Theorem enomni 7094

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 6389 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 7093	. 2 |- ( A ~~ B -> ( A e. Omni -> B e. Omni ) )
2		ensym 6738	. . 3 |- ( A ~~ B -> B ~~ A )
3		enomnilem 7093	. . 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 128	1 |- ( A ~~ B -> ( A e. Omni <-> B e. Omni ) )

Syntax hints:   -> wi 4   <-> wb 104   e. wcel 2135   class class class wbr 3976   ~~ cen 6695  Omnicomni 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 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-13 2137  ax-14 2138  ax-ext 2146  ax-sep 4094  ax-nul 4102  ax-pow 4147  ax-pr 4181  ax-un 4405  ax-setind 4508

This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-fal 1348  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ne 2335  df-ral 2447  df-rex 2448  df-v 2723  df-sbc 2947  df-dif 3113  df-un 3115  df-in 3117  df-ss 3124  df-nul 3405  df-pw 3555  df-sn 3576  df-pr 3577  df-op 3579  df-uni 3784  df-int 3819  df-br 3977  df-opab 4038  df-id 4265  df-suc 4343  df-iom 4562  df-xp 4604  df-rel 4605  df-cnv 4606  df-co 4607  df-dm 4608  df-rn 4609  df-res 4610  df-ima 4611  df-iota 5147  df-fun 5184  df-fn 5185  df-f 5186  df-f1 5187  df-fo 5188  df-f1o 5189  df-fv 5190  df-ov 5839  df-oprab 5840  df-mpo 5841  df-1o 6375  df-2o 6376  df-er 6492  df-map 6607  df-en 6698  df-omni 7090

This theorem is referenced by:  exmidunben  12296  trilpo  13756