Theorem bdeq0 15072

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 15070	. . 3 |- BOUNDED (/)
2	1	bdss 15069	. 2 |- BOUNDED x C_ (/)
3		0ss 3476	. . 3 |- (/) C_ x
4		eqss 3185	. . 3 |- ( x = (/) <-> ( x C_ (/) /\ (/) C_ x ) )
5	3, 4	mpbiran2 943	. 2 |- ( x = (/) <-> x C_ (/) )
6	2, 5	bd0r 15030	1 |- BOUNDED x = (/)

Syntax hints:   = wceq 1364   C_ wss 3144   (/)c0 3437  BOUNDED wbd 15017

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171  ax-bd0 15018  ax-bdim 15019  ax-bdn 15022  ax-bdal 15023  ax-bdeq 15025

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-v 2754  df-dif 3146  df-in 3150  df-ss 3157  df-nul 3438  df-bdc 15046

This theorem is referenced by:  bj-bd0el  15073  bj-nn0suc0  15155