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Theorem nnaddcld 7701
Description: Closure of addition of positive integers. (Contributed by Mario Carneiro, 27-May-2016.)
Hypotheses
Ref Expression
nnge1d.1 (φA ℕ)
nnmulcld.2 (φB ℕ)
Assertion
Ref Expression
nnaddcld (φ → (A + B) ℕ)

Proof of Theorem nnaddcld
StepHypRef Expression
1 nnge1d.1 . 2 (φA ℕ)
2 nnmulcld.2 . 2 (φB ℕ)
3 nnaddcl 7675 . 2 ((A B ℕ) → (A + B) ℕ)
41, 2, 3syl2anc 391 1 (φ → (A + B) ℕ)
Colors of variables: wff set class
Syntax hints:  wi 4   wcel 1390  (class class class)co 5455   + caddc 6674  cn 7655
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-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-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-cnex 6734  ax-resscn 6735  ax-1cn 6736  ax-1re 6737  ax-addrcl 6740  ax-addass 6745
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-rab 2309  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-br 3756  df-iota 4810  df-fv 4853  df-ov 5458  df-inn 7656
This theorem is referenced by: (None)
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