Theorem enomni 7343

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 6602 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 7342	. 2 ⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ Omni → 𝐵 ∈ Omni))
2		ensym 6960	. . 3 ⊢ (𝐴 ≈ 𝐵 → 𝐵 ≈ 𝐴)
3		enomnilem 7342	. . 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 2201   class class class wbr 4089   ≈ cen 6912  Omnicomni 7338

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2203  ax-14 2204  ax-ext 2212  ax-sep 4208  ax-nul 4216  ax-pow 4266  ax-pr 4301  ax-un 4532  ax-setind 4637

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1810  df-eu 2081  df-mo 2082  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ne 2402  df-ral 2514  df-rex 2515  df-v 2803  df-sbc 3031  df-dif 3201  df-un 3203  df-in 3205  df-ss 3212  df-nul 3494  df-pw 3655  df-sn 3676  df-pr 3677  df-op 3679  df-uni 3895  df-int 3930  df-br 4090  df-opab 4152  df-id 4392  df-suc 4470  df-iom 4691  df-xp 4733  df-rel 4734  df-cnv 4735  df-co 4736  df-dm 4737  df-rn 4738  df-res 4739  df-ima 4740  df-iota 5288  df-fun 5330  df-fn 5331  df-f 5332  df-f1 5333  df-fo 5334  df-f1o 5335  df-fv 5336  df-ov 6026  df-oprab 6027  df-mpo 6028  df-1o 6587  df-2o 6588  df-er 6707  df-map 6824  df-en 6915  df-omni 7339

This theorem is referenced by:  exmidunben  13070  nnnninfen  16686  trilpo  16714