Theorem bdeq0 13065

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 13063	. . 3 |- BOUNDED (/)
2	1	bdss 13062	. 2 |- BOUNDED x C_ (/)
3		0ss 3401	. . 3 |- (/) C_ x
4		eqss 3112	. . 3 |- ( x = (/) <-> ( x C_ (/) /\ (/) C_ x ) )
5	3, 4	mpbiran2 925	. 2 |- ( x = (/) <-> x C_ (/) )
6	2, 5	bd0r 13023	1 |- BOUNDED x = (/)

Syntax hints:   = wceq 1331   C_ wss 3071   (/)c0 3363  BOUNDED wbd 13010

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-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-bd0 13011  ax-bdim 13012  ax-bdn 13015  ax-bdal 13016  ax-bdeq 13018

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-v 2688  df-dif 3073  df-in 3077  df-ss 3084  df-nul 3364  df-bdc 13039

This theorem is referenced by:  bj-bd0el  13066  bj-nn0suc0  13148