Theorem enwomni 7325

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 6566 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 7324	. 2 |- ( A ~~ B -> ( A e. WOmni -> B e. WOmni ) )
2		ensym 6923	. . 3 |- ( A ~~ B -> B ~~ A )
3		enwomnilem 7324	. . 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 129	1 |- ( A ~~ B -> ( A e. WOmni <-> B e. WOmni ) )

Syntax hints:   -> wi 4   <-> wb 105   e. wcel 2200   class class class wbr 4082   ~~ cen 6875  WOmnicwomni 7318

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4521  ax-setind 4626

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-br 4083  df-opab 4145  df-id 4381  df-suc 4459  df-iom 4680  df-xp 4722  df-rel 4723  df-cnv 4724  df-co 4725  df-dm 4726  df-rn 4727  df-res 4728  df-ima 4729  df-iota 5274  df-fun 5316  df-fn 5317  df-f 5318  df-f1 5319  df-fo 5320  df-f1o 5321  df-fv 5322  df-ov 5997  df-oprab 5998  df-mpo 5999  df-1o 6552  df-2o 6553  df-er 6670  df-map 6787  df-en 6878  df-womni 7319

This theorem is referenced by:  redcwlpo  16354  nconstwlpo  16365