Theorem enwomni 7475

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 6676 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 7474	. 2 |- ( A ~~ B -> ( A e. WOmni -> B e. WOmni ) )
2		ensym 7035	. . 3 |- ( A ~~ B -> B ~~ A )
3		enwomnilem 7474	. . 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 2205   class class class wbr 4115   ~~ cen 6987  WOmnicwomni 7468

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 2207  ax-14 2208  ax-ext 2216  ax-sep 4234  ax-nul 4242  ax-pow 4293  ax-pr 4328  ax-un 4560  ax-setind 4665

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-v 2817  df-sbc 3046  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3677  df-sn 3701  df-pr 3702  df-op 3704  df-uni 3921  df-int 3956  df-br 4116  df-opab 4178  df-id 4420  df-suc 4498  df-iom 4719  df-xp 4761  df-rel 4762  df-cnv 4763  df-co 4764  df-dm 4765  df-rn 4766  df-res 4767  df-ima 4768  df-iota 5318  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 6062  df-oprab 6063  df-mpo 6064  df-1o 6661  df-2o 6662  df-er 6781  df-map 6898  df-en 6990  df-womni 7469

This theorem is referenced by:  redcwlpo  16981  nconstwlpo  16992