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Theorem bdsnss 15486
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 15459 . 2 BOUNDED 𝑥𝐴
3 vex 2766 . . 3 𝑥 ∈ V
43snss 3757 . 2 (𝑥𝐴 ↔ {𝑥} ⊆ 𝐴)
52, 4bd0 15437 1 BOUNDED {𝑥} ⊆ 𝐴
Colors of variables: wff set class
Syntax hints:  wcel 2167  wss 3157  {csn 3622  BOUNDED wbd 15425  BOUNDED wbdc 15453
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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178  ax-bd0 15426
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-in 3163  df-ss 3170  df-sn 3628  df-bdc 15454
This theorem is referenced by:  bdvsn  15487  bdeqsuc  15494
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