Theorem sseq0 2304

Description: A subclass of an empty class is empty.

Assertion

Ref	Expression
sseq0	|- ((A (_ B /\ B = (/)) -> A = (/))

Proof of Theorem sseq0

Step	Hyp	Ref	Expression
1		sseq2 2083	. . 3 |- (B = (/) -> (A (_ B <-> A (_ (/)))
2	1	biimpac 418	. 2 |- ((A (_ B /\ B = (/)) -> A (_ (/))
3		ss0b 2302	. 2 |- (A (_ (/) <-> A = (/))
4	2, 3	sylib 198	1 |- ((A (_ B /\ B = (/)) -> A = (/))

Syntax hints:   -> wi 3   /\ wa 223   = wceq 956   (_ wss 2047  (/)c0 2280

This theorem is referenced by:  ssne0 2305  sncld 7787  lpbl 7880

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

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-v 1812  df-dif 2049  df-in 2051  df-ss 2053  df-nul 2281

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