Theorem nn0eln0 4284

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 4283	. 2 ⊢ (A ∈ 𝜔 → (A = ∅ ∨ ∅ ∈ A))
2		noel 3222	. . . . 5 ⊢ ¬ ∅ ∈ ∅
3		eleq2 2098	. . . . 5 ⊢ (A = ∅ → (∅ ∈ A ↔ ∅ ∈ ∅))
4	2, 3	mtbiri 599	. . . 4 ⊢ (A = ∅ → ¬ ∅ ∈ A)
5		nner 2207	. . . 4 ⊢ (A = ∅ → ¬ A ≠ ∅)
6	4, 5	2falsed 617	. . 3 ⊢ (A = ∅ → (∅ ∈ A ↔ A ≠ ∅))
7		id 19	. . . 4 ⊢ (∅ ∈ A → ∅ ∈ A)
8		ne0i 3224	. . . 4 ⊢ (∅ ∈ A → A ≠ ∅)
9	7, 8	2thd 164	. . 3 ⊢ (∅ ∈ A → (∅ ∈ A ↔ A ≠ ∅))
10	6, 9	jaoi 635	. 2 ⊢ ((A = ∅ ∨ ∅ ∈ A) → (∅ ∈ A ↔ A ≠ ∅))
11	1, 10	syl 14	1 ⊢ (A ∈ 𝜔 → (∅ ∈ A ↔ A ≠ ∅))

Syntax hints:   → wi 4   ↔ wb 98   ∨ wo 628   = wceq 1242   ∈ wcel 1390   ≠ wne 2201  ∅c0 3218  𝜔com 4256

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-iinf 4254

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-uni 3572  df-int 3607  df-suc 4074  df-iom 4257

This theorem is referenced by:  nnmord  6026