Theorem bdceqir 15336

Description: A class equal to a bounded one is bounded. Stated with a commuted (compared with bdceqi 15335) equality in the hypothesis, to work better with definitions (𝐵 is the definiendum that one wants to prove bounded; see comment of bd0r 15317). (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 2197	. 2 ⊢ 𝐴 = 𝐵
4	1, 3	bdceqi 15335	1 ⊢ BOUNDED 𝐵

Syntax hints:   = wceq 1364  BOUNDED wbdc 15332

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 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-4 1521  ax-17 1537  ax-ial 1545  ax-ext 2175  ax-bd0 15305

This theorem depends on definitions:  df-bi 117  df-cleq 2186  df-clel 2189  df-bdc 15333

This theorem is referenced by:  bdcrab  15344  bdccsb  15352  bdcdif  15353  bdcun  15354  bdcin  15355  bdcnulALT  15358  bdcpw  15361  bdcsn  15362  bdcpr  15363  bdctp  15364  bdcuni  15368  bdcint  15369  bdciun  15370  bdciin  15371  bdcsuc  15372  bdcriota  15375