Theorem nn0eln0 4653

Description: A natural number is nonempty iff it contains the empty set. Although in constructive mathematics it is generally more natural to work with inhabited sets and ignore the whole concept of nonempty sets, in the specific case of natural numbers this theorem may be helpful in converting proofs which were written assuming excluded middle. (Contributed by Jim Kingdon, 28-Aug-2019.)

Assertion

Ref	Expression
nn0eln0	⊢ (𝐴 ∈ ω → (∅ ∈ 𝐴 ↔ 𝐴 ≠ ∅))

Proof of Theorem nn0eln0

Step	Hyp	Ref	Expression
1		0elnn 4652	. 2 ⊢ (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∅ ∈ 𝐴))
2		noel 3451	. . . . 5 ⊢ ¬ ∅ ∈ ∅
3		eleq2 2257	. . . . 5 ⊢ (𝐴 = ∅ → (∅ ∈ 𝐴 ↔ ∅ ∈ ∅))
4	2, 3	mtbiri 676	. . . 4 ⊢ (𝐴 = ∅ → ¬ ∅ ∈ 𝐴)
5		nner 2368	. . . 4 ⊢ (𝐴 = ∅ → ¬ 𝐴 ≠ ∅)
6	4, 5	2falsed 703	. . 3 ⊢ (𝐴 = ∅ → (∅ ∈ 𝐴 ↔ 𝐴 ≠ ∅))
7		id 19	. . . 4 ⊢ (∅ ∈ 𝐴 → ∅ ∈ 𝐴)
8		ne0i 3454	. . . 4 ⊢ (∅ ∈ 𝐴 → 𝐴 ≠ ∅)
9	7, 8	2thd 175	. . 3 ⊢ (∅ ∈ 𝐴 → (∅ ∈ 𝐴 ↔ 𝐴 ≠ ∅))
10	6, 9	jaoi 717	. 2 ⊢ ((𝐴 = ∅ ∨ ∅ ∈ 𝐴) → (∅ ∈ 𝐴 ↔ 𝐴 ≠ ∅))
11	1, 10	syl 14	1 ⊢ (𝐴 ∈ ω → (∅ ∈ 𝐴 ↔ 𝐴 ≠ ∅))

Syntax hints:   → wi 4   ↔ wb 105   ∨ wo 709   = wceq 1364   ∈ wcel 2164   ≠ wne 2364  ∅c0 3447  ωcom 4623

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-nul 4156  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-iinf 4621

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-v 2762  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-pw 3604  df-sn 3625  df-pr 3626  df-uni 3837  df-int 3872  df-suc 4403  df-iom 4624

This theorem is referenced by:  nnmord  6572  bj-charfunr  15372