Theorem bdceqir 15574

Description: A class equal to a bounded one is bounded. Stated with a commuted (compared with bdceqi 15573) equality in the hypothesis, to work better with definitions ( B is the definiendum that one wants to prove bounded; see comment of bd0r 15555). (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 2200	. 2 |- A = B
4	1, 3	bdceqi 15573	1 |- BOUNDED B

Syntax hints:   = wceq 1364  BOUNDED wbdc 15570

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 1461  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-4 1524  ax-17 1540  ax-ial 1548  ax-ext 2178  ax-bd0 15543

This theorem depends on definitions:  df-bi 117  df-cleq 2189  df-clel 2192  df-bdc 15571

This theorem is referenced by:  bdcrab  15582  bdccsb  15590  bdcdif  15591  bdcun  15592  bdcin  15593  bdcnulALT  15596  bdcpw  15599  bdcsn  15600  bdcpr  15601  bdctp  15602  bdcuni  15606  bdcint  15607  bdciun  15608  bdciin  15609  bdcsuc  15610  bdcriota  15613