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Theorem 0elnn 4283
 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 2043 . . 3 (x = ∅ → (x = ∅ ↔ ∅ = ∅))
2 eleq2 2098 . . 3 (x = ∅ → (∅ x ↔ ∅ ∅))
31, 2orbi12d 706 . 2 (x = ∅ → ((x = ∅ x) ↔ (∅ = ∅ ∅)))
4 eqeq1 2043 . . 3 (x = y → (x = ∅ ↔ y = ∅))
5 eleq2 2098 . . 3 (x = y → (∅ x ↔ ∅ y))
64, 5orbi12d 706 . 2 (x = y → ((x = ∅ x) ↔ (y = ∅ y)))
7 eqeq1 2043 . . 3 (x = suc y → (x = ∅ ↔ suc y = ∅))
8 eleq2 2098 . . 3 (x = suc y → (∅ x ↔ ∅ suc y))
97, 8orbi12d 706 . 2 (x = suc y → ((x = ∅ x) ↔ (suc y = ∅ suc y)))
10 eqeq1 2043 . . 3 (x = A → (x = ∅ ↔ A = ∅))
11 eleq2 2098 . . 3 (x = A → (∅ x ↔ ∅ A))
1210, 11orbi12d 706 . 2 (x = A → ((x = ∅ x) ↔ (A = ∅ A)))
13 eqid 2037 . . 3 ∅ = ∅
1413orci 649 . 2 (∅ = ∅ ∅)
15 0ex 3875 . . . . . . 7 V
1615sucid 4120 . . . . . 6 suc ∅
17 suceq 4105 . . . . . 6 (y = ∅ → suc y = suc ∅)
1816, 17syl5eleqr 2124 . . . . 5 (y = ∅ → ∅ suc y)
1918a1i 9 . . . 4 (y 𝜔 → (y = ∅ → ∅ suc y))
20 sssucid 4118 . . . . . 6 y ⊆ suc y
2120a1i 9 . . . . 5 (y 𝜔 → y ⊆ suc y)
2221sseld 2938 . . . 4 (y 𝜔 → (∅ y → ∅ suc y))
2319, 22jaod 636 . . 3 (y 𝜔 → ((y = ∅ y) → ∅ suc y))
24 olc 631 . . 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 4266 1 (A 𝜔 → (A = ∅ A))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∨ wo 628   = wceq 1242   ∈ wcel 1390   ⊆ wss 2911  ∅c0 3218  suc csuc 4068  𝜔com 4256 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 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-iinf 4254 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-uni 3572  df-int 3607  df-suc 4074  df-iom 4257 This theorem is referenced by:  nn0eln0  4284  nnsucsssuc  6010  nntri3or  6011  nnm00  6038  ssfiexmid  6254  elni2  6298  enq0tr  6416
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