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Theorem sseqin2 2219
Description: A relationship between subclass and intersection. Similar to Exercise 9 of [TakeutiZaring] p. 18.
Assertion
Ref Expression
sseqin2 |- (A (_ B <-> (B i^i A) = A)
Proof of Theorem sseqin2
StepHypRef Expression
1 df-ss 2043 . 2 |- (A (_ B <-> (A i^i B) = A)
2 incom 2198 . . 3 |- (A i^i B) = (B i^i A)
32eqeq1i 1474 . 2 |- ((A i^i B) = A <-> (B i^i A) = A)
41, 3bitr 173 1 |- (A (_ B <-> (B i^i A) = A)
Colors of variables: wff set class
Syntax hints:   <-> wb 146   = wceq 953   i^i cin 2036   (_ wss 2037
This theorem is referenced by:  dfss4 2232  onfr 2976  resabs1 3372  pw2en 4426  fiint 4534  cmcmlem 9451  pjvect 9558  pjocvect 9559  ssmd2 10147  mdslmd4 10168  irredlem2 10226  irredlem3 10227
This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452
This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-v 1803  df-in 2041  df-ss 2043
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