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Theorem bdsnss 15063
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 15036 . 2 BOUNDED 𝑥𝐴
3 vex 2755 . . 3 𝑥 ∈ V
43snss 3742 . 2 (𝑥𝐴 ↔ {𝑥} ⊆ 𝐴)
52, 4bd0 15014 1 BOUNDED {𝑥} ⊆ 𝐴
Colors of variables: wff set class
Syntax hints:  wcel 2160  wss 3144  {csn 3607  BOUNDED wbd 15002  BOUNDED wbdc 15030
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-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171  ax-bd0 15003
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-in 3150  df-ss 3157  df-sn 3613  df-bdc 15031
This theorem is referenced by:  bdvsn  15064  bdeqsuc  15071
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