Theorem enomni 7473

Description: Omniscience is invariant with respect to equinumerosity. For example, this means that we can express the Limited Principle of Omniscience as either ω ∈ Omni or ℕ₀ ∈ Omni. The former is a better match to conventional notation in the sense that df2o3 6696 says that 2_o = {∅, 1_o} whereas the corresponding relationship does not exist between 2 and {0, 1}. (Contributed by Jim Kingdon, 13-Jul-2022.)

Assertion

Ref	Expression
enomni	⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ Omni ↔ 𝐵 ∈ Omni))

Proof of Theorem enomni

Step	Hyp	Ref	Expression
1		enomnilem 7472	. 2 ⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ Omni → 𝐵 ∈ Omni))
2		ensym 7062	. . 3 ⊢ (𝐴 ≈ 𝐵 → 𝐵 ≈ 𝐴)
3		enomnilem 7472	. . 3 ⊢ (𝐵 ≈ 𝐴 → (𝐵 ∈ Omni → 𝐴 ∈ Omni))
4	2, 3	syl 14	. 2 ⊢ (𝐴 ≈ 𝐵 → (𝐵 ∈ Omni → 𝐴 ∈ Omni))
5	1, 4	impbid 129	1 ⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ Omni ↔ 𝐵 ∈ Omni))

Syntax hints:   → wi 4   ↔ wb 105   ∈ wcel 2209   class class class wbr 4128   ≈ cen 7014  Omnicomni 7468

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 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 4247  ax-nul 4257  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682

This theorem 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 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-br 4129  df-opab 4191  df-id 4436  df-suc 4514  df-iom 4736  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-ov 6082  df-oprab 6083  df-mpo 6084  df-1o 6681  df-2o 6682  df-er 6801  df-map 6918  df-en 7017  df-omni 7469

This theorem is referenced by:  exmidunben  13300  nnnninfen  17038  trilpo  17066