 𝑗 → {∅} ∈ (∪ 𝑗 ∖ {∅}))
20	19	ne0d 4269	. . . . . . . . . . . . . . . . 17 ⊢ (2o ∈ 𝑗 → (∪ 𝑗 ∖ {∅}) ≠ ∅)
21	7, 20	2thd 264	. . . . . . . . . . . . . . . 16 ⊢ (2o ∈ 𝑗 → (¬ ∅ ∈ (∪ 𝑗 ∖ {∅}) ↔ (∪ 𝑗 ∖ {∅}) ≠ ∅))
22		nbbn 385	. . . . . . . . . . . . . . . 16 ⊢ ((¬ ∅ ∈ (∪ 𝑗 ∖ {∅}) ↔ (∪ 𝑗 ∖ {∅}) ≠ ∅) ↔ ¬ (∅ ∈ (∪ 𝑗 ∖ {∅}) ↔ (∪ 𝑗 ∖ {∅}) ≠ ∅))
23	21, 22	sylib 217	. . . . . . . . . . . . . . 15 ⊢ (2o ∈ 𝑗 → ¬ (∅ ∈ (∪ 𝑗 ∖ {∅}) ↔ (∪ 𝑗 ∖ {∅}) ≠ ∅))
24		on0eln0 6321	. . . . . . . . . . . . . . 15 ⊢ ((∪ 𝑗 ∖ {∅}) ∈ On → (∅ ∈ (∪ 𝑗 ∖ {∅}) ↔ (∪ 𝑗 ∖ {∅}) ≠ ∅))
25	23, 24	nsyl 140	. . . . . . . . . . . . . 14 ⊢ (2o ∈ 𝑗 → ¬ (∪ 𝑗 ∖ {∅}) ∈ On)
26	6, 25	nsyli 157	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ On → (2o ∈ 𝑗 → ¬ (∪ 𝑗 ∖ {∅}) ∈ 𝑗))
27	26	imp 407	. . . . . . . . . . . 12 ⊢ ((𝑗 ∈ On ∧ 2o ∈ 𝑗) → ¬ (∪ 𝑗 ∖ {∅}) ∈ 𝑗)
28		0ex 5231	. . . . . . . . . . . . . . . . . 18 ⊢ ∅ ∈ V
29	28	prid1 4698	. . . . . . . . . . . . . . . . 17 ⊢ ∅ ∈ {∅, {∅}}
30	29, 10	eleqtrri 2838	. . . . . . . . . . . . . . . 16 ⊢ ∅ ∈ 2o
31		elunii 4844	. . . . . . . . . . . . . . . 16 ⊢ ((∅ ∈ 2o ∧ 2o ∈ 𝑗) → ∅ ∈ ∪ 𝑗)
32	30, 31	mpan 687	. . . . . . . . . . . . . . 15 ⊢ (2o ∈ 𝑗 → ∅ ∈ ∪ 𝑗)
33	32	adantl 482	. . . . . . . . . . . . . 14 ⊢ ((𝑗 ∈ On ∧ 2o ∈ 𝑗) → ∅ ∈ ∪ 𝑗)
34		simpr 485	. . . . . . . . . . . . . . . 16 ⊢ (((𝑗 ∈ On ∧ 2o ∈ 𝑗) ∧ 𝑎 = ∅) → 𝑎 = ∅)
35	34	sneqd 4573	. . . . . . . . . . . . . . 15 ⊢ (((𝑗 ∈ On ∧ 2o ∈ 𝑗) ∧ 𝑎 = ∅) → {𝑎} = {∅})
36	35	eleq1d 2823	. . . . . . . . . . . . . 14 ⊢ (((𝑗 ∈ On ∧ 2o ∈ 𝑗) ∧ 𝑎 = ∅) → ({𝑎} ∈ (Clsd‘𝑗) ↔ {∅} ∈ (Clsd‘𝑗)))
37	33, 36	rspcdv 3553	. . . . . . . . . . . . 13 ⊢ ((𝑗 ∈ On ∧ 2o ∈ 𝑗) → (∀𝑎 ∈ ∪ 𝑗{𝑎} ∈ (Clsd‘𝑗) → {∅} ∈ (Clsd‘𝑗)))
38	2	cldopn 22182	. . . . . . . . . . . . 13 ⊢ ({∅} ∈ (Clsd‘𝑗) → (∪ 𝑗 ∖ {∅}) ∈ 𝑗)
39	37, 38	syl6 35	. . . . . . . . . . . 12 ⊢ ((𝑗 ∈ On ∧ 2o ∈ 𝑗) → (∀𝑎 ∈ ∪ 𝑗{𝑎} ∈ (Clsd‘𝑗) → (∪ 𝑗 ∖ {∅}) ∈ 𝑗))
40	27, 39	mtod 197	. . . . . . . . . . 11 ⊢ ((𝑗 ∈ On ∧ 2o ∈ 𝑗) → ¬ ∀𝑎 ∈ ∪ 𝑗{𝑎} ∈ (Clsd‘𝑗))
41	40	ex 413	. . . . . . . . . 10 ⊢ (𝑗 ∈ On → (2o ∈ 𝑗 → ¬ ∀𝑎 ∈ ∪ 𝑗{𝑎} ∈ (Clsd‘𝑗)))
42	41	con2d 134	. . . . . . . . 9 ⊢ (𝑗 ∈ On → (∀𝑎 ∈ ∪ 𝑗{𝑎} ∈ (Clsd‘𝑗) → ¬ 2o ∈ 𝑗))
43	4, 42	syl5 34	. . . . . . . 8 ⊢ (𝑗 ∈ On → (𝑗 ∈ Fre → ¬ 2o ∈ 𝑗))
44		2on 8311	. . . . . . . . 9 ⊢ 2o ∈ On
45		ontri1 6300	. . . . . . . . . 10 ⊢ ((𝑗 ∈ On ∧ 2o ∈ On) → (𝑗 ⊆ 2o ↔ ¬ 2o ∈ 𝑗))
46		onsssuc 6353	. . . . . . . . . 10 ⊢ ((𝑗 ∈ On ∧ 2o ∈ On) → (𝑗 ⊆ 2o ↔ 𝑗 ∈ suc 2o))
47	45, 46	bitr3d 280	. . . . . . . . 9 ⊢ ((𝑗 ∈ On ∧ 2o ∈ On) → (¬ 2o ∈ 𝑗 ↔ 𝑗 ∈ suc 2o))
48	44, 47	mpan2 688	. . . . . . . 8 ⊢ (𝑗 ∈ On → (¬ 2o ∈ 𝑗 ↔ 𝑗 ∈ suc 2o))
49	43, 48	sylibd 238	. . . . . . 7 ⊢ (𝑗 ∈ On → (𝑗 ∈ Fre → 𝑗 ∈ suc 2o))
50	49	imp 407	. . . . . 6 ⊢ ((𝑗 ∈ On ∧ 𝑗 ∈ Fre) → 𝑗 ∈ suc 2o)
51		0ntop 22054	. . . . . . . . . 10 ⊢ ¬ ∅ ∈ Top
52		t1top 22481	. . . . . . . . . 10 ⊢ (∅ ∈ Fre → ∅ ∈ Top)
53	51, 52	mto 196	. . . . . . . . 9 ⊢ ¬ ∅ ∈ Fre
54		nelneq 2863	. . . . . . . . 9 ⊢ ((𝑗 ∈ Fre ∧ ¬ ∅ ∈ Fre) → ¬ 𝑗 = ∅)
55	53, 54	mpan2 688	. . . . . . . 8 ⊢ (𝑗 ∈ Fre → ¬ 𝑗 = ∅)
56		elsni 4578	. . . . . . . 8 ⊢ (𝑗 ∈ {∅} → 𝑗 = ∅)
57	55, 56	nsyl 140	. . . . . . 7 ⊢ (𝑗 ∈ Fre → ¬ 𝑗 ∈ {∅})
58	57	adantl 482	. . . . . 6 ⊢ ((𝑗 ∈ On ∧ 𝑗 ∈ Fre) → ¬ 𝑗 ∈ {∅})
59	50, 58	eldifd 3898	. . . . 5 ⊢ ((𝑗 ∈ On ∧ 𝑗 ∈ Fre) → 𝑗 ∈ (suc 2o ∖ {∅}))
60	1, 59	sylbi 216	. . . 4 ⊢ (𝑗 ∈ (On ∩ Fre) → 𝑗 ∈ (suc 2o ∖ {∅}))
61	60	ssriv 3925	. . 3 ⊢ (On ∩ Fre) ⊆ (suc 2o ∖ {∅})
62		df-suc 6272	. . . . . 6 ⊢ suc 2o = (2o ∪ {2o})
63	62	difeq1i 4053	. . . . 5 ⊢ (suc 2o ∖ {∅}) = ((2o ∪ {2o}) ∖ {∅})
64		difundir 4214	. . . . 5 ⊢ ((2o ∪ {2o}) ∖ {∅}) = ((2o ∖ {∅}) ∪ ({2o} ∖ {∅}))
65	63, 64	eqtri 2766	. . . 4 ⊢ (suc 2o ∖ {∅}) = ((2o ∖ {∅}) ∪ ({2o} ∖ {∅}))
66		df-pr 4564	. . . . 5 ⊢ {1o, 2o} = ({1o} ∪ {2o})
67		df2o3 8305	. . . . . . . . 9 ⊢ 2o = {∅, 1o}
68		df-pr 4564	. . . . . . . . 9 ⊢ {∅, 1o} = ({∅} ∪ {1o})
69	67, 68	eqtri 2766	. . . . . . . 8 ⊢ 2o = ({∅} ∪ {1o})
70	69	difeq1i 4053	. . . . . . 7 ⊢ (2o ∖ {∅}) = (({∅} ∪ {1o}) ∖ {∅})
71		difundir 4214	. . . . . . 7 ⊢ (({∅} ∪ {1o}) ∖ {∅}) = (({∅} ∖ {∅}) ∪ ({1o} ∖ {∅}))
72		difid 4304	. . . . . . . . 9 ⊢ ({∅} ∖ {∅}) = ∅
73		1n0 8318	. . . . . . . . . . . 12 ⊢ 1o ≠ ∅
74		disjsn2 4648	. . . . . . . . . . . 12 ⊢ (1o ≠ ∅ → ({1o} ∩ {∅}) = ∅)
75	73, 74	ax-mp 5	. . . . . . . . . . 11 ⊢ ({1o} ∩ {∅}) = ∅
76	75	difeq2i 4054	. . . . . . . . . 10 ⊢ ({1o} ∖ ({1o} ∩ {∅})) = ({1o} ∖ ∅)
77		difin 4195	. . . . . . . . . 10 ⊢ ({1o} ∖ ({1o} ∩ {∅})) = ({1o} ∖ {∅})
78		dif0 4306	. . . . . . . . . 10 ⊢ ({1o} ∖ ∅) = {1o}
79	76, 77, 78	3eqtr3i 2774	. . . . . . . . 9 ⊢ ({1o} ∖ {∅}) = {1o}
80	72, 79	uneq12i 4095	. . . . . . . 8 ⊢ (({∅} ∖ {∅}) ∪ ({1o} ∖ {∅})) = (∅ ∪ {1o})
81		uncom 4087	. . . . . . . 8 ⊢ (∅ ∪ {1o}) = ({1o} ∪ ∅)
82		un0 4324	. . . . . . . 8 ⊢ ({1o} ∪ ∅) = {1o}
83	80, 81, 82	3eqtri 2770	. . . . . . 7 ⊢ (({∅} ∖ {∅}) ∪ ({1o} ∖ {∅})) = {1o}
84	70, 71, 83	3eqtri 2770	. . . . . 6 ⊢ (2o ∖ {∅}) = {1o}
85		2on0 8313	. . . . . . . . 9 ⊢ 2o ≠ ∅
86		disjsn2 4648	. . . . . . . . 9 ⊢ (2o ≠ ∅ → ({2o} ∩ {∅}) = ∅)
87	85, 86	ax-mp 5	. . . . . . . 8 ⊢ ({2o} ∩ {∅}) = ∅
88	87	difeq2i 4054	. . . . . . 7 ⊢ ({2o} ∖ ({2o} ∩ {∅})) = ({2o} ∖ ∅)
89		difin 4195	. . . . . . 7 ⊢ ({2o} ∖ ({2o} ∩ {∅})) = ({2o} ∖ {∅})
90		dif0 4306	. . . . . . 7 ⊢ ({2o} ∖ ∅) = {2o}
91	88, 89, 90	3eqtr3i 2774	. . . . . 6 ⊢ ({2o} ∖ {∅}) = {2o}
92	84, 91	uneq12i 4095	. . . . 5 ⊢ ((2o ∖ {∅}) ∪ ({2o} ∖ {∅})) = ({1o} ∪ {2o})
93	66, 92	eqtr4i 2769	. . . 4 ⊢ {1o, 2o} = ((2o ∖ {∅}) ∪ ({2o} ∖ {∅}))
94	65, 93	eqtr4i 2769	. . 3 ⊢ (suc 2o ∖ {∅}) = {1o, 2o}
95	61, 94	sseqtri 3957	. 2 ⊢ (On ∩ Fre) ⊆ {1o, 2o}
96		ssoninhaus 34637	. . 3 ⊢ {1o, 2o} ⊆ (On ∩ Haus)
97		haust1 22503	. . . . 5 ⊢ (𝑗 ∈ Haus → 𝑗 ∈ Fre)
98	97	ssriv 3925	. . . 4 ⊢ Haus ⊆ Fre
99		sslin 4168	. . . 4 ⊢ (Haus ⊆ Fre → (On ∩ Haus) ⊆ (On ∩ Fre))
100	98, 99	ax-mp 5	. . 3 ⊢ (On ∩ Haus) ⊆ (On ∩ Fre)
101	96, 100	sstri 3930	. 2 ⊢ {1o, 2o} ⊆ (On ∩ Fre)
102	95, 101	eqssi 3937	1 ⊢ (On ∩ Fre) = {1o, 2o}

Syntax hints:  ¬ wn 3   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106   ≠ wne 2943  ∀wral 3064   ∖ cdif 3884   ∪ cun 3885   ∩ cin 3886   ⊆ wss 3887  ∅c0 4256  {csn 4561  {cpr 4563  ∪ cuni 4839  Oncon0 6266  suc csuc 6268  ‘cfv 6433  1_oc1o 8290  2_oc2o 8291  Topctop 22042  Clsdccld 22167  Frect1 22458  Hauscha 22459

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-ord 6269  df-on 6270  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-fv 6441  df-1o 8297  df-2o 8298  df-topgen 17154  df-top 22043  df-topon 22060  df-cld 22170  df-t1 22465  df-haus 22466

This theorem is referenced by:  oninhaus  34639