Theorem enmkv 7221

Description: Being Markov is invariant with respect to equinumerosity. For example, this means that we can express the Markov's Principle as either om e. Markov or NN0 e. Markov. The former is a better match to conventional notation in the sense that df2o3 6483 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	|- ( A ~~ B -> ( A e. Markov <-> B e. Markov ) )

Proof of Theorem enmkv

Step	Hyp	Ref	Expression
1		enmkvlem 7220	. 2 |- ( A ~~ B -> ( A e. Markov -> B e. Markov ) )
2		ensym 6835	. . 3 |- ( A ~~ B -> B ~~ A )
3		enmkvlem 7220	. . 3 |- ( B ~~ A -> ( B e. Markov -> A e. Markov ) )
4	2, 3	syl 14	. 2 |- ( A ~~ B -> ( B e. Markov -> A e. Markov ) )
5	1, 4	impbid 129	1 |- ( A ~~ B -> ( A e. Markov <-> B e. Markov ) )

Syntax hints:   -> wi 4   <-> wb 105   e. wcel 2164   class class class wbr 4029   ~~ cen 6792  Markovcmarkov 7210

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2986  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-br 4030  df-opab 4091  df-id 4324  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1o 6469  df-2o 6470  df-er 6587  df-map 6704  df-en 6795  df-markov 7211

This theorem is referenced by:  neapmkv  15558