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Theorem 0top 17050
 Description: The singleton of the empty set is the only topology possible for an empty underlying set. (Contributed by NM, 9-Sep-2006.)
Assertion
Ref Expression
0top

Proof of Theorem 0top
StepHypRef Expression
1 olc 375 . . 3
2 0opn 16979 . . . . . 6
3 n0i 3635 . . . . . 6
42, 3syl 16 . . . . 5
54pm2.21d 101 . . . 4
6 idd 23 . . . 4
75, 6jaod 371 . . 3
81, 7impbid2 197 . 2
9 uni0b 4042 . . 3
10 sssn 3959 . . 3
119, 10bitr2i 243 . 2
128, 11syl6rbb 255 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wb 178   wo 359   wceq 1653   wcel 1726   wss 3322  c0 3630  csn 3816  cuni 4017  ctop 16960 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-v 2960  df-dif 3325  df-in 3329  df-ss 3336  df-nul 3631  df-pw 3803  df-sn 3822  df-uni 4018  df-top 16965
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