Theorem 0top 7635

Description: The singleton of the empty set is the only topology possible for an empty underlying set.

Assertion

Ref	Expression
0top	|- (J e. Top -> (U.J = (/) <-> J = {(/)}))

Proof of Theorem 0top

Step	Hyp	Ref	Expression
1		olc 268	. . 3 |- (J = {(/)} -> (J = (/) \/ J = {(/)}))
2		0opnt 7601	. . . . . 6 |- (J e. Top -> (/) e. J)
3		n0i 2285	. . . . . 6 |- ((/) e. J -> -. J = (/))
4	2, 3	syl 10	. . . . 5 |- (J e. Top -> -. J = (/))
5	4	pm2.21d 78	. . . 4 |- (J e. Top -> (J = (/) -> J = {(/)}))
6		idd 61	. . . 4 |- (J e. Top -> (J = {(/)} -> J = {(/)}))
7	5, 6	jaod 424	. . 3 |- (J e. Top -> ((J = (/) \/ J = {(/)}) -> J = {(/)}))
8	1, 7	impbid2 518	. 2 |- (J e. Top -> (J = {(/)} <-> (J = (/) \/ J = {(/)})))
9		uni0b 2523	. . 3 |- (U.J = (/) <-> J (_ {(/)})
10		sssn 2473	. . 3 |- (J (_ {(/)} <-> (J = (/) \/ J = {(/)}))
11	9, 10	bitr2 174	. 2 |- ((J = (/) \/ J = {(/)}) <-> U.J = (/))
12	8, 11	syl6rbb 537	1 |- (J e. Top -> (U.J = (/) <-> J = {(/)}))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   \/ wo 222   = wceq 956   e. wcel 958   (_ wss 2047  (/)c0 2280  {csn 2409  U.cuni 2503  Topctop 7588

This theorem is referenced by:  top2ind 10548

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-uni 2504  df-top 7592

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