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Theorem 0elnn 4539
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 2147 . . 3 (𝑥 = ∅ → (𝑥 = ∅ ↔ ∅ = ∅))
2 eleq2 2204 . . 3 (𝑥 = ∅ → (∅ ∈ 𝑥 ↔ ∅ ∈ ∅))
31, 2orbi12d 783 . 2 (𝑥 = ∅ → ((𝑥 = ∅ ∨ ∅ ∈ 𝑥) ↔ (∅ = ∅ ∨ ∅ ∈ ∅)))
4 eqeq1 2147 . . 3 (𝑥 = 𝑦 → (𝑥 = ∅ ↔ 𝑦 = ∅))
5 eleq2 2204 . . 3 (𝑥 = 𝑦 → (∅ ∈ 𝑥 ↔ ∅ ∈ 𝑦))
64, 5orbi12d 783 . 2 (𝑥 = 𝑦 → ((𝑥 = ∅ ∨ ∅ ∈ 𝑥) ↔ (𝑦 = ∅ ∨ ∅ ∈ 𝑦)))
7 eqeq1 2147 . . 3 (𝑥 = suc 𝑦 → (𝑥 = ∅ ↔ suc 𝑦 = ∅))
8 eleq2 2204 . . 3 (𝑥 = suc 𝑦 → (∅ ∈ 𝑥 ↔ ∅ ∈ suc 𝑦))
97, 8orbi12d 783 . 2 (𝑥 = suc 𝑦 → ((𝑥 = ∅ ∨ ∅ ∈ 𝑥) ↔ (suc 𝑦 = ∅ ∨ ∅ ∈ suc 𝑦)))
10 eqeq1 2147 . . 3 (𝑥 = 𝐴 → (𝑥 = ∅ ↔ 𝐴 = ∅))
11 eleq2 2204 . . 3 (𝑥 = 𝐴 → (∅ ∈ 𝑥 ↔ ∅ ∈ 𝐴))
1210, 11orbi12d 783 . 2 (𝑥 = 𝐴 → ((𝑥 = ∅ ∨ ∅ ∈ 𝑥) ↔ (𝐴 = ∅ ∨ ∅ ∈ 𝐴)))
13 eqid 2140 . . 3 ∅ = ∅
1413orci 721 . 2 (∅ = ∅ ∨ ∅ ∈ ∅)
15 0ex 4062 . . . . . . 7 ∅ ∈ V
1615sucid 4346 . . . . . 6 ∅ ∈ suc ∅
17 suceq 4331 . . . . . 6 (𝑦 = ∅ → suc 𝑦 = suc ∅)
1816, 17eleqtrrid 2230 . . . . 5 (𝑦 = ∅ → ∅ ∈ suc 𝑦)
1918a1i 9 . . . 4 (𝑦 ∈ ω → (𝑦 = ∅ → ∅ ∈ suc 𝑦))
20 sssucid 4344 . . . . . 6 𝑦 ⊆ suc 𝑦
2120a1i 9 . . . . 5 (𝑦 ∈ ω → 𝑦 ⊆ suc 𝑦)
2221sseld 3100 . . . 4 (𝑦 ∈ ω → (∅ ∈ 𝑦 → ∅ ∈ suc 𝑦))
2319, 22jaod 707 . . 3 (𝑦 ∈ ω → ((𝑦 = ∅ ∨ ∅ ∈ 𝑦) → ∅ ∈ suc 𝑦))
24 olc 701 . . 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 4521 1 (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∅ ∈ 𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wo 698   = wceq 1332  wcel 1481  wss 3075  c0 3367  suc csuc 4294  ωcom 4511
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 4053  ax-nul 4061  ax-pow 4105  ax-pr 4138  ax-un 4362  ax-iinf 4509
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-ral 2422  df-rex 2423  df-v 2691  df-dif 3077  df-un 3079  df-in 3081  df-ss 3088  df-nul 3368  df-pw 3516  df-sn 3537  df-pr 3538  df-uni 3744  df-int 3779  df-suc 4300  df-iom 4512
This theorem is referenced by:  nn0eln0  4540  nnsucsssuc  6395  nntri3or  6396  nnm00  6432  ssfilem  6776  diffitest  6788  fiintim  6824  enumct  7007  elni2  7145  enq0tr  7265  nninfalllemn  13375
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