Theorem sseqin2 2225

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

Step	Hyp	Ref	Expression
1		df-ss 2049	. 2 |- (A (_ B <-> (A i^i B) = A)
2		incom 2204	. . 3 |- (A i^i B) = (B i^i A)
3	2	eqeq1i 1479	. 2 |- ((A i^i B) = A <-> (B i^i A) = A)
4	1, 3	bitr 173	1 |- (A (_ B <-> (B i^i A) = A)

Syntax hints:   <-> wb 146   = wceq 954   i^i cin 2042   (_ wss 2043

This theorem is referenced by:  dfss4 2238  onfr 2981  resabs1 3380  pw2en 4432  fiint 4540  cmcmlem 9474  pjvect 9581  pjocvect 9582  ssmd2 10176  mdslmd4 10197  irredlem2 10255  irredlem3 10256

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-v 1808  df-in 2047  df-ss 2049

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