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

Theorem nndcel 6364
Description: Set membership between two natural numbers is decidable. (Contributed by Jim Kingdon, 6-Sep-2019.)
Assertion
Ref Expression
nndcel ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → DECID 𝐴𝐵)

Proof of Theorem nndcel
StepHypRef Expression
1 nntri3or 6357 . . 3 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵𝐴 = 𝐵𝐵𝐴))
2 orc 686 . . . 4 (𝐴𝐵 → (𝐴𝐵 ∨ ¬ 𝐴𝐵))
3 elirr 4426 . . . . . 6 ¬ 𝐵𝐵
4 eleq1 2180 . . . . . 6 (𝐴 = 𝐵 → (𝐴𝐵𝐵𝐵))
53, 4mtbiri 649 . . . . 5 (𝐴 = 𝐵 → ¬ 𝐴𝐵)
65olcd 708 . . . 4 (𝐴 = 𝐵 → (𝐴𝐵 ∨ ¬ 𝐴𝐵))
7 en2lp 4439 . . . . . 6 ¬ (𝐵𝐴𝐴𝐵)
87imnani 665 . . . . 5 (𝐵𝐴 → ¬ 𝐴𝐵)
98olcd 708 . . . 4 (𝐵𝐴 → (𝐴𝐵 ∨ ¬ 𝐴𝐵))
102, 6, 93jaoi 1266 . . 3 ((𝐴𝐵𝐴 = 𝐵𝐵𝐴) → (𝐴𝐵 ∨ ¬ 𝐴𝐵))
111, 10syl 14 . 2 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 ∨ ¬ 𝐴𝐵))
12 df-dc 805 . 2 (DECID 𝐴𝐵 ↔ (𝐴𝐵 ∨ ¬ 𝐴𝐵))
1311, 12sylibr 133 1 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → DECID 𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wo 682  DECID wdc 804  w3o 946   = wceq 1316  wcel 1465  ωcom 4474
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-iinf 4472
This theorem depends on definitions:  df-bi 116  df-dc 805  df-3or 948  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rex 2399  df-v 2662  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-uni 3707  df-int 3742  df-tr 3997  df-iord 4258  df-on 4260  df-suc 4263  df-iom 4475
This theorem is referenced by:  enumctlemm  6967  nnnninf  6991  ltdcpi  7099  nninfalllemn  13129
  Copyright terms: Public domain W3C validator