Theorem enwomni 7412

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 6640 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 7411	. 2 |- ( A ~~ B -> ( A e. WOmni -> B e. WOmni ) )
2		ensym 6998	. . 3 |- ( A ~~ B -> B ~~ A )
3		enwomnilem 7411	. . 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 2202   class class class wbr 4093   ~~ cen 6950  WOmnicwomni 7405

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 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641

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 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-v 2805  df-sbc 3033  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-br 4094  df-opab 4156  df-id 4396  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1o 6625  df-2o 6626  df-er 6745  df-map 6862  df-en 6953  df-womni 7406

This theorem is referenced by:  redcwlpo  16768  nconstwlpo  16779