Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  enmkv GIF version

Theorem enmkv 7055
 Description: Being Markov is invariant with respect to equinumerosity. For example, this means that we can express the Markov's Principle as either ω ∈ Markov or ℕ0 ∈ Markov. The former is a better match to conventional notation in the sense that df2o3 6339 says that 2o = {∅, 1o} whereas the corresponding relationship does not exist between 2 and {0, 1}. (Contributed by Jim Kingdon, 24-Jun-2024.)
Assertion
Ref Expression
enmkv (𝐴𝐵 → (𝐴 ∈ Markov ↔ 𝐵 ∈ Markov))

Proof of Theorem enmkv
StepHypRef Expression
1 enmkvlem 7054 . 2 (𝐴𝐵 → (𝐴 ∈ Markov → 𝐵 ∈ Markov))
2 ensym 6687 . . 3 (𝐴𝐵𝐵𝐴)
3 enmkvlem 7054 . . 3 (𝐵𝐴 → (𝐵 ∈ Markov → 𝐴 ∈ Markov))
42, 3syl 14 . 2 (𝐴𝐵 → (𝐵 ∈ Markov → 𝐴 ∈ Markov))
51, 4impbid 128 1 (𝐴𝐵 → (𝐴 ∈ Markov ↔ 𝐵 ∈ Markov))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 104   ∈ wcel 2112   class class class wbr 3939   ≈ cen 6644  Markovcmarkov 7044 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 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-13 2114  ax-14 2115  ax-ext 2123  ax-sep 4056  ax-nul 4064  ax-pow 4108  ax-pr 4142  ax-un 4366  ax-setind 4463 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1732  df-eu 1993  df-mo 1994  df-clab 2128  df-cleq 2134  df-clel 2137  df-nfc 2272  df-ne 2311  df-ral 2423  df-rex 2424  df-v 2693  df-sbc 2916  df-dif 3080  df-un 3082  df-in 3084  df-ss 3091  df-nul 3371  df-pw 3519  df-sn 3540  df-pr 3541  df-op 3543  df-uni 3747  df-int 3782  df-br 3940  df-opab 4000  df-id 4226  df-suc 4304  df-iom 4516  df-xp 4557  df-rel 4558  df-cnv 4559  df-co 4560  df-dm 4561  df-rn 4562  df-res 4563  df-ima 4564  df-iota 5100  df-fun 5137  df-fn 5138  df-f 5139  df-f1 5140  df-fo 5141  df-f1o 5142  df-fv 5143  df-ov 5789  df-oprab 5790  df-mpo 5791  df-1o 6325  df-2o 6326  df-er 6441  df-map 6556  df-en 6647  df-markov 7045 This theorem is referenced by:  neapmkv  13481
 Copyright terms: Public domain W3C validator