Theorem bdceqir 16439

Description: A class equal to a bounded one is bounded. Stated with a commuted (compared with bdceqi 16438) equality in the hypothesis, to work better with definitions (𝐵 is the definiendum that one wants to prove bounded; see comment of bd0r 16420). (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 16438	1 ⊢ BOUNDED 𝐵

Syntax hints:   = wceq 1397  BOUNDED wbdc 16435

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 16408

This theorem depends on definitions:  df-bi 117  df-cleq 2224  df-clel 2227  df-bdc 16436

This theorem is referenced by:  bdcrab  16447  bdccsb  16455  bdcdif  16456  bdcun  16457  bdcin  16458  bdcnulALT  16461  bdcpw  16464  bdcsn  16465  bdcpr  16466  bdctp  16467  bdcuni  16471  bdcint  16472  bdciun  16473  bdciin  16474  bdcsuc  16475  bdcriota  16478