Theorem ssalel 3225

Description: Alternate definition of the subclass relationship between two classes. Definition 5.9 of [TakeutiZaring] p. 17. (Contributed by NM, 8-Jan-2002.)

Assertion

Ref	Expression
ssalel	|- ( A C_ B <-> A. x ( x e. A -> x e. B ) )

Distinct variable groups:   x, A   x, B

Proof of Theorem ssalel

Step	Hyp	Ref	Expression
1		dfss 3224	. . 3 |- ( A C_ B <-> A = ( A i^i B ) )
2		df-in 3216	. . . 4 |- ( A i^i B ) = { x | ( x e. A /\ x e. B ) }
3	2	eqeq2i 2243	. . 3 |- ( A = ( A i^i B ) <-> A = { x | ( x e. A /\ x e. B ) } )
4		abeq2 2341	. . 3 |- ( A = { x | ( x e. A /\ x e. B ) } <-> A. x ( x e. A <-> ( x e. A /\ x e. B ) ) )
5	1, 3, 4	3bitri 206	. 2 |- ( A C_ B <-> A. x ( x e. A <-> ( x e. A /\ x e. B ) ) )
6		pm4.71 389	. . 3 |- ( ( x e. A -> x e. B ) <-> ( x e. A <-> ( x e. A /\ x e. B ) ) )
7	6	albii 1519	. 2 |- ( A. x ( x e. A -> x e. B ) <-> A. x ( x e. A <-> ( x e. A /\ x e. B ) ) )
8	5, 7	bitr4i 187	1 |- ( A C_ B <-> A. x ( x e. A -> x e. B ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   A.wal 1396   = wceq 1398   e. wcel 2203   {cab 2218   i^i cin 3209   C_ wss 3210

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-11 1555  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-in 3216  df-ss 3223

This theorem is referenced by:  dfss3  3226  dfss2f  3228  ssel  3231  ssriv  3241  ssrdv  3243  sstr2  3244  eqss  3252  nssr  3297  rabss2  3320  ssconb  3351  ssequn1  3388  unss  3392  ssin  3442  ssddif  3454  reldisj  3559  ssdif0im  3572  inssdif0im  3575  ssundifim  3592  sbcssg  3617  pwss  3687  snssOLD  3818  snssb  3826  snsssn  3864  ssuni  3935  unissb  3943  intss  3969  iunss  4031  dftr2  4209  axpweq  4283  axpow2  4288  ssextss  4335  ordunisuc2r  4635  setind  4660  zfregfr  4695  tfi  4703  ssrel  4837  ssrel2  4839  ssrelrel  4849  reliun  4872  relop  4904  issref  5144  funimass4  5726  isprm2  12810  bj-inf2vnlem3  16734  bj-inf2vnlem4  16735