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Theorem onsucssi 6905
Description: A set belongs to an ordinal number iff its successor is a subset of the ordinal number. Exercise 8 of [TakeutiZaring] p. 42 and its converse. (Contributed by NM, 16-Sep-1995.)
Hypotheses
Ref Expression
onssi.1 𝐴 ∈ On
onsucssi.2 𝐵 ∈ On
Assertion
Ref Expression
onsucssi (𝐴𝐵 ↔ suc 𝐴𝐵)

Proof of Theorem onsucssi
StepHypRef Expression
1 onssi.1 . 2 𝐴 ∈ On
2 onsucssi.2 . . 3 𝐵 ∈ On
32onordi 5730 . 2 Ord 𝐵
4 ordelsuc 6884 . 2 ((𝐴 ∈ On ∧ Ord 𝐵) → (𝐴𝐵 ↔ suc 𝐴𝐵))
51, 3, 4mp2an 703 1 (𝐴𝐵 ↔ suc 𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wb 194  wcel 1975  wss 3534  Ord word 5620  Oncon0 5621  suc csuc 5623
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2227  ax-ext 2584  ax-sep 4698  ax-nul 4707  ax-pr 4823  ax-un 6819
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2456  df-mo 2457  df-clab 2591  df-cleq 2597  df-clel 2600  df-nfc 2734  df-ne 2776  df-ral 2895  df-rex 2896  df-rab 2899  df-v 3169  df-sbc 3397  df-dif 3537  df-un 3539  df-in 3541  df-ss 3548  df-pss 3550  df-nul 3869  df-if 4031  df-sn 4120  df-pr 4122  df-tp 4124  df-op 4126  df-uni 4362  df-br 4573  df-opab 4633  df-tr 4670  df-eprel 4934  df-po 4944  df-so 4945  df-fr 4982  df-we 4984  df-ord 5624  df-on 5625  df-suc 5627
This theorem is referenced by:  omopthlem1  7594  rankval4  8585  rankc1  8588  rankc2  8589  rankxplim  8597  rankxplim3  8599  onsucsuccmpi  31413
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