Theorem nn0eln0 4676

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 4675	. 2 ⊢ (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∅ ∈ 𝐴))
2		noel 3468	. . . . 5 ⊢ ¬ ∅ ∈ ∅
3		eleq2 2270	. . . . 5 ⊢ (𝐴 = ∅ → (∅ ∈ 𝐴 ↔ ∅ ∈ ∅))
4	2, 3	mtbiri 677	. . . 4 ⊢ (𝐴 = ∅ → ¬ ∅ ∈ 𝐴)
5		nner 2381	. . . 4 ⊢ (𝐴 = ∅ → ¬ 𝐴 ≠ ∅)
6	4, 5	2falsed 704	. . 3 ⊢ (𝐴 = ∅ → (∅ ∈ 𝐴 ↔ 𝐴 ≠ ∅))
7		id 19	. . . 4 ⊢ (∅ ∈ 𝐴 → ∅ ∈ 𝐴)
8		ne0i 3471	. . . 4 ⊢ (∅ ∈ 𝐴 → 𝐴 ≠ ∅)
9	7, 8	2thd 175	. . 3 ⊢ (∅ ∈ 𝐴 → (∅ ∈ 𝐴 ↔ 𝐴 ≠ ∅))
10	6, 9	jaoi 718	. 2 ⊢ ((𝐴 = ∅ ∨ ∅ ∈ 𝐴) → (∅ ∈ 𝐴 ↔ 𝐴 ≠ ∅))
11	1, 10	syl 14	1 ⊢ (𝐴 ∈ ω → (∅ ∈ 𝐴 ↔ 𝐴 ≠ ∅))

Syntax hints:   → wi 4   ↔ wb 105   ∨ wo 710   = wceq 1373   ∈ wcel 2177   ≠ wne 2377  ∅c0 3464  ωcom 4646

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4170  ax-nul 4178  ax-pow 4226  ax-pr 4261  ax-un 4488  ax-iinf 4644

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-ral 2490  df-rex 2491  df-v 2775  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-pw 3623  df-sn 3644  df-pr 3645  df-uni 3857  df-int 3892  df-suc 4426  df-iom 4647

This theorem is referenced by:  nnmord  6616  bj-charfunr  15884