Theorem enmkv 7228

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 6488 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 7227	. 2 |- ( A ~~ B -> ( A e. Markov -> B e. Markov ) )
2		ensym 6840	. . 3 |- ( A ~~ B -> B ~~ A )
3		enmkvlem 7227	. . 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 2167   class class class wbr 4033   ~~ cen 6797  Markovcmarkov 7217

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-v 2765  df-sbc 2990  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-br 4034  df-opab 4095  df-id 4328  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1o 6474  df-2o 6475  df-er 6592  df-map 6709  df-en 6800  df-markov 7218

This theorem is referenced by:  neapmkv  15712