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Theorem 0elnn 4267
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 2028 . . 3 (x = ∅ → (x = ∅ ↔ ∅ = ∅))
2 eleq2 2083 . . 3 (x = ∅ → (∅ x ↔ ∅ ∅))
31, 2orbi12d 694 . 2 (x = ∅ → ((x = ∅ x) ↔ (∅ = ∅ ∅)))
4 eqeq1 2028 . . 3 (x = y → (x = ∅ ↔ y = ∅))
5 eleq2 2083 . . 3 (x = y → (∅ x ↔ ∅ y))
64, 5orbi12d 694 . 2 (x = y → ((x = ∅ x) ↔ (y = ∅ y)))
7 eqeq1 2028 . . 3 (x = suc y → (x = ∅ ↔ suc y = ∅))
8 eleq2 2083 . . 3 (x = suc y → (∅ x ↔ ∅ suc y))
97, 8orbi12d 694 . 2 (x = suc y → ((x = ∅ x) ↔ (suc y = ∅ suc y)))
10 eqeq1 2028 . . 3 (x = A → (x = ∅ ↔ A = ∅))
11 eleq2 2083 . . 3 (x = A → (∅ x ↔ ∅ A))
1210, 11orbi12d 694 . 2 (x = A → ((x = ∅ x) ↔ (A = ∅ A)))
13 eqid 2022 . . 3 ∅ = ∅
1413orci 637 . 2 (∅ = ∅ ∅)
15 0ex 3858 . . . . . . 7 V
1615sucid 4103 . . . . . 6 suc ∅
17 suceq 4088 . . . . . 6 (y = ∅ → suc y = suc ∅)
1816, 17syl5eleqr 2109 . . . . 5 (y = ∅ → ∅ suc y)
1918a1i 9 . . . 4 (y 𝜔 → (y = ∅ → ∅ suc y))
20 sssucid 4101 . . . . . 6 y ⊆ suc y
2120a1i 9 . . . . 5 (y 𝜔 → y ⊆ suc y)
2221sseld 2921 . . . 4 (y 𝜔 → (∅ y → ∅ suc y))
2319, 22jaod 624 . . 3 (y 𝜔 → ((y = ∅ y) → ∅ suc y))
24 olc 619 . . 3 (∅ suc y → (suc y = ∅ suc y))
2523, 24syl6 29 . 2 (y 𝜔 → ((y = ∅ y) → (suc y = ∅ suc y)))
263, 6, 9, 12, 14, 25finds 4250 1 (A 𝜔 → (A = ∅ A))
Colors of variables: wff set class
Syntax hints:  wi 4   wo 616   = wceq 1228   wcel 1374  wss 2894  c0 3201  suc csuc 4051  𝜔com 4240
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-13 1385  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-nul 3857  ax-pow 3901  ax-pr 3918  ax-un 4120  ax-iinf 4238
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-dif 2897  df-un 2899  df-in 2901  df-ss 2908  df-nul 3202  df-pw 3336  df-sn 3356  df-pr 3357  df-uni 3555  df-int 3590  df-suc 4057  df-iom 4241
This theorem is referenced by:  nn0eln0  4268  nnsucsssuc  5986  nntri3or  5987  nnm00  6013  elni2  6174  enq0tr  6289
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