Theorem bj-bdcel 13024

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 𝐴 ∈ 𝑥

Distinct variable groups:   𝑥,𝑦   𝑦,𝐴

Allowed substitution hint:   𝐴(𝑥)

Proof of Theorem bj-bdcel

Step	Hyp	Ref	Expression
1		bj-bdcel.bd	. . 3 ⊢ BOUNDED 𝑦 = 𝐴
2	1	ax-bdex 13006	. 2 ⊢ BOUNDED ∃𝑦 ∈ 𝑥 𝑦 = 𝐴
3		risset 2461	. 2 ⊢ (𝐴 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝑥 𝑦 = 𝐴)
4	2, 3	bd0r 13012	1 ⊢ BOUNDED 𝐴 ∈ 𝑥

Syntax hints:   = wceq 1331   ∈ wcel 1480  ∃wrex 2415  BOUNDED wbd 12999

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-5 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-ial 1514  ax-bd0 13000  ax-bdex 13006

This theorem depends on definitions:  df-bi 116  df-clel 2133  df-rex 2420

This theorem is referenced by:  bj-bd0el  13055  bj-bdsucel  13069