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 ω ∈ Markov or ℕ₀ ∈ Markov. The former is a better match to conventional notation in the sense that df2o3 6702 says that 2_o = {∅, 1_o} 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 ∈ Markov ↔ 𝐵 ∈ Markov))

Proof of Theorem enmkv

Step	Hyp	Ref	Expression
1		enmkvlem 7501	. 2 ⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ Markov → 𝐵 ∈ Markov))
2		ensym 7068	. . 3 ⊢ (𝐴 ≈ 𝐵 → 𝐵 ≈ 𝐴)
3		enmkvlem 7501	. . 3 ⊢ (𝐵 ≈ 𝐴 → (𝐵 ∈ Markov → 𝐴 ∈ Markov))
4	2, 3	syl 14	. 2 ⊢ (𝐴 ≈ 𝐵 → (𝐵 ∈ Markov → 𝐴 ∈ Markov))
5	1, 4	impbid 129	1 ⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ Markov ↔ 𝐵 ∈ Markov))

This proof depends on syntax axioms:   → wi 4   ↔ wb 105   ∈ 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