Theorem enmkv 7290

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 6539 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 7289	. 2 |- ( A ~~ B -> ( A e. Markov -> B e. Markov ) )
2		ensym 6896	. . 3 |- ( A ~~ B -> B ~~ A )
3		enmkvlem 7289	. . 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 2178   class class class wbr 4059   ~~ cen 6848  Markovcmarkov 7279

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-ral 2491  df-rex 2492  df-v 2778  df-sbc 3006  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-br 4060  df-opab 4122  df-id 4358  df-suc 4436  df-iom 4657  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1o 6525  df-2o 6526  df-er 6643  df-map 6760  df-en 6851  df-markov 7280

This theorem is referenced by:  neapmkv  16209