Theorem topopn 7602

Description: The underlying set of a topology is an open set.

Hypothesis

Ref	Expression
1open.1	|- X = U.J

Assertion

Ref	Expression
topopn	|- (J e. Top -> X e. J)

Proof of Theorem topopn

Step	Hyp	Ref	Expression
1		ssid 2080	. . 3 |- J (_ J
2		uniopnt 7598	. . 3 |- ((J e. Top /\ J (_ J) -> U.J e. J)
3	1, 2	mpan2 696	. 2 |- (J e. Top -> U.J e. J)
4		1open.1	. 2 |- X = U.J
5	3, 4	syl5eqel 1552	1 |- (J e. Top -> X e. J)

Syntax hints:   -> wi 3   = wceq 956   e. wcel 958   (_ wss 2047  U.cuni 2503  Topctop 7588

This theorem is referenced by:  tpsex 7605  istps2 7607  basgen2t 7639  0cld 7678  ntrtop 7701  tpnei 7734  cnconst 7780  mapudiscn 10512  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-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-ral 1649  df-v 1812  df-in 2051  df-ss 2053  df-pw 2402  df-uni 2504  df-top 7592

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