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Theorem bj-bdcel 11166
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 11148 . 2 BOUNDED𝑦𝑥 𝑦 = 𝐴
3 risset 2402 . 2 (𝐴𝑥 ↔ ∃𝑦𝑥 𝑦 = 𝐴)
42, 3bd0r 11154 1 BOUNDED 𝐴𝑥
Colors of variables: wff set class
Syntax hints:   = wceq 1287  wcel 1436  wrex 2356  BOUNDED wbd 11141
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1379  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-4 1443  ax-ial 1470  ax-bd0 11142  ax-bdex 11148
This theorem depends on definitions:  df-bi 115  df-clel 2081  df-rex 2361
This theorem is referenced by:  bj-bd0el  11197  bj-bdsucel  11211
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