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Theorem nndcel 6362
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 6355 . . 3 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵𝐴 = 𝐵𝐵𝐴))
2 orc 684 . . . 4 (𝐴𝐵 → (𝐴𝐵 ∨ ¬ 𝐴𝐵))
3 elirr 4424 . . . . . 6 ¬ 𝐵𝐵
4 eleq1 2178 . . . . . 6 (𝐴 = 𝐵 → (𝐴𝐵𝐵𝐵))
53, 4mtbiri 647 . . . . 5 (𝐴 = 𝐵 → ¬ 𝐴𝐵)
65olcd 706 . . . 4 (𝐴 = 𝐵 → (𝐴𝐵 ∨ ¬ 𝐴𝐵))
7 en2lp 4437 . . . . . 6 ¬ (𝐵𝐴𝐴𝐵)
87imnani 663 . . . . 5 (𝐵𝐴 → ¬ 𝐴𝐵)
98olcd 706 . . . 4 (𝐵𝐴 → (𝐴𝐵 ∨ ¬ 𝐴𝐵))
102, 6, 93jaoi 1264 . . 3 ((𝐴𝐵𝐴 = 𝐵𝐵𝐴) → (𝐴𝐵 ∨ ¬ 𝐴𝐵))
111, 10syl 14 . 2 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 ∨ ¬ 𝐴𝐵))
12 df-dc 803 . 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 680  DECID wdc 802  w3o 944   = wceq 1314  wcel 1463  ωcom 4472
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-iinf 4470
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-ral 2396  df-rex 2397  df-v 2660  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501  df-pr 3502  df-uni 3705  df-int 3740  df-tr 3995  df-iord 4256  df-on 4258  df-suc 4261  df-iom 4473
This theorem is referenced by:  enumctlemm  6965  nnnninf  6989  ltdcpi  7095  nninfalllemn  13036
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