Theorem nn0eln0 4602

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 4601	. 2 ⊢ (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∅ ∈ 𝐴))
2		noel 3418	. . . . 5 ⊢ ¬ ∅ ∈ ∅
3		eleq2 2234	. . . . 5 ⊢ (𝐴 = ∅ → (∅ ∈ 𝐴 ↔ ∅ ∈ ∅))
4	2, 3	mtbiri 670	. . . 4 ⊢ (𝐴 = ∅ → ¬ ∅ ∈ 𝐴)
5		nner 2344	. . . 4 ⊢ (𝐴 = ∅ → ¬ 𝐴 ≠ ∅)
6	4, 5	2falsed 697	. . 3 ⊢ (𝐴 = ∅ → (∅ ∈ 𝐴 ↔ 𝐴 ≠ ∅))
7		id 19	. . . 4 ⊢ (∅ ∈ 𝐴 → ∅ ∈ 𝐴)
8		ne0i 3420	. . . 4 ⊢ (∅ ∈ 𝐴 → 𝐴 ≠ ∅)
9	7, 8	2thd 174	. . 3 ⊢ (∅ ∈ 𝐴 → (∅ ∈ 𝐴 ↔ 𝐴 ≠ ∅))
10	6, 9	jaoi 711	. 2 ⊢ ((𝐴 = ∅ ∨ ∅ ∈ 𝐴) → (∅ ∈ 𝐴 ↔ 𝐴 ≠ ∅))
11	1, 10	syl 14	1 ⊢ (𝐴 ∈ ω → (∅ ∈ 𝐴 ↔ 𝐴 ≠ ∅))

Syntax hints:   → wi 4   ↔ wb 104   ∨ wo 703   = wceq 1348   ∈ wcel 2141   ≠ wne 2340  ∅c0 3414  ωcom 4572

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-nul 4113  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-iinf 4570

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3566  df-sn 3587  df-pr 3588  df-uni 3795  df-int 3830  df-suc 4354  df-iom 4573

This theorem is referenced by:  nnmord  6493  bj-charfunr  13767