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Theorem ordelsuc 6889
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 6887 . . 3 (Ord 𝐵 → (𝐴𝐵 → suc 𝐴𝐵))
21adantl 480 . 2 ((𝐴𝐶 ∧ Ord 𝐵) → (𝐴𝐵 → suc 𝐴𝐵))
3 sucssel 5722 . . 3 (𝐴𝐶 → (suc 𝐴𝐵𝐴𝐵))
43adantr 479 . 2 ((𝐴𝐶 ∧ Ord 𝐵) → (suc 𝐴𝐵𝐴𝐵))
52, 4impbid 200 1 ((𝐴𝐶 ∧ Ord 𝐵) → (𝐴𝐵 ↔ suc 𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382  wcel 1976  wss 3539  Ord word 5625  suc csuc 5628
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pr 4828  ax-un 6824
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 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-br 4578  df-opab 4638  df-tr 4675  df-eprel 4939  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-ord 5629  df-on 5630  df-suc 5632
This theorem is referenced by:  onsucmin  6890  onsucssi  6910  tfindsg2  6930  ordgt0ge1  7441  onomeneq  8012  cantnflem1  8446  r1ordg  8501  r1val1  8509  rankonidlem  8551  rankxplim3  8604
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