Theorem bdceqir 16165

Description: A class equal to a bounded one is bounded. Stated with a commuted (compared with bdceqi 16164) equality in the hypothesis, to work better with definitions (𝐵 is the definiendum that one wants to prove bounded; see comment of bd0r 16146). (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 16164	1 ⊢ BOUNDED 𝐵

Syntax hints:   = wceq 1395  BOUNDED wbdc 16161

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 16134

This theorem depends on definitions:  df-bi 117  df-cleq 2222  df-clel 2225  df-bdc 16162

This theorem is referenced by:  bdcrab  16173  bdccsb  16181  bdcdif  16182  bdcun  16183  bdcin  16184  bdcnulALT  16187  bdcpw  16190  bdcsn  16191  bdcpr  16192  bdctp  16193  bdcuni  16197  bdcint  16198  bdciun  16199  bdciin  16200  bdcsuc  16201  bdcriota  16204