Theorem sseqin2 2232

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 2056	. 2 |- (A (_ B <-> (A i^i B) = A)
2		incom 2211	. . 3 |- (A i^i B) = (B i^i A)
3	2	eqeq1i 1485	. 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 958   i^i cin 2049   (_ wss 2050

This theorem is referenced by:  dfss4 2245  onfr 2992  resabs1 3394  pw2en 4452  fiint 4572  fiintOLD 4573  cmcmlem 9529  pjvect 9636  pjocvect 9637  ssmd2 10234  mdslmd4 10255  irredlem2 10313  irredlem3 10314  dmdbr7at 10346

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-v 1815  df-in 2054  df-ss 2056

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