Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  enwomni Unicode version

Theorem enwomni 7096
 Description: Weak omniscience is invariant with respect to equinumerosity. For example, this means that we can express the Weak Limited Principle of Omniscience as either WOmni or WOmni. The former is a better match to conventional notation in the sense that df2o3 6371 says that whereas the corresponding relationship does not exist between and . (Contributed by Jim Kingdon, 20-Jun-2024.)
Assertion
Ref Expression
enwomni WOmni WOmni

Proof of Theorem enwomni
StepHypRef Expression
1 enwomnilem 7095 . 2 WOmni WOmni
2 ensym 6719 . . 3
3 enwomnilem 7095 . . 3 WOmni WOmni
42, 3syl 14 . 2 WOmni WOmni
51, 4impbid 128 1 WOmni WOmni
 Colors of variables: wff set class Syntax hints:   wi 4   wb 104   wcel 2128   class class class wbr 3965   cen 6676  WOmnicwomni 7089 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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-nul 4090  ax-pow 4134  ax-pr 4168  ax-un 4392  ax-setind 4494 This theorem depends on definitions:  df-bi 116  df-dc 821  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-v 2714  df-sbc 2938  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-br 3966  df-opab 4026  df-id 4252  df-suc 4330  df-iom 4548  df-xp 4589  df-rel 4590  df-cnv 4591  df-co 4592  df-dm 4593  df-rn 4594  df-res 4595  df-ima 4596  df-iota 5132  df-fun 5169  df-fn 5170  df-f 5171  df-f1 5172  df-fo 5173  df-f1o 5174  df-fv 5175  df-ov 5821  df-oprab 5822  df-mpo 5823  df-1o 6357  df-2o 6358  df-er 6473  df-map 6588  df-en 6679  df-womni 7090 This theorem is referenced by:  redcwlpo  13588  nconstwlpo  13598
 Copyright terms: Public domain W3C validator