Theorem bj-bd0el 15878

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 15877	. 2 ⊢ BOUNDED 𝑦 = ∅
2	1	bj-bdcel 15847	1 ⊢ BOUNDED ∅ ∈ 𝑥

Syntax hints:   ∈ wcel 2177  ∅c0 3461  BOUNDED wbd 15822

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188  ax-bd0 15823  ax-bdim 15824  ax-bdn 15827  ax-bdal 15828  ax-bdex 15829  ax-bdeq 15830

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-v 2775  df-dif 3169  df-in 3173  df-ss 3180  df-nul 3462  df-bdc 15851

This theorem is referenced by:  bj-d0clsepcl  15935  bj-bdind  15940