Theorem bdeq0 15803

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 15801	. . 3 |- BOUNDED (/)
2	1	bdss 15800	. 2 |- BOUNDED x C_ (/)
3		0ss 3499	. . 3 |- (/) C_ x
4		eqss 3208	. . 3 |- ( x = (/) <-> ( x C_ (/) /\ (/) C_ x ) )
5	3, 4	mpbiran2 944	. 2 |- ( x = (/) <-> x C_ (/) )
6	2, 5	bd0r 15761	1 |- BOUNDED x = (/)

Syntax hints:   = wceq 1373   C_ wss 3166   (/)c0 3460  BOUNDED wbd 15748

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187  ax-bd0 15749  ax-bdim 15750  ax-bdn 15753  ax-bdal 15754  ax-bdeq 15756

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-v 2774  df-dif 3168  df-in 3172  df-ss 3179  df-nul 3461  df-bdc 15777

This theorem is referenced by:  bj-bd0el  15804  bj-nn0suc0  15886