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Mirrors > Home > ILE Home > Th. List > Mathboxes > bdceqir | GIF version |
Description: A class equal to a bounded one is bounded. Stated with a commuted (compared with bdceqi 13030) equality in the hypothesis, to work better with definitions (𝐵 is the definiendum that one wants to prove bounded; see comment of bd0r 13012). (Contributed by BJ, 3-Oct-2019.) |
Ref | Expression |
---|---|
bdceqir.min | ⊢ BOUNDED 𝐴 |
bdceqir.maj | ⊢ 𝐵 = 𝐴 |
Ref | Expression |
---|---|
bdceqir | ⊢ BOUNDED 𝐵 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bdceqir.min | . 2 ⊢ BOUNDED 𝐴 | |
2 | bdceqir.maj | . . 3 ⊢ 𝐵 = 𝐴 | |
3 | 2 | eqcomi 2141 | . 2 ⊢ 𝐴 = 𝐵 |
4 | 1, 3 | bdceqi 13030 | 1 ⊢ BOUNDED 𝐵 |
Colors of variables: wff set class |
Syntax hints: = wceq 1331 BOUNDED wbdc 13027 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1423 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-4 1487 ax-17 1506 ax-ial 1514 ax-ext 2119 ax-bd0 13000 |
This theorem depends on definitions: df-bi 116 df-cleq 2130 df-clel 2133 df-bdc 13028 |
This theorem is referenced by: bdcrab 13039 bdccsb 13047 bdcdif 13048 bdcun 13049 bdcin 13050 bdcnulALT 13053 bdcpw 13056 bdcsn 13057 bdcpr 13058 bdctp 13059 bdcuni 13063 bdcint 13064 bdciun 13065 bdciin 13066 bdcsuc 13067 bdcriota 13070 |
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