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Theorem peano1OLD 7827
Description: Obsolete version of peano1 7826 as of 29-Nov-2024. (Contributed by NM, 15-May-1994.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
peano1OLD ∅ ∈ ω

Proof of Theorem peano1OLD
StepHypRef Expression
1 limom 7819 . 2 Lim ω
2 0ellim 6381 . 2 (Lim ω → ∅ ∈ ω)
31, 2ax-mp 5 1 ∅ ∈ ω
Colors of variables: wff setvar class
Syntax hints:  wcel 2107  c0 4283  Lim wlim 6319  ωcom 7803
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-tr 5224  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-om 7804
This theorem is referenced by: (None)
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