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Theorem onsucb 4504
Description: A class is an ordinal number if and only if its successor is an ordinal number. Biconditional form of onsuc 4502. (Contributed by NM, 9-Sep-2003.)
Assertion
Ref Expression
onsucb (𝐴 ∈ On ↔ suc 𝐴 ∈ On)

Proof of Theorem onsucb
StepHypRef Expression
1 onsuc 4502 . 2 (𝐴 ∈ On → suc 𝐴 ∈ On)
2 eloni 4377 . . 3 (suc 𝐴 ∈ On → Ord suc 𝐴)
3 elex 2750 . . . . 5 (suc 𝐴 ∈ On → suc 𝐴 ∈ V)
4 sucexb 4498 . . . . 5 (𝐴 ∈ V ↔ suc 𝐴 ∈ V)
53, 4sylibr 134 . . . 4 (suc 𝐴 ∈ On → 𝐴 ∈ V)
6 elong 4375 . . . . 5 (𝐴 ∈ V → (𝐴 ∈ On ↔ Ord 𝐴))
7 ordsucg 4503 . . . . 5 (𝐴 ∈ V → (Ord 𝐴 ↔ Ord suc 𝐴))
86, 7bitrd 188 . . . 4 (𝐴 ∈ V → (𝐴 ∈ On ↔ Ord suc 𝐴))
95, 8syl 14 . . 3 (suc 𝐴 ∈ On → (𝐴 ∈ On ↔ Ord suc 𝐴))
102, 9mpbird 167 . 2 (suc 𝐴 ∈ On → 𝐴 ∈ On)
111, 10impbii 126 1 (𝐴 ∈ On ↔ suc 𝐴 ∈ On)
Colors of variables: wff set class
Syntax hints:  wb 105  wcel 2148  Vcvv 2739  Ord word 4364  Oncon0 4365  suc csuc 4367
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-uni 3812  df-tr 4104  df-iord 4368  df-on 4370  df-suc 4373
This theorem is referenced by:  onsucmin  4508  onsucuni2  4565
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