Theorem bdeq0 16002

Description: Boundedness of the formula expressing that a setvar is equal to the empty class. (Contributed by BJ, 21-Nov-2019.)

Assertion

Ref	Expression
bdeq0	|- BOUNDED x = (/)

Proof of Theorem bdeq0

Step	Hyp	Ref	Expression
1		bdcnul 16000	. . 3 |- BOUNDED (/)
2	1	bdss 15999	. 2 |- BOUNDED x C_ (/)
3		0ss 3507	. . 3 |- (/) C_ x
4		eqss 3216	. . 3 |- ( x = (/) <-> ( x C_ (/) /\ (/) C_ x ) )
5	3, 4	mpbiran2 944	. 2 |- ( x = (/) <-> x C_ (/) )
6	2, 5	bd0r 15960	1 |- BOUNDED x = (/)

Syntax hints:   = wceq 1373   C_ wss 3174   (/)c0 3468  BOUNDED wbd 15947

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 2189  ax-bd0 15948  ax-bdim 15949  ax-bdn 15952  ax-bdal 15953  ax-bdeq 15955

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-v 2778  df-dif 3176  df-in 3180  df-ss 3187  df-nul 3469  df-bdc 15976

This theorem is referenced by:  bj-bd0el  16003  bj-nn0suc0  16085