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Theorem bj-bdcel 9957
Description: Boundedness of a membership formula. (Contributed by BJ, 8-Dec-2019.)
Hypothesis
Ref Expression
bj-bdcel.bd BOUNDED 𝑦 = 𝐴
Assertion
Ref Expression
bj-bdcel BOUNDED 𝐴𝑥
Distinct variable groups:   𝑥,𝑦   𝑦,𝐴
Allowed substitution hint:   𝐴(𝑥)

Proof of Theorem bj-bdcel
StepHypRef Expression
1 bj-bdcel.bd . . 3 BOUNDED 𝑦 = 𝐴
21ax-bdex 9939 . 2 BOUNDED𝑦𝑥 𝑦 = 𝐴
3 risset 2352 . 2 (𝐴𝑥 ↔ ∃𝑦𝑥 𝑦 = 𝐴)
42, 3bd0r 9945 1 BOUNDED 𝐴𝑥
Colors of variables: wff set class
Syntax hints:   = wceq 1243  wcel 1393  wrex 2307  BOUNDED wbd 9932
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-ial 1427  ax-bd0 9933  ax-bdex 9939
This theorem depends on definitions:  df-bi 110  df-clel 2036  df-rex 2312
This theorem is referenced by:  bj-bd0el  9988  bj-bdsucel  10002
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