Theorem enomni 7337

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 6596 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 7336	. 2 |- ( A ~~ B -> ( A e. Omni -> B e. Omni ) )
2		ensym 6954	. . 3 |- ( A ~~ B -> B ~~ A )
3		enomnilem 7336	. . 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 2202   class class class wbr 4088   ~~ cen 6906  Omnicomni 7332

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-v 2804  df-sbc 3032  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-id 4390  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1o 6581  df-2o 6582  df-er 6701  df-map 6818  df-en 6909  df-omni 7333

This theorem is referenced by:  exmidunben  13046  nnnninfen  16623  trilpo  16647