Theorem suc11 4481

Description: The successor operation behaves like a one-to-one function. Compare Exercise 16 of [Enderton] p. 194. (Contributed by NM, 3-Sep-2003.)

Assertion

Ref	Expression
suc11	⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (suc 𝐴 = suc 𝐵 ↔ 𝐴 = 𝐵))

Proof of Theorem suc11

Step	Hyp	Ref	Expression
1		suc11g 4480	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (suc 𝐴 = suc 𝐵 ↔ 𝐴 = 𝐵))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1332   ∈ wcel 1481  Oncon0 4293  suc csuc 4295

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-setind 4460

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-v 2691  df-dif 3078  df-un 3080  df-sn 3538  df-pr 3539  df-suc 4301

This theorem is referenced by:  omp1eomlem  6987