Theorem peano4 3159

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 e. om /\ B e. om) -> (suc A = suc B <-> A = B))

Proof of Theorem peano4

Step	Hyp	Ref	Expression
1		suc11 3100	. 2 |- ((A e. On /\ B e. On) -> (suc A = suc B <-> A = B))
2		nnont 3145	. 2 |- (A e. om -> A e. On)
3		nnont 3145	. 2 |- (B e. om -> B e. On)
4	1, 2, 3	syl2an 456	1 |- ((A e. om /\ B e. om) -> (suc A = suc B <-> A = B))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 958   e. wcel 960  Oncon0 2955  suc csuc 2957  omcom 3138

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2709  ax-pow 2749  ax-pr 2786

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-v 1815  df-dif 2053  df-un 2054  df-in 2055  df-ss 2057  df-nul 2285  df-pw 2407  df-sn 2417  df-pr 2418  df-op 2421  df-uni 2509  df-br 2626  df-opab 2673  df-tr 2687  df-eprel 2839  df-po 2847  df-so 2857  df-fr 2924  df-we 2941  df-ord 2958  df-on 2959  df-suc 2961  df-om 3139

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