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Theorem dfss3 3051
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 dfss2 3050 . 2 (𝐴𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
2 df-ral 2393 . 2 (∀𝑥𝐴 𝑥𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
31, 2bitr4i 186 1 (𝐴𝐵 ↔ ∀𝑥𝐴 𝑥𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  wal 1310  wcel 1461  wral 2388  wss 3035
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-11 1465  ax-4 1468  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095
This theorem depends on definitions:  df-bi 116  df-nf 1418  df-sb 1717  df-clab 2100  df-cleq 2106  df-clel 2109  df-ral 2393  df-in 3041  df-ss 3048
This theorem is referenced by:  ssrab  3139  eqsnm  3646  uni0b  3725  uni0c  3726  ssint  3751  ssiinf  3826  sspwuni  3861  dftr3  3988  tfis  4455  rninxp  4938  fnres  5195  eqfnfv3  5472  funimass3  5488  ffvresb  5535  tfrlemibxssdm  6176  tfr1onlembxssdm  6192  tfrcllembxssdm  6205  isbasis2g  12049  tgval2  12057  eltg2b  12060  tgss2  12085  basgen2  12087  bastop1  12089  unicld  12122  neipsm  12160  ssidcn  12215  bdss  12745
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