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 om e. WOmni or NN0 e. WOmni. The former is a better match to conventional notation in the sense that df2o3 6371 says that 2o = { (/) , 1o } whereas the corresponding relationship does not exist between 2 and { 0 , 1 }. (Contributed by Jim Kingdon, 20-Jun-2024.)

Assertion

Ref	Expression
enwomni	|- ( A ~~ B -> ( A e. WOmni <-> B e. WOmni ) )

Proof of Theorem enwomni

Step	Hyp	Ref	Expression
1		enwomnilem 7095	. 2 |- ( A ~~ B -> ( A e. WOmni -> B e. WOmni ) )
2		ensym 6719	. . 3 |- ( A ~~ B -> B ~~ A )
3		enwomnilem 7095	. . 3 |- ( B ~~ A -> ( B e. WOmni -> A e. WOmni ) )
4	2, 3	syl 14	. 2 |- ( A ~~ B -> ( B e. WOmni -> A e. WOmni ) )
5	1, 4	impbid 128	1 |- ( A ~~ B -> ( A e. WOmni <-> B e. WOmni ) )

Syntax hints:   -> wi 4   <-> wb 104   e. 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