Theorem bdeq0 16462

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 16460	. . 3 |- BOUNDED (/)
2	1	bdss 16459	. 2 |- BOUNDED x C_ (/)
3		0ss 3533	. . 3 |- (/) C_ x
4		eqss 3242	. . 3 |- ( x = (/) <-> ( x C_ (/) /\ (/) C_ x ) )
5	3, 4	mpbiran2 949	. 2 |- ( x = (/) <-> x C_ (/) )
6	2, 5	bd0r 16420	1 |- BOUNDED x = (/)

Syntax hints:   = wceq 1397   C_ wss 3200   (/)c0 3494  BOUNDED wbd 16407

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213  ax-bd0 16408  ax-bdim 16409  ax-bdn 16412  ax-bdal 16413  ax-bdeq 16415

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-v 2804  df-dif 3202  df-in 3206  df-ss 3213  df-nul 3495  df-bdc 16436

This theorem is referenced by:  bj-bd0el  16463  bj-nn0suc0  16545