Theorem peano4 2393

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 2341	. 2 ⊢ ((A ∈ On ∧ B ∈ On) → (suc A = suc B ↔ A = B))
2		nnont 2379	. 2 ⊢ (A ∈ ω → A ∈ On)
3		nnont 2379	. 2 ⊢ (B ∈ ω → B ∈ On)
4	1, 2, 3	syl2an 349	1 ⊢ ((A ∈ ω ∧ B ∈ ω) → (suc A = suc B ↔ A = B))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196   = wceq 1091   ∈ wcel 1092  Oncon0 2199  suc csuc 2201  ωcom 2372

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3an 583  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-suc 2205  df-om 2373

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