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| Mirrors > Home > MPE Home > Th. List > pweqi | Structured version Visualization version GIF version | ||
| Description: Equality inference for power class. (Contributed by NM, 27-Nov-2013.) | 
| Ref | Expression | 
|---|---|
| pweqi.1 | ⊢ 𝐴 = 𝐵 | 
| Ref | Expression | 
|---|---|
| pweqi | ⊢ 𝒫 𝐴 = 𝒫 𝐵 | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | pweqi.1 | . 2 ⊢ 𝐴 = 𝐵 | |
| 2 | pweq 4613 | . 2 ⊢ (𝐴 = 𝐵 → 𝒫 𝐴 = 𝒫 𝐵) | |
| 3 | 1, 2 | ax-mp 5 | 1 ⊢ 𝒫 𝐴 = 𝒫 𝐵 | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1539 𝒫 cpw 4599 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1542 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-v 3481 df-ss 3967 df-pw 4601 | 
| This theorem is referenced by: rankxplim 9920 pwdju1 10232 fin23lem17 10379 mnfnre 11305 qtopres 23707 hmphdis 23805 ust0 24229 made0 27913 umgrpredgv 29158 issubgr 29289 uhgrissubgr 29293 cusgredg 29442 cffldtocusgr 29465 cffldtocusgrOLD 29466 konigsbergiedgw 30268 shsspwh 31266 circtopn 33837 lfuhgr 35124 rankeq1o 36173 onsucsuccmpi 36445 bj-unirel 37053 elrfi 42710 islmodfg 43086 clsk1indlem4 44062 clsk1indlem1 44063 clsk1independent 44064 omef 46516 caragensplit 46520 caragenelss 46521 carageneld 46522 omeunile 46525 caragensspw 46529 0ome 46549 isomennd 46551 ovn02 46588 uspgrimprop 47878 isuspgrimlem 47879 grtri 47912 usgrexmpl1lem 47985 usgrexmpl2lem 47990 lcoop 48333 lincvalsc0 48343 linc0scn0 48345 lincdifsn 48346 linc1 48347 lspsslco 48359 lincresunit3lem2 48402 lincresunit3 48403 | 
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