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Theorem dfss2 2907
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
dfss2 (ABx(x Ax B))
Distinct variable groups:   x,A   x,B

Proof of Theorem dfss2
StepHypRef Expression
1 dfss 2905 . . 3 (ABA = (AB))
2 df-in 2897 . . . 4 (AB) = {x ∣ (x A x B)}
32eqeq2i 2028 . . 3 (A = (AB) ↔ A = {x ∣ (x A x B)})
4 abeq2 2124 . . 3 (A = {x ∣ (x A x B)} ↔ x(x A ↔ (x A x B)))
51, 3, 43bitri 195 . 2 (ABx(x A ↔ (x A x B)))
6 pm4.71 369 . . 3 ((x Ax B) ↔ (x A ↔ (x A x B)))
76albii 1335 . 2 (x(x Ax B) ↔ x(x A ↔ (x A x B)))
85, 7bitr4i 176 1 (ABx(x Ax B))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98  wal 1224   = wceq 1226   wcel 1370  {cab 2004  cin 2889  wss 2890
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-11 1374  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000
This theorem depends on definitions:  df-bi 110  df-nf 1326  df-sb 1624  df-clab 2005  df-cleq 2011  df-clel 2014  df-in 2897  df-ss 2904
This theorem is referenced by:  dfss3  2908  dfss2f  2909  ssel  2912  ssriv  2922  ssrdv  2924  sstr2  2925  eqss  2933  nssr  2976  rabss2  2996  ssconb  3049  ssequn1  3086  unss  3090  ssin  3132  ssddif  3144  reldisj  3244  ssdif0im  3259  inssdif0im  3264  ssundifim  3279  sbcssg  3305  pwss  3345  snss  3464  snsssn  3502  ssuni  3572  unissb  3580  intss  3606  iunss  3668  dftr2  3826  axpweq  3894  axpow2  3899  ssextss  3926  ordunisuc2r  4185  setind  4202  tfi  4228  ssrel  4351  ssrel2  4353  ssrelrel  4363  reliun  4381  relop  4409  issref  4630  funimass4  5145  bj-inf2vnlem3  7329  bj-inf2vnlem4  7330
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