Theorem peano4 3142

Description: Two natural numbers are equal iff their successors are equal, i.e. the successor function is one-to-one. One of Peano's 5 postulates for arithmetic. Proposition 7.30(4) of [TakeutiZaring] p. 43.

Assertion

Ref	Expression
peano4	⊢ ((A ∈ ω ⋀ B ∈ ω) → (suc A = suc B ↔ A = B))

Proof of Theorem peano4

Step	Hyp	Ref	Expression
1		suc11 3083	. 2 ⊢ ((A ∈ On ⋀ B ∈ On) → (suc A = suc B ↔ A = B))
2		nnont 3128	. 2 ⊢ (A ∈ ω → A ∈ On)
3		nnont 3128	. 2 ⊢ (B ∈ ω → B ∈ On)
4	1, 2, 3	syl2an 454	1 ⊢ ((A ∈ ω ⋀ B ∈ ω) → (suc A = suc B ↔ A = B))

Syntax hints:   → wi 3   ↔ wb 146   ⋀ wa 223   = wceq 953   ∈ wcel 955  Oncon0 2938  suc csuc 2940  ωcom 3121

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732  ax-pr 2769

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-uni 2494  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-suc 2944  df-om 3122

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