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Theorem nn0eln0 4539
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 → (∅ ∈ 𝐴𝐴 ≠ ∅))

Proof of Theorem nn0eln0
StepHypRef Expression
1 0elnn 4538 . 2 (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∅ ∈ 𝐴))
2 noel 3370 . . . . 5 ¬ ∅ ∈ ∅
3 eleq2 2204 . . . . 5 (𝐴 = ∅ → (∅ ∈ 𝐴 ↔ ∅ ∈ ∅))
42, 3mtbiri 665 . . . 4 (𝐴 = ∅ → ¬ ∅ ∈ 𝐴)
5 nner 2313 . . . 4 (𝐴 = ∅ → ¬ 𝐴 ≠ ∅)
64, 52falsed 692 . . 3 (𝐴 = ∅ → (∅ ∈ 𝐴𝐴 ≠ ∅))
7 id 19 . . . 4 (∅ ∈ 𝐴 → ∅ ∈ 𝐴)
8 ne0i 3372 . . . 4 (∅ ∈ 𝐴𝐴 ≠ ∅)
97, 82thd 174 . . 3 (∅ ∈ 𝐴 → (∅ ∈ 𝐴𝐴 ≠ ∅))
106, 9jaoi 706 . 2 ((𝐴 = ∅ ∨ ∅ ∈ 𝐴) → (∅ ∈ 𝐴𝐴 ≠ ∅))
111, 10syl 14 1 (𝐴 ∈ ω → (∅ ∈ 𝐴𝐴 ≠ ∅))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  wo 698   = wceq 1332  wcel 1481  wne 2309  c0 3366  ωcom 4510
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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4052  ax-nul 4060  ax-pow 4104  ax-pr 4137  ax-un 4361  ax-iinf 4508
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-v 2691  df-dif 3076  df-un 3078  df-in 3080  df-ss 3087  df-nul 3367  df-pw 3515  df-sn 3536  df-pr 3537  df-uni 3743  df-int 3778  df-suc 4299  df-iom 4511
This theorem is referenced by:  nnmord  6419  nnnninf  7029
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