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Theorem nn0xnn0d 11787
 Description: A standard nonnegative integer is an extended nonnegative integer, deduction form. (Contributed by AV, 10-Dec-2020.)
Hypothesis
Ref Expression
nn0xnn0d.1 (𝜑𝐴 ∈ ℕ0)
Assertion
Ref Expression
nn0xnn0d (𝜑𝐴 ∈ ℕ0*)

Proof of Theorem nn0xnn0d
StepHypRef Expression
1 nn0ssxnn0 11781 . 2 0 ⊆ ℕ0*
2 nn0xnn0d.1 . 2 (𝜑𝐴 ∈ ℕ0)
31, 2sseldi 3851 1 (𝜑𝐴 ∈ ℕ0*)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2051  ℕ0cn0 11706  ℕ0*cxnn0 11778 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-ext 2745 This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-clab 2754  df-cleq 2766  df-clel 2841  df-nfc 2913  df-v 3412  df-un 3829  df-in 3831  df-ss 3838  df-xnn0 11779 This theorem is referenced by:  xnn0xaddcl  12444  fusgrn0eqdrusgr  27071  cusgrrusgr  27082
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