Theorem enomni 7430

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 6662 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 7429	. 2 |- ( A ~~ B -> ( A e. Omni -> B e. Omni ) )
2		ensym 7021	. . 3 |- ( A ~~ B -> B ~~ A )
3		enomnilem 7429	. . 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 2203   class class class wbr 4109   ~~ cen 6973  Omnicomni 7425

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-v 2815  df-sbc 3043  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-br 4110  df-opab 4172  df-id 4414  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1o 6647  df-2o 6648  df-er 6767  df-map 6884  df-en 6976  df-omni 7426

This theorem is referenced by:  exmidunben  13177  nnnninfen  16799  trilpo  16827