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Theorem nndcel 5991
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 A B)

Proof of Theorem nndcel
StepHypRef Expression
1 nntri3or 5987 . . 3 ((A 𝜔 B 𝜔) → (A B A = B B A))
2 orc 620 . . . 4 (A B → (A B ¬ A B))
3 elirr 4208 . . . . . 6 ¬ B B
4 eleq1 2082 . . . . . 6 (A = B → (A BB B))
53, 4mtbiri 587 . . . . 5 (A = B → ¬ A B)
65olcd 640 . . . 4 (A = B → (A B ¬ A B))
7 en2lp 4216 . . . . . 6 ¬ (B A A B)
87imnani 612 . . . . 5 (B A → ¬ A B)
98olcd 640 . . . 4 (B A → (A B ¬ A B))
102, 6, 93jaoi 1184 . . 3 ((A B A = B B A) → (A B ¬ A B))
111, 10syl 14 . 2 ((A 𝜔 B 𝜔) → (A B ¬ A B))
12 df-dc 734 . 2 (DECID A B ↔ (A B ¬ A B))
1311, 12sylibr 137 1 ((A 𝜔 B 𝜔) → DECID A B)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97   wo 616  DECID wdc 733   w3o 872   = wceq 1228   wcel 1374  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-setind 4204  ax-iinf 4238
This theorem depends on definitions:  df-bi 110  df-dc 734  df-3or 874  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-ne 2188  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-tr 3829  df-iord 4052  df-on 4054  df-suc 4057  df-iom 4241
This theorem is referenced by:  ltdcpi  6183
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