Theorem enomni 7256

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 6529 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 7255	. 2 |- ( A ~~ B -> ( A e. Omni -> B e. Omni ) )
2		ensym 6886	. . 3 |- ( A ~~ B -> B ~~ A )
3		enomnilem 7255	. . 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 2177   class class class wbr 4051   ~~ cen 6838  Omnicomni 7251

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 4170  ax-nul 4178  ax-pow 4226  ax-pr 4261  ax-un 4488  ax-setind 4593

This theorem depends on definitions:  df-bi 117  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 3003  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3857  df-int 3892  df-br 4052  df-opab 4114  df-id 4348  df-suc 4426  df-iom 4647  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-rn 4694  df-res 4695  df-ima 4696  df-iota 5241  df-fun 5282  df-fn 5283  df-f 5284  df-f1 5285  df-fo 5286  df-f1o 5287  df-fv 5288  df-ov 5960  df-oprab 5961  df-mpo 5962  df-1o 6515  df-2o 6516  df-er 6633  df-map 6750  df-en 6841  df-omni 7252

This theorem is referenced by:  exmidunben  12872  nnnninfen  16099  trilpo  16123