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Theorem dfss3 3181
Description: Alternate definition of subclass relationship. (Contributed by NM, 14-Oct-1999.)
Assertion
Ref Expression
dfss3 (𝐴𝐵 ↔ ∀𝑥𝐴 𝑥𝐵)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem dfss3
StepHypRef Expression
1 ssalel 3180 . 2 (𝐴𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
2 df-ral 2488 . 2 (∀𝑥𝐴 𝑥𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
31, 2bitr4i 187 1 (𝐴𝐵 ↔ ∀𝑥𝐴 𝑥𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105  wal 1370  wcel 2175  wral 2483  wss 3165
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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-11 1528  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186
This theorem depends on definitions:  df-bi 117  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-ral 2488  df-in 3171  df-ss 3178
This theorem is referenced by:  ssrab  3270  eqsnm  3795  uni0b  3874  uni0c  3875  ssint  3900  ssiinf  3976  sspwuni  4011  dftr3  4145  tfis  4630  rninxp  5125  fnres  5391  eqfnfv3  5678  funimass3  5695  ffvresb  5742  tfrlemibxssdm  6412  tfr1onlembxssdm  6428  tfrcllembxssdm  6441  exmidontriimlem3  7334  suplocsr  7921  4sqlem19  12674  imasaddfnlemg  13088  isbasis2g  14459  tgval2  14465  eltg2b  14468  tgss2  14493  basgen2  14495  bastop1  14497  unicld  14530  neipsm  14568  ssidcn  14624  bdss  15733
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