ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  0elnn GIF version

Theorem 0elnn 4636
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 2196 . . 3 (𝑥 = ∅ → (𝑥 = ∅ ↔ ∅ = ∅))
2 eleq2 2253 . . 3 (𝑥 = ∅ → (∅ ∈ 𝑥 ↔ ∅ ∈ ∅))
31, 2orbi12d 794 . 2 (𝑥 = ∅ → ((𝑥 = ∅ ∨ ∅ ∈ 𝑥) ↔ (∅ = ∅ ∨ ∅ ∈ ∅)))
4 eqeq1 2196 . . 3 (𝑥 = 𝑦 → (𝑥 = ∅ ↔ 𝑦 = ∅))
5 eleq2 2253 . . 3 (𝑥 = 𝑦 → (∅ ∈ 𝑥 ↔ ∅ ∈ 𝑦))
64, 5orbi12d 794 . 2 (𝑥 = 𝑦 → ((𝑥 = ∅ ∨ ∅ ∈ 𝑥) ↔ (𝑦 = ∅ ∨ ∅ ∈ 𝑦)))
7 eqeq1 2196 . . 3 (𝑥 = suc 𝑦 → (𝑥 = ∅ ↔ suc 𝑦 = ∅))
8 eleq2 2253 . . 3 (𝑥 = suc 𝑦 → (∅ ∈ 𝑥 ↔ ∅ ∈ suc 𝑦))
97, 8orbi12d 794 . 2 (𝑥 = suc 𝑦 → ((𝑥 = ∅ ∨ ∅ ∈ 𝑥) ↔ (suc 𝑦 = ∅ ∨ ∅ ∈ suc 𝑦)))
10 eqeq1 2196 . . 3 (𝑥 = 𝐴 → (𝑥 = ∅ ↔ 𝐴 = ∅))
11 eleq2 2253 . . 3 (𝑥 = 𝐴 → (∅ ∈ 𝑥 ↔ ∅ ∈ 𝐴))
1210, 11orbi12d 794 . 2 (𝑥 = 𝐴 → ((𝑥 = ∅ ∨ ∅ ∈ 𝑥) ↔ (𝐴 = ∅ ∨ ∅ ∈ 𝐴)))
13 eqid 2189 . . 3 ∅ = ∅
1413orci 732 . 2 (∅ = ∅ ∨ ∅ ∈ ∅)
15 0ex 4145 . . . . . . 7 ∅ ∈ V
1615sucid 4435 . . . . . 6 ∅ ∈ suc ∅
17 suceq 4420 . . . . . 6 (𝑦 = ∅ → suc 𝑦 = suc ∅)
1816, 17eleqtrrid 2279 . . . . 5 (𝑦 = ∅ → ∅ ∈ suc 𝑦)
1918a1i 9 . . . 4 (𝑦 ∈ ω → (𝑦 = ∅ → ∅ ∈ suc 𝑦))
20 sssucid 4433 . . . . . 6 𝑦 ⊆ suc 𝑦
2120a1i 9 . . . . 5 (𝑦 ∈ ω → 𝑦 ⊆ suc 𝑦)
2221sseld 3169 . . . 4 (𝑦 ∈ ω → (∅ ∈ 𝑦 → ∅ ∈ suc 𝑦))
2319, 22jaod 718 . . 3 (𝑦 ∈ ω → ((𝑦 = ∅ ∨ ∅ ∈ 𝑦) → ∅ ∈ suc 𝑦))
24 olc 712 . . 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 4617 1 (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∅ ∈ 𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wo 709   = wceq 1364  wcel 2160  wss 3144  c0 3437  suc csuc 4383  ωcom 4607
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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-iinf 4605
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-uni 3825  df-int 3860  df-suc 4389  df-iom 4608
This theorem is referenced by:  nn0eln0  4637  nnsucsssuc  6516  nntri3or  6517  nnm00  6554  ssfilem  6902  diffitest  6914  fiintim  6956  enumct  7143  nnnninfeq  7155  elni2  7342  enq0tr  7462  bj-charfunr  15015
  Copyright terms: Public domain W3C validator