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Theorem 0elnn 4386
Description: A natural number is either the empty set or has the empty set as an element. (Contributed by Jim Kingdon, 23-Aug-2019.)
Assertion
Ref Expression
0elnn (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∅ ∈ 𝐴))

Proof of Theorem 0elnn
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqeq1 2089 . . 3 (𝑥 = ∅ → (𝑥 = ∅ ↔ ∅ = ∅))
2 eleq2 2146 . . 3 (𝑥 = ∅ → (∅ ∈ 𝑥 ↔ ∅ ∈ ∅))
31, 2orbi12d 740 . 2 (𝑥 = ∅ → ((𝑥 = ∅ ∨ ∅ ∈ 𝑥) ↔ (∅ = ∅ ∨ ∅ ∈ ∅)))
4 eqeq1 2089 . . 3 (𝑥 = 𝑦 → (𝑥 = ∅ ↔ 𝑦 = ∅))
5 eleq2 2146 . . 3 (𝑥 = 𝑦 → (∅ ∈ 𝑥 ↔ ∅ ∈ 𝑦))
64, 5orbi12d 740 . 2 (𝑥 = 𝑦 → ((𝑥 = ∅ ∨ ∅ ∈ 𝑥) ↔ (𝑦 = ∅ ∨ ∅ ∈ 𝑦)))
7 eqeq1 2089 . . 3 (𝑥 = suc 𝑦 → (𝑥 = ∅ ↔ suc 𝑦 = ∅))
8 eleq2 2146 . . 3 (𝑥 = suc 𝑦 → (∅ ∈ 𝑥 ↔ ∅ ∈ suc 𝑦))
97, 8orbi12d 740 . 2 (𝑥 = suc 𝑦 → ((𝑥 = ∅ ∨ ∅ ∈ 𝑥) ↔ (suc 𝑦 = ∅ ∨ ∅ ∈ suc 𝑦)))
10 eqeq1 2089 . . 3 (𝑥 = 𝐴 → (𝑥 = ∅ ↔ 𝐴 = ∅))
11 eleq2 2146 . . 3 (𝑥 = 𝐴 → (∅ ∈ 𝑥 ↔ ∅ ∈ 𝐴))
1210, 11orbi12d 740 . 2 (𝑥 = 𝐴 → ((𝑥 = ∅ ∨ ∅ ∈ 𝑥) ↔ (𝐴 = ∅ ∨ ∅ ∈ 𝐴)))
13 eqid 2083 . . 3 ∅ = ∅
1413orci 683 . 2 (∅ = ∅ ∨ ∅ ∈ ∅)
15 0ex 3925 . . . . . . 7 ∅ ∈ V
1615sucid 4200 . . . . . 6 ∅ ∈ suc ∅
17 suceq 4185 . . . . . 6 (𝑦 = ∅ → suc 𝑦 = suc ∅)
1816, 17syl5eleqr 2172 . . . . 5 (𝑦 = ∅ → ∅ ∈ suc 𝑦)
1918a1i 9 . . . 4 (𝑦 ∈ ω → (𝑦 = ∅ → ∅ ∈ suc 𝑦))
20 sssucid 4198 . . . . . 6 𝑦 ⊆ suc 𝑦
2120a1i 9 . . . . 5 (𝑦 ∈ ω → 𝑦 ⊆ suc 𝑦)
2221sseld 3007 . . . 4 (𝑦 ∈ ω → (∅ ∈ 𝑦 → ∅ ∈ suc 𝑦))
2319, 22jaod 670 . . 3 (𝑦 ∈ ω → ((𝑦 = ∅ ∨ ∅ ∈ 𝑦) → ∅ ∈ suc 𝑦))
24 olc 665 . . 3 (∅ ∈ suc 𝑦 → (suc 𝑦 = ∅ ∨ ∅ ∈ suc 𝑦))
2523, 24syl6 33 . 2 (𝑦 ∈ ω → ((𝑦 = ∅ ∨ ∅ ∈ 𝑦) → (suc 𝑦 = ∅ ∨ ∅ ∈ suc 𝑦)))
263, 6, 9, 12, 14, 25finds 4369 1 (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∅ ∈ 𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wo 662   = wceq 1285  wcel 1434  wss 2982  c0 3267  suc csuc 4148  ωcom 4359
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-nul 3924  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-iinf 4357
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2612  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-nul 3268  df-pw 3402  df-sn 3422  df-pr 3423  df-uni 3622  df-int 3657  df-suc 4154  df-iom 4360
This theorem is referenced by:  nn0eln0  4387  nnsucsssuc  6156  nntri3or  6157  nnm00  6189  ssfilem  6431  diffitest  6443  elni2  6618  enq0tr  6738
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