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Theorem bdsnss 13908
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 13881 . 2 BOUNDED 𝑥𝐴
3 vex 2733 . . 3 𝑥 ∈ V
43snss 3709 . 2 (𝑥𝐴 ↔ {𝑥} ⊆ 𝐴)
52, 4bd0 13859 1 BOUNDED {𝑥} ⊆ 𝐴
Colors of variables: wff set class
Syntax hints:  wcel 2141  wss 3121  {csn 3583  BOUNDED wbd 13847  BOUNDED wbdc 13875
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152  ax-bd0 13848
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-in 3127  df-ss 3134  df-sn 3589  df-bdc 13876
This theorem is referenced by:  bdvsn  13909  bdeqsuc  13916
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