Theorem enomni 7134

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 6428 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

Step	Hyp	Ref	Expression
1		enomnilem 7133	. 2 |- ( A ~~ B -> ( A e. Omni -> B e. Omni ) )
2		ensym 6778	. . 3 |- ( A ~~ B -> B ~~ A )
3		enomnilem 7133	. . 3 |- ( B ~~ A -> ( B e. Omni -> A e. Omni ) )
4	2, 3	syl 14	. 2 |- ( A ~~ B -> ( B e. Omni -> A e. Omni ) )
5	1, 4	impbid 129	1 |- ( A ~~ B -> ( A e. Omni <-> B e. Omni ) )

Syntax hints:   -> wi 4   <-> wb 105   e. wcel 2148   class class class wbr 4002   ~~ cen 6735  Omnicomni 7129

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4120  ax-nul 4128  ax-pow 4173  ax-pr 4208  ax-un 4432  ax-setind 4535

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-br 4003  df-opab 4064  df-id 4292  df-suc 4370  df-iom 4589  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-rn 4636  df-res 4637  df-ima 4638  df-iota 5177  df-fun 5217  df-fn 5218  df-f 5219  df-f1 5220  df-fo 5221  df-f1o 5222  df-fv 5223  df-ov 5875  df-oprab 5876  df-mpo 5877  df-1o 6414  df-2o 6415  df-er 6532  df-map 6647  df-en 6738  df-omni 7130

This theorem is referenced by:  exmidunben  12419  trilpo  14651