Theorem bj-bd0el 9988

Description: Boundedness of the formula "the empty set belongs to the setvar 𝑥". (Contributed by BJ, 30-Nov-2019.)

Assertion

Ref	Expression
bj-bd0el	⊢ BOUNDED ∅ ∈ 𝑥

Proof of Theorem bj-bd0el

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bdeq0 9987	. 2 ⊢ BOUNDED 𝑦 = ∅
2	1	bj-bdcel 9957	1 ⊢ BOUNDED ∅ ∈ 𝑥

Syntax hints:   ∈ wcel 1393  ∅c0 3224  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-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-bd0 9933  ax-bdim 9934  ax-bdn 9937  ax-bdal 9938  ax-bdex 9939  ax-bdeq 9940

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-dif 2920  df-in 2924  df-ss 2931  df-nul 3225  df-bdc 9961

This theorem is referenced by:  bj-d0clsepcl  10049  bj-bdind  10054