Theorem peano4 7899

Description: Two natural numbers are equal iff their successors are equal, i.e. the successor function is one-to-one. One of Peano's five postulates for arithmetic. Proposition 7.30(4) of [TakeutiZaring] p. 43. (Contributed by NM, 3-Sep-2003.)

Assertion

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

Proof of Theorem peano4

Step	Hyp	Ref	Expression
1		nnon 7877	. 2 ⊢ (𝐴 ∈ ω → 𝐴 ∈ On)
2		nnon 7877	. 2 ⊢ (𝐵 ∈ ω → 𝐵 ∈ On)
3		suc11 6478	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (suc 𝐴 = suc 𝐵 ↔ 𝐴 = 𝐵))
4	1, 2, 3	syl2an 594	1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (suc 𝐴 = suc 𝐵 ↔ 𝐴 = 𝐵))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   = wceq 1533   ∈ wcel 2098  Oncon0 6371  suc csuc 6373  ωcom 7871

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-tr 5267  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-ord 6374  df-on 6375  df-suc 6377  df-om 7872

This theorem is referenced by:  dif1ennnALT  9302  fseqdom  10051  finxpreclem4  37001