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Theorem onsucuni 6920
 Description: A class of ordinal numbers is a subclass of the successor of its union. Similar to Proposition 7.26 of [TakeutiZaring] p. 41. (Contributed by NM, 19-Sep-2003.)
Assertion
Ref Expression
onsucuni (𝐴 ⊆ On → 𝐴 ⊆ suc 𝐴)

Proof of Theorem onsucuni
StepHypRef Expression
1 ssorduni 6877 . 2 (𝐴 ⊆ On → Ord 𝐴)
2 ssid 3587 . . 3 𝐴 𝐴
3 ordunisssuc 5747 . . 3 ((𝐴 ⊆ On ∧ Ord 𝐴) → ( 𝐴 𝐴𝐴 ⊆ suc 𝐴))
42, 3mpbii 222 . 2 ((𝐴 ⊆ On ∧ Ord 𝐴) → 𝐴 ⊆ suc 𝐴)
51, 4mpdan 699 1 (𝐴 ⊆ On → 𝐴 ⊆ suc 𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ⊆ wss 3540  ∪ cuni 4372  Ord word 5639  Oncon0 5640  suc csuc 5642 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-tr 4681  df-eprel 4949  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-ord 5643  df-on 5644  df-suc 5646 This theorem is referenced by:  ordsucuni  6921  tz9.12lem3  8535  onssnum  8746  dfac12lem2  8849  ackbij1lem16  8940  cfslb2n  8973  hsmexlem1  9131
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