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Theorem enomni 7021
 Description: Omniscience is invariant with respect to equinumerosity. For example, this means that we can express the Limited Principle of Omniscience as either ω ∈ Omni or ℕ0 ∈ Omni. The former is a better match to conventional notation in the sense that df2o3 6336 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 (𝐴𝐵 → (𝐴 ∈ Omni ↔ 𝐵 ∈ Omni))

Proof of Theorem enomni
StepHypRef Expression
1 enomnilem 7020 . 2 (𝐴𝐵 → (𝐴 ∈ Omni → 𝐵 ∈ Omni))
2 ensym 6684 . . 3 (𝐴𝐵𝐵𝐴)
3 enomnilem 7020 . . 3 (𝐵𝐴 → (𝐵 ∈ Omni → 𝐴 ∈ Omni))
42, 3syl 14 . 2 (𝐴𝐵 → (𝐵 ∈ Omni → 𝐴 ∈ Omni))
51, 4impbid 128 1 (𝐴𝐵 → (𝐴 ∈ Omni ↔ 𝐵 ∈ Omni))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 104   ∈ wcel 1481   class class class wbr 3938   ≈ cen 6641  Omnicomni 7014 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4055  ax-nul 4063  ax-pow 4107  ax-pr 4140  ax-un 4364  ax-setind 4461 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-v 2692  df-sbc 2915  df-dif 3079  df-un 3081  df-in 3083  df-ss 3090  df-nul 3370  df-pw 3518  df-sn 3539  df-pr 3540  df-op 3542  df-uni 3746  df-int 3781  df-br 3939  df-opab 3999  df-id 4224  df-suc 4302  df-iom 4514  df-xp 4554  df-rel 4555  df-cnv 4556  df-co 4557  df-dm 4558  df-rn 4559  df-res 4560  df-ima 4561  df-iota 5097  df-fun 5134  df-fn 5135  df-f 5136  df-f1 5137  df-fo 5138  df-f1o 5139  df-fv 5140  df-ov 5786  df-oprab 5787  df-mpo 5788  df-1o 6322  df-2o 6323  df-er 6438  df-map 6553  df-en 6644  df-omni 7016 This theorem is referenced by:  exmidunben  11995  trilpo  13432
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