Theorem enwomni 7284

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 6526 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 7283	. 2 |- ( A ~~ B -> ( A e. WOmni -> B e. WOmni ) )
2		ensym 6883	. . 3 |- ( A ~~ B -> B ~~ A )
3		enwomnilem 7283	. . 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 2177   class class class wbr 4048   ~~ cen 6835  WOmnicwomni 7277

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4167  ax-nul 4175  ax-pow 4223  ax-pr 4258  ax-un 4485  ax-setind 4590

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-ral 2490  df-rex 2491  df-v 2775  df-sbc 3001  df-dif 3170  df-un 3172  df-in 3174  df-ss 3181  df-nul 3463  df-pw 3620  df-sn 3641  df-pr 3642  df-op 3644  df-uni 3854  df-int 3889  df-br 4049  df-opab 4111  df-id 4345  df-suc 4423  df-iom 4644  df-xp 4686  df-rel 4687  df-cnv 4688  df-co 4689  df-dm 4690  df-rn 4691  df-res 4692  df-ima 4693  df-iota 5238  df-fun 5279  df-fn 5280  df-f 5281  df-f1 5282  df-fo 5283  df-f1o 5284  df-fv 5285  df-ov 5957  df-oprab 5958  df-mpo 5959  df-1o 6512  df-2o 6513  df-er 6630  df-map 6747  df-en 6838  df-womni 7278

This theorem is referenced by:  redcwlpo  16109  nconstwlpo  16120