Theorem enmkv 7502

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 6702 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 7501	. 2 |- ( A ~~ B -> ( A e. Markov -> B e. Markov ) )
2		ensym 7068	. . 3 |- ( A ~~ B -> B ~~ A )
3		enmkvlem 7501	. . 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 ) )

This proof depends on syntax axioms:   -> wi 4   <-> wb 105   e. wcel 2209   class class class wbr 4130   ~~ cen 7020  Markovcmarkov 7491

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-br 4131  df-opab 4193  df-id 4438  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1o 6687  df-2o 6688  df-er 6807  df-map 6924  df-en 7023  df-markov 7492

This theorem is used by:  neapmkv  17118