Mathbox for Jonathan Ben-Naim < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj21 Structured version   Visualization version   GIF version

Theorem bnj21 30037
 Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypothesis
Ref Expression
bnj21.1 𝐵 = {𝑥𝐴𝜑}
Assertion
Ref Expression
bnj21 𝐵𝐴
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)

Proof of Theorem bnj21
StepHypRef Expression
1 bnj21.1 . 2 𝐵 = {𝑥𝐴𝜑}
2 ssrab2 3650 . 2 {𝑥𝐴𝜑} ⊆ 𝐴
31, 2eqsstri 3598 1 𝐵𝐴
 Colors of variables: wff setvar class Syntax hints:   = wceq 1475  {crab 2900   ⊆ wss 3540 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905  df-in 3547  df-ss 3554 This theorem is referenced by:  bnj1212  30124  bnj213  30206  bnj1286  30341  bnj1312  30380  bnj1523  30393
 Copyright terms: Public domain W3C validator