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Theorem ordelsuc 7535
 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 7533 . . 3 (Ord 𝐵 → (𝐴𝐵 → suc 𝐴𝐵))
21adantl 486 . 2 ((𝐴𝐶 ∧ Ord 𝐵) → (𝐴𝐵 → suc 𝐴𝐵))
3 sucssel 6262 . . 3 (𝐴𝐶 → (suc 𝐴𝐵𝐴𝐵))
43adantr 485 . 2 ((𝐴𝐶 ∧ Ord 𝐵) → (suc 𝐴𝐵𝐴𝐵))
52, 4impbid 215 1 ((𝐴𝐶 ∧ Ord 𝐵) → (𝐴𝐵 ↔ suc 𝐴𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∈ wcel 2112   ⊆ wss 3859  Ord word 6169  suc csuc 6172 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 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5170  ax-nul 5177  ax-pr 5299  ax-un 7460 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-ral 3076  df-rex 3077  df-rab 3080  df-v 3412  df-sbc 3698  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-pss 3878  df-nul 4227  df-if 4422  df-sn 4524  df-pr 4526  df-tp 4528  df-op 4530  df-uni 4800  df-br 5034  df-opab 5096  df-tr 5140  df-eprel 5436  df-po 5444  df-so 5445  df-fr 5484  df-we 5486  df-ord 6173  df-on 6174  df-suc 6176 This theorem is referenced by:  onsucmin  7536  onsucssi  7556  tfindsg2  7576  ordgt0ge1  8133  onomeneq  8731  cantnflem1  9178  r1ordg  9233  r1val1  9241  rankonidlem  9283  rankxplim3  9336
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