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Theorem enomni 7015
Description: Omniscience is invariant with respect to equinumerosity. For example, this means that we can express the Limited Principle of Omniscience as either  om  e. Omni or  NN0  e. Omni. The former is a better match to conventional notation in the sense that df2o3 6331 says that  2o  =  { (/)
,  1o } whereas the corresponding relationship does not exist between  2 and  { 0 ,  1 }. (Contributed by Jim Kingdon, 13-Jul-2022.)
Assertion
Ref Expression
enomni  |-  ( A 
~~  B  ->  ( A  e. Omni  <->  B  e. Omni ) )

Proof of Theorem enomni
StepHypRef Expression
1 enomnilem 7014 . 2  |-  ( A 
~~  B  ->  ( A  e. Omni  ->  B  e. Omni
) )
2 ensym 6679 . . 3  |-  ( A 
~~  B  ->  B  ~~  A )
3 enomnilem 7014 . . 3  |-  ( B 
~~  A  ->  ( B  e. Omni  ->  A  e. Omni
) )
42, 3syl 14 . 2  |-  ( A 
~~  B  ->  ( B  e. Omni  ->  A  e. Omni
) )
51, 4impbid 128 1  |-  ( A 
~~  B  ->  ( A  e. Omni  <->  B  e. Omni ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    e. wcel 1481   class class class wbr 3933    ~~ cen 6636  Omnicomni 7008
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-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4050  ax-nul 4058  ax-pow 4102  ax-pr 4135  ax-un 4359  ax-setind 4456
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-v 2689  df-sbc 2911  df-dif 3074  df-un 3076  df-in 3078  df-ss 3085  df-nul 3365  df-pw 3513  df-sn 3534  df-pr 3535  df-op 3537  df-uni 3741  df-int 3776  df-br 3934  df-opab 3994  df-id 4219  df-suc 4297  df-iom 4509  df-xp 4549  df-rel 4550  df-cnv 4551  df-co 4552  df-dm 4553  df-rn 4554  df-res 4555  df-ima 4556  df-iota 5092  df-fun 5129  df-fn 5130  df-f 5131  df-f1 5132  df-fo 5133  df-f1o 5134  df-fv 5135  df-ov 5781  df-oprab 5782  df-mpo 5783  df-1o 6317  df-2o 6318  df-er 6433  df-map 6548  df-en 6639  df-omni 7010
This theorem is referenced by:  exmidunben  11966  trilpo  13394
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