Theorem enwomni 7236

Description: Weak omniscience is invariant with respect to equinumerosity. For example, this means that we can express the Weak Limited Principle of Omniscience as either ω ∈ WOmni or ℕ₀ ∈ WOmni. The former is a better match to conventional notation in the sense that df2o3 6488 says that 2_o = {∅, 1_o} whereas the corresponding relationship does not exist between 2 and {0, 1}. (Contributed by Jim Kingdon, 20-Jun-2024.)

Assertion

Ref	Expression
enwomni	⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ WOmni ↔ 𝐵 ∈ WOmni))

Proof of Theorem enwomni

Step	Hyp	Ref	Expression
1		enwomnilem 7235	. 2 ⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ WOmni → 𝐵 ∈ WOmni))
2		ensym 6840	. . 3 ⊢ (𝐴 ≈ 𝐵 → 𝐵 ≈ 𝐴)
3		enwomnilem 7235	. . 3 ⊢ (𝐵 ≈ 𝐴 → (𝐵 ∈ WOmni → 𝐴 ∈ WOmni))
4	2, 3	syl 14	. 2 ⊢ (𝐴 ≈ 𝐵 → (𝐵 ∈ WOmni → 𝐴 ∈ WOmni))
5	1, 4	impbid 129	1 ⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ WOmni ↔ 𝐵 ∈ WOmni))

Syntax hints:   → wi 4   ↔ wb 105   ∈ wcel 2167   class class class wbr 4033   ≈ cen 6797  WOmnicwomni 7229

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-v 2765  df-sbc 2990  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-br 4034  df-opab 4095  df-id 4328  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1o 6474  df-2o 6475  df-er 6592  df-map 6709  df-en 6800  df-womni 7230

This theorem is referenced by:  redcwlpo  15699  nconstwlpo  15710