Theorem bdceqir 16490

Description: A class equal to a bounded one is bounded. Stated with a commuted (compared with bdceqi 16489) equality in the hypothesis, to work better with definitions (𝐵 is the definiendum that one wants to prove bounded; see comment of bd0r 16471). (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 2235	. 2 ⊢ 𝐴 = 𝐵
4	1, 3	bdceqi 16489	1 ⊢ BOUNDED 𝐵

Syntax hints:   = wceq 1397  BOUNDED wbdc 16486

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 1495  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-4 1558  ax-17 1574  ax-ial 1582  ax-ext 2213  ax-bd0 16459

This theorem depends on definitions:  df-bi 117  df-cleq 2224  df-clel 2227  df-bdc 16487

This theorem is referenced by:  bdcrab  16498  bdccsb  16506  bdcdif  16507  bdcun  16508  bdcin  16509  bdcnulALT  16512  bdcpw  16515  bdcsn  16516  bdcpr  16517  bdctp  16518  bdcuni  16522  bdcint  16523  bdciun  16524  bdciin  16525  bdcsuc  16526  bdcriota  16529