Theorem nn0eln0 4634

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 4633	. 2 ⊢ (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∅ ∈ 𝐴))
2		noel 3441	. . . . 5 ⊢ ¬ ∅ ∈ ∅
3		eleq2 2253	. . . . 5 ⊢ (𝐴 = ∅ → (∅ ∈ 𝐴 ↔ ∅ ∈ ∅))
4	2, 3	mtbiri 676	. . . 4 ⊢ (𝐴 = ∅ → ¬ ∅ ∈ 𝐴)
5		nner 2364	. . . 4 ⊢ (𝐴 = ∅ → ¬ 𝐴 ≠ ∅)
6	4, 5	2falsed 703	. . 3 ⊢ (𝐴 = ∅ → (∅ ∈ 𝐴 ↔ 𝐴 ≠ ∅))
7		id 19	. . . 4 ⊢ (∅ ∈ 𝐴 → ∅ ∈ 𝐴)
8		ne0i 3444	. . . 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 2160   ≠ wne 2360  ∅c0 3437  ωcom 4604

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 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-nul 4144  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-iinf 4602

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-v 2754  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-uni 3825  df-int 3860  df-suc 4386  df-iom 4605

This theorem is referenced by:  nnmord  6537  bj-charfunr  15000