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Theorem elpwpwel 4407
 Description: A class belongs to a double power class if and only if its union belongs to the power class. (Contributed by BJ, 22-Jan-2023.)
Assertion
Ref Expression
elpwpwel

Proof of Theorem elpwpwel
StepHypRef Expression
1 uniexb 4405 . . 3
21anbi1i 454 . 2
3 elpwpw 3909 . 2
4 elpwb 3527 . 2
52, 3, 43bitr4i 211 1
 Colors of variables: wff set class Syntax hints:   wa 103   wb 104   wcel 2112  cvv 2691   wss 3078  cpw 3517  cuni 3746 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-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-13 2114  ax-14 2115  ax-ext 2123  ax-sep 4056  ax-pow 4108  ax-un 4366 This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1732  df-clab 2128  df-cleq 2134  df-clel 2137  df-nfc 2272  df-ral 2423  df-rex 2424  df-v 2693  df-in 3084  df-ss 3091  df-pw 3519  df-uni 3747 This theorem is referenced by: (None)
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