Theorem bdeq0 16398

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 16396	. . 3 |- BOUNDED (/)
2	1	bdss 16395	. 2 |- BOUNDED x C_ (/)
3		0ss 3531	. . 3 |- (/) C_ x
4		eqss 3240	. . 3 |- ( x = (/) <-> ( x C_ (/) /\ (/) C_ x ) )
5	3, 4	mpbiran2 947	. 2 |- ( x = (/) <-> x C_ (/) )
6	2, 5	bd0r 16356	1 |- BOUNDED x = (/)

Syntax hints:   = wceq 1395   C_ wss 3198   (/)c0 3492  BOUNDED wbd 16343

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211  ax-bd0 16344  ax-bdim 16345  ax-bdn 16348  ax-bdal 16349  ax-bdeq 16351

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-v 2802  df-dif 3200  df-in 3204  df-ss 3211  df-nul 3493  df-bdc 16372

This theorem is referenced by:  bj-bd0el  16399  bj-nn0suc0  16481