Mathbox for BJ < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >   Mathboxes  >  bdsnss GIF version

Theorem bdsnss 12882
 Description: Inclusion of a singleton of a setvar in a bounded class is a bounded formula. (Contributed by BJ, 16-Oct-2019.)
Hypothesis
Ref Expression
bdsnss.1 BOUNDED 𝐴
Assertion
Ref Expression
bdsnss BOUNDED {𝑥} ⊆ 𝐴
Distinct variable group:   𝑥,𝐴

Proof of Theorem bdsnss
StepHypRef Expression
1 bdsnss.1 . . 3 BOUNDED 𝐴
21bdeli 12855 . 2 BOUNDED 𝑥𝐴
3 vex 2661 . . 3 𝑥 ∈ V
43snss 3617 . 2 (𝑥𝐴 ↔ {𝑥} ⊆ 𝐴)
52, 4bd0 12833 1 BOUNDED {𝑥} ⊆ 𝐴
 Colors of variables: wff set class Syntax hints:   ∈ wcel 1463   ⊆ wss 3039  {csn 3495  BOUNDED wbd 12821  BOUNDED wbdc 12849 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-bd0 12822 This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660  df-in 3045  df-ss 3052  df-sn 3501  df-bdc 12850 This theorem is referenced by:  bdvsn  12883  bdeqsuc  12890
 Copyright terms: Public domain W3C validator