Theorem sn0top 7607

Description: The singleton of the empty set is a topology. (Contributed by Stefan Allan, 3-Mar-2006.)

Assertion

Ref	Expression
sn0top	⊢ {∅} ∈ Top

Proof of Theorem sn0top

Step	Hyp	Ref	Expression
1		p0ex 2766	. . 3 ⊢ {∅} ∈ V
2		istopg 7556	. . 3 ⊢ ({∅} ∈ V → ({∅} ∈ Top ↔ (∀x(x ⊆ {∅} → ∪x ∈ {∅}) ⋀ ∀x ∈ {∅}∀y ∈ {∅} (x ∩ y) ∈ {∅})))
3	1, 2	ax-mp 7	. 2 ⊢ ({∅} ∈ Top ↔ (∀x(x ⊆ {∅} → ∪x ∈ {∅}) ⋀ ∀x ∈ {∅}∀y ∈ {∅} (x ∩ y) ∈ {∅}))
4		sssn 2470	. . . 4 ⊢ (x ⊆ {∅} ↔ (x = ∅ ⋁ x = {∅}))
5		unieq 2506	. . . . . 6 ⊢ (x = ∅ → ∪x = ∪∅)
6		uni0 2521	. . . . . . 7 ⊢ ∪∅ = ∅
7		0ex 2707	. . . . . . . 8 ⊢ ∅ ∈ V
8	7	elsnc2 2434	. . . . . . 7 ⊢ (∪∅ ∈ {∅} ↔ ∪∅ = ∅)
9	6, 8	mpbir 190	. . . . . 6 ⊢ ∪∅ ∈ {∅}
10	5, 9	syl6eqel 1554	. . . . 5 ⊢ (x = ∅ → ∪x ∈ {∅})
11		unieq 2506	. . . . . 6 ⊢ (x = {∅} → ∪x = ∪{∅})
12	7	unisn 2513	. . . . . . . 8 ⊢ ∪{∅} = ∅
13		eqtrt 1490	. . . . . . . 8 ⊢ ((∪x = ∪{∅} ⋀ ∪{∅} = ∅) → ∪x = ∅)
14	12, 13	mpan2 695	. . . . . . 7 ⊢ (∪x = ∪{∅} → ∪x = ∅)
15		visset 1810	. . . . . . . . 9 ⊢ x ∈ V
16	15	uniex 2866	. . . . . . . 8 ⊢ ∪x ∈ V
17	16	elsnc 2428	. . . . . . 7 ⊢ (∪x ∈ {∅} ↔ ∪x = ∅)
18	14, 17	sylibr 200	. . . . . 6 ⊢ (∪x = ∪{∅} → ∪x ∈ {∅})
19	11, 18	syl 10	. . . . 5 ⊢ (x = {∅} → ∪x ∈ {∅})
20	10, 19	jaoi 341	. . . 4 ⊢ ((x = ∅ ⋁ x = {∅}) → ∪x ∈ {∅})
21	4, 20	sylbi 199	. . 3 ⊢ (x ⊆ {∅} → ∪x ∈ {∅})
22	21	ax-gen 962	. 2 ⊢ ∀x(x ⊆ {∅} → ∪x ∈ {∅})
23		elsn 2418	. . . . 5 ⊢ (y ∈ {∅} ↔ y = ∅)
24		ineq2 2208	. . . . . . 7 ⊢ (y = ∅ → (x ∩ y) = (x ∩ ∅))
25		in0 2295	. . . . . . . . 9 ⊢ (x ∩ ∅) = ∅
26	25	eqeq2i 1483	. . . . . . . 8 ⊢ ((x ∩ y) = (x ∩ ∅) ↔ (x ∩ y) = ∅)
27	26	biimp 151	. . . . . . 7 ⊢ ((x ∩ y) = (x ∩ ∅) → (x ∩ y) = ∅)
28	24, 27	syl 10	. . . . . 6 ⊢ (y = ∅ → (x ∩ y) = ∅)
29	15	inex1 2712	. . . . . . . 8 ⊢ (x ∩ y) ∈ V
30	29	elsnc 2428	. . . . . . 7 ⊢ ((x ∩ y) ∈ {∅} ↔ (x ∩ y) = ∅)
31	30	biimpr 152	. . . . . 6 ⊢ ((x ∩ y) = ∅ → (x ∩ y) ∈ {∅})
32	28, 31	syl 10	. . . . 5 ⊢ (y = ∅ → (x ∩ y) ∈ {∅})
33	23, 32	sylbi 199	. . . 4 ⊢ (y ∈ {∅} → (x ∩ y) ∈ {∅})
34	33	adantl 388	. . 3 ⊢ ((x ∈ {∅} ⋀ y ∈ {∅}) → (x ∩ y) ∈ {∅})
35	34	rgen2a 1697	. 2 ⊢ ∀x ∈ {∅}∀y ∈ {∅} (x ∩ y) ∈ {∅}
36	3, 22, 35	mpbir2an 729	1 ⊢ {∅} ∈ Top

Syntax hints:   → wi 3   ↔ wb 146   ⋁ wo 222   ⋀ wa 223  ∀wal 953   = wceq 955   ∈ wcel 957  ∀wral 1643  Vcvv 1808   ∩ cin 2043   ⊆ wss 2044  ∅c0 2277  {csn 2406  ∪cuni 2499  Topctop 7548

This theorem is referenced by:  limfillem2 10524

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-sep 2699  ax-nul 2706  ax-pow 2738  ax-un 2862

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-ral 1647  df-rex 1648  df-v 1809  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-nul 2278  df-pw 2399  df-sn 2409  df-pr 2410  df-uni 2500  df-top 7552

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