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Theorem bdsnss 13435
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 13408 . 2 BOUNDED 𝑥𝐴
3 vex 2715 . . 3 𝑥 ∈ V
43snss 3685 . 2 (𝑥𝐴 ↔ {𝑥} ⊆ 𝐴)
52, 4bd0 13386 1 BOUNDED {𝑥} ⊆ 𝐴
Colors of variables: wff set class
Syntax hints:  wcel 2128  wss 3102  {csn 3560  BOUNDED wbd 13374  BOUNDED wbdc 13402
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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139  ax-bd0 13375
This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-in 3108  df-ss 3115  df-sn 3566  df-bdc 13403
This theorem is referenced by:  bdvsn  13436  bdeqsuc  13443
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