Theorem ss0b 2299

Description: Any subset of the empty set is empty. Theorem 5 of [Suppes] p. 23 and its converse.

Assertion

Ref	Expression
ss0b	|- (A (_ (/) <-> A = (/))

Proof of Theorem ss0b

Step	Hyp	Ref	Expression
1		eqss 2074	. . 3 |- (A = (/) <-> (A (_ (/) /\ (/) (_ A))
2		0ss 2298	. . 3 |- (/) (_ A
3	1, 2	mpbiran2 728	. 2 |- (A = (/) <-> A (_ (/))
4	3	bicomi 172	1 |- (A (_ (/) <-> A = (/))

Syntax hints:   <-> wb 146   = wceq 955   (_ wss 2044  (/)c0 2277

This theorem is referenced by:  ss0 2300  sseq0 2301  un00 2303  ssdisj 2315  pw0 2465  undom 4427  kmlem5 4752  card0 4806  cf0 4893  infxpidmlem11 7522  0nnei 7686  esnnei 10454

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-12 967  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

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1171  df-clab 1463  df-cleq 1468  df-clel 1471  df-v 1809  df-dif 2046  df-in 2048  df-ss 2050  df-nul 2278

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