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Theorem 0elnn 4367
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 2062 . . 3 (𝑥 = ∅ → (𝑥 = ∅ ↔ ∅ = ∅))
2 eleq2 2117 . . 3 (𝑥 = ∅ → (∅ ∈ 𝑥 ↔ ∅ ∈ ∅))
31, 2orbi12d 717 . 2 (𝑥 = ∅ → ((𝑥 = ∅ ∨ ∅ ∈ 𝑥) ↔ (∅ = ∅ ∨ ∅ ∈ ∅)))
4 eqeq1 2062 . . 3 (𝑥 = 𝑦 → (𝑥 = ∅ ↔ 𝑦 = ∅))
5 eleq2 2117 . . 3 (𝑥 = 𝑦 → (∅ ∈ 𝑥 ↔ ∅ ∈ 𝑦))
64, 5orbi12d 717 . 2 (𝑥 = 𝑦 → ((𝑥 = ∅ ∨ ∅ ∈ 𝑥) ↔ (𝑦 = ∅ ∨ ∅ ∈ 𝑦)))
7 eqeq1 2062 . . 3 (𝑥 = suc 𝑦 → (𝑥 = ∅ ↔ suc 𝑦 = ∅))
8 eleq2 2117 . . 3 (𝑥 = suc 𝑦 → (∅ ∈ 𝑥 ↔ ∅ ∈ suc 𝑦))
97, 8orbi12d 717 . 2 (𝑥 = suc 𝑦 → ((𝑥 = ∅ ∨ ∅ ∈ 𝑥) ↔ (suc 𝑦 = ∅ ∨ ∅ ∈ suc 𝑦)))
10 eqeq1 2062 . . 3 (𝑥 = 𝐴 → (𝑥 = ∅ ↔ 𝐴 = ∅))
11 eleq2 2117 . . 3 (𝑥 = 𝐴 → (∅ ∈ 𝑥 ↔ ∅ ∈ 𝐴))
1210, 11orbi12d 717 . 2 (𝑥 = 𝐴 → ((𝑥 = ∅ ∨ ∅ ∈ 𝑥) ↔ (𝐴 = ∅ ∨ ∅ ∈ 𝐴)))
13 eqid 2056 . . 3 ∅ = ∅
1413orci 660 . 2 (∅ = ∅ ∨ ∅ ∈ ∅)
15 0ex 3911 . . . . . . 7 ∅ ∈ V
1615sucid 4181 . . . . . 6 ∅ ∈ suc ∅
17 suceq 4166 . . . . . 6 (𝑦 = ∅ → suc 𝑦 = suc ∅)
1816, 17syl5eleqr 2143 . . . . 5 (𝑦 = ∅ → ∅ ∈ suc 𝑦)
1918a1i 9 . . . 4 (𝑦 ∈ ω → (𝑦 = ∅ → ∅ ∈ suc 𝑦))
20 sssucid 4179 . . . . . 6 𝑦 ⊆ suc 𝑦
2120a1i 9 . . . . 5 (𝑦 ∈ ω → 𝑦 ⊆ suc 𝑦)
2221sseld 2971 . . . 4 (𝑦 ∈ ω → (∅ ∈ 𝑦 → ∅ ∈ suc 𝑦))
2319, 22jaod 647 . . 3 (𝑦 ∈ ω → ((𝑦 = ∅ ∨ ∅ ∈ 𝑦) → ∅ ∈ suc 𝑦))
24 olc 642 . . 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 4350 1 (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∅ ∈ 𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wo 639   = wceq 1259  wcel 1409  wss 2944  c0 3251  suc csuc 4129  ωcom 4340
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-nul 3910  ax-pow 3954  ax-pr 3971  ax-un 4197  ax-iinf 4338
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-dif 2947  df-un 2949  df-in 2951  df-ss 2958  df-nul 3252  df-pw 3388  df-sn 3408  df-pr 3409  df-uni 3608  df-int 3643  df-suc 4135  df-iom 4341
This theorem is referenced by:  nn0eln0  4368  nnsucsssuc  6101  nntri3or  6102  nnm00  6132  ssfiexmid  6366  diffitest  6374  elni2  6469  enq0tr  6589
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