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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 4540 | . 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 1528 𝒫 cpw 4537 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2800 df-cleq 2814 df-clel 2893 df-in 3942 df-ss 3951 df-pw 4539 |
This theorem is referenced by: pwfi 8808 rankxplim 9297 pwdju1 9605 fin23lem17 9749 mnfnre 10673 qtopres 22236 hmphdis 22334 ust0 22757 umgrpredgv 26853 issubgr 26981 uhgrissubgr 26985 cusgredg 27134 cffldtocusgr 27157 konigsbergiedgw 27955 shsspwh 28951 circtopn 31001 lfuhgr 32262 rankeq1o 33530 onsucsuccmpi 33689 elrfi 39171 islmodfg 39549 clsk1indlem4 40274 clsk1indlem1 40275 clsk1independent 40276 omef 42659 caragensplit 42663 caragenelss 42664 carageneld 42665 omeunile 42668 caragensspw 42672 0ome 42692 isomennd 42694 ovn02 42731 lcoop 44364 lincvalsc0 44374 linc0scn0 44376 lincdifsn 44377 linc1 44378 lspsslco 44390 lincresunit3lem2 44433 lincresunit3 44434 |
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