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Theorem sucelon 7059
Description: The successor of an ordinal number is an ordinal number. (Contributed by NM, 9-Sep-2003.)
Assertion
Ref Expression
sucelon (𝐴 ∈ On ↔ suc 𝐴 ∈ On)

Proof of Theorem sucelon
StepHypRef Expression
1 ordsuc 7056 . . 3 (Ord 𝐴 ↔ Ord suc 𝐴)
2 sucexb 7051 . . 3 (𝐴 ∈ V ↔ suc 𝐴 ∈ V)
31, 2anbi12i 733 . 2 ((Ord 𝐴𝐴 ∈ V) ↔ (Ord suc 𝐴 ∧ suc 𝐴 ∈ V))
4 elon2 5772 . 2 (𝐴 ∈ On ↔ (Ord 𝐴𝐴 ∈ V))
5 elon2 5772 . 2 (suc 𝐴 ∈ On ↔ (Ord suc 𝐴 ∧ suc 𝐴 ∈ V))
63, 4, 53bitr4i 292 1 (𝐴 ∈ On ↔ suc 𝐴 ∈ On)
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383  wcel 2030  Vcvv 3231  Ord word 5760  Oncon0 5761  suc csuc 5763
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-tr 4786  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-ord 5764  df-on 5765  df-suc 5767
This theorem is referenced by:  onsucmin  7063  tfindsg2  7103  oaordi  7671  oalimcl  7685  omlimcl  7703  omeulem1  7707  oeordsuc  7719  infensuc  8179  cantnflem1b  8621  cantnflem1  8624  r1ordg  8679  alephnbtwn  8932  cfsuc  9117  alephsuc3  9440  alephreg  9442  bdayimaon  31968  nosupbnd1lem1  31979  nosupbnd1  31985  nosupbnd2lem1  31986  nosupbnd2  31987
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