MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ordelsuc Structured version   Visualization version   GIF version

Theorem ordelsuc 7760
Description: A set belongs to an ordinal iff its successor is a subset of the ordinal. Exercise 8 of [TakeutiZaring] p. 42 and its converse. (Contributed by NM, 29-Nov-2003.)
Assertion
Ref Expression
ordelsuc ((𝐴𝐶 ∧ Ord 𝐵) → (𝐴𝐵 ↔ suc 𝐴𝐵))

Proof of Theorem ordelsuc
StepHypRef Expression
1 ordsucss 7758 . . 3 (Ord 𝐵 → (𝐴𝐵 → suc 𝐴𝐵))
21adantl 483 . 2 ((𝐴𝐶 ∧ Ord 𝐵) → (𝐴𝐵 → suc 𝐴𝐵))
3 sucssel 6417 . . 3 (𝐴𝐶 → (suc 𝐴𝐵𝐴𝐵))
43adantr 482 . 2 ((𝐴𝐶 ∧ Ord 𝐵) → (suc 𝐴𝐵𝐴𝐵))
52, 4impbid 211 1 ((𝐴𝐶 ∧ Ord 𝐵) → (𝐴𝐵 ↔ suc 𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  wcel 2107  wss 3915  Ord word 6321  suc csuc 6324
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-tr 5228  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-ord 6325  df-on 6326  df-suc 6328
This theorem is referenced by:  onsucmin  7761  onsucssi  7782  tfindsg2  7803  ordgt0ge1  8444  onomeneqOLD  9180  cantnflem1  9632  ttrcltr  9659  dmttrcl  9664  r1ordg  9721  r1val1  9729  rankonidlem  9771  rankxplim3  9824  ordeldifsucon  41623  ordeldif1o  41624  oaltom  41751  omltoe  41753
  Copyright terms: Public domain W3C validator