Theorem enmkv 7508

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 6706 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 7507	. 2 |- ( A ~~ B -> ( A e. Markov -> B e. Markov ) )
2		ensym 7072	. . 3 |- ( A ~~ B -> B ~~ A )
3		enmkvlem 7507	. . 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 4131   ~~ cen 7024  Markovcmarkov 7497

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 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686

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 4132  df-opab 4194  df-id 4440  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1o 6691  df-2o 6692  df-er 6811  df-map 6928  df-en 7027  df-markov 7498

This theorem is used by:  neapmkv  17325