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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 14531) equality in the hypothesis, to work better with definitions (𝐵 is the definiendum that one wants to prove bounded; see comment of bd0r 14513). (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 2181 | . 2 ⊢ 𝐴 = 𝐵 |
4 | 1, 3 | bdceqi 14531 | 1 ⊢ BOUNDED 𝐵 |
Colors of variables: wff set class |
Syntax hints: = wceq 1353 BOUNDED wbdc 14528 |
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 1447 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-4 1510 ax-17 1526 ax-ial 1534 ax-ext 2159 ax-bd0 14501 |
This theorem depends on definitions: df-bi 117 df-cleq 2170 df-clel 2173 df-bdc 14529 |
This theorem is referenced by: bdcrab 14540 bdccsb 14548 bdcdif 14549 bdcun 14550 bdcin 14551 bdcnulALT 14554 bdcpw 14557 bdcsn 14558 bdcpr 14559 bdctp 14560 bdcuni 14564 bdcint 14565 bdciun 14566 bdciin 14567 bdcsuc 14568 bdcriota 14571 |
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