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Theorem elni2 6298
Description: Membership in the class of positive integers. (Contributed by NM, 27-Nov-1995.)
Assertion
Ref Expression
elni2 (A N ↔ (A 𝜔 A))

Proof of Theorem elni2
StepHypRef Expression
1 pinn 6293 . . 3 (A NA 𝜔)
2 0npi 6297 . . . . . 6 ¬ ∅ N
3 eleq1 2097 . . . . . 6 (A = ∅ → (A N ↔ ∅ N))
42, 3mtbiri 599 . . . . 5 (A = ∅ → ¬ A N)
54con2i 557 . . . 4 (A N → ¬ A = ∅)
6 0elnn 4283 . . . . . 6 (A 𝜔 → (A = ∅ A))
71, 6syl 14 . . . . 5 (A N → (A = ∅ A))
87ord 642 . . . 4 (A N → (¬ A = ∅ → ∅ A))
95, 8mpd 13 . . 3 (A N → ∅ A)
101, 9jca 290 . 2 (A N → (A 𝜔 A))
11 nndceq0 4282 . . . . . 6 (A 𝜔 → DECID A = ∅)
12 df-dc 742 . . . . . 6 (DECID A = ∅ ↔ (A = ∅ ¬ A = ∅))
1311, 12sylib 127 . . . . 5 (A 𝜔 → (A = ∅ ¬ A = ∅))
1413anim1i 323 . . . 4 ((A 𝜔 A) → ((A = ∅ ¬ A = ∅) A))
15 ancom 253 . . . . 5 ((∅ A (A = ∅ ¬ A = ∅)) ↔ ((A = ∅ ¬ A = ∅) A))
16 andi 730 . . . . 5 ((∅ A (A = ∅ ¬ A = ∅)) ↔ ((∅ A A = ∅) (∅ A ¬ A = ∅)))
1715, 16bitr3i 175 . . . 4 (((A = ∅ ¬ A = ∅) A) ↔ ((∅ A A = ∅) (∅ A ¬ A = ∅)))
1814, 17sylib 127 . . 3 ((A 𝜔 A) → ((∅ A A = ∅) (∅ A ¬ A = ∅)))
19 noel 3222 . . . . . . . . 9 ¬ ∅
20 eleq2 2098 . . . . . . . . 9 (A = ∅ → (∅ A ↔ ∅ ∅))
2119, 20mtbiri 599 . . . . . . . 8 (A = ∅ → ¬ ∅ A)
2221pm2.21d 549 . . . . . . 7 (A = ∅ → (∅ AA N))
2322impcom 116 . . . . . 6 ((∅ A A = ∅) → A N)
2423a1i 9 . . . . 5 (A 𝜔 → ((∅ A A = ∅) → A N))
25 df-ne 2203 . . . . . . 7 (A ≠ ∅ ↔ ¬ A = ∅)
26 elni 6292 . . . . . . . 8 (A N ↔ (A 𝜔 A ≠ ∅))
2726simplbi2 367 . . . . . . 7 (A 𝜔 → (A ≠ ∅ → A N))
2825, 27syl5bir 142 . . . . . 6 (A 𝜔 → (¬ A = ∅ → A N))
2928adantld 263 . . . . 5 (A 𝜔 → ((∅ A ¬ A = ∅) → A N))
3024, 29jaod 636 . . . 4 (A 𝜔 → (((∅ A A = ∅) (∅ A ¬ A = ∅)) → A N))
3130adantr 261 . . 3 ((A 𝜔 A) → (((∅ A A = ∅) (∅ A ¬ A = ∅)) → A N))
3218, 31mpd 13 . 2 ((A 𝜔 A) → A N)
3310, 32impbii 117 1 (A N ↔ (A 𝜔 A))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97  wb 98   wo 628  DECID wdc 741   = wceq 1242   wcel 1390  wne 2201  c0 3218  𝜔com 4256  Ncnpi 6256
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-dc 742  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-ne 2203  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  df-ni 6288
This theorem is referenced by:  addclpi  6311  mulclpi  6312  mulcanpig  6319  addnidpig  6320  ltexpi  6321  ltmpig  6323  nnppipi  6327  archnqq  6400  enq0tr  6416
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