Theorem bdceqir 16290

Description: A class equal to a bounded one is bounded. Stated with a commuted (compared with bdceqi 16289) equality in the hypothesis, to work better with definitions ( B is the definiendum that one wants to prove bounded; see comment of bd0r 16271). (Contributed by BJ, 3-Oct-2019.)

Hypotheses

Ref	Expression
bdceqir.min	|- BOUNDED A
bdceqir.maj	|- B = A

Assertion

Ref	Expression
bdceqir	|- BOUNDED B

Proof of Theorem bdceqir

Step	Hyp	Ref	Expression
1		bdceqir.min	. 2 |- BOUNDED A
2		bdceqir.maj	. . 3 |- B = A
3	2	eqcomi 2233	. 2 |- A = B
4	1, 3	bdceqi 16289	1 |- BOUNDED B

Syntax hints:   = wceq 1395  BOUNDED wbdc 16286

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-5 1493  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-4 1556  ax-17 1572  ax-ial 1580  ax-ext 2211  ax-bd0 16259

This theorem depends on definitions:  df-bi 117  df-cleq 2222  df-clel 2225  df-bdc 16287

This theorem is referenced by:  bdcrab  16298  bdccsb  16306  bdcdif  16307  bdcun  16308  bdcin  16309  bdcnulALT  16312  bdcpw  16315  bdcsn  16316  bdcpr  16317  bdctp  16318  bdcuni  16322  bdcint  16323  bdciun  16324  bdciin  16325  bdcsuc  16326  bdcriota  16329