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Theorem onsucuni 7837
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 7787 . 2 (𝐴 ⊆ On → Ord 𝐴)
2 ssid 4004 . . 3 𝐴 𝐴
3 ordunisssuc 6480 . . 3 ((𝐴 ⊆ On ∧ Ord 𝐴) → ( 𝐴 𝐴𝐴 ⊆ suc 𝐴))
42, 3mpbii 232 . 2 ((𝐴 ⊆ On ∧ Ord 𝐴) → 𝐴 ⊆ suc 𝐴)
51, 4mpdan 685 1 (𝐴 ⊆ On → 𝐴 ⊆ suc 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  wss 3949   cuni 4912  Ord word 6373  Oncon0 6374  suc csuc 6376
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-tr 5270  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-ord 6377  df-on 6378  df-suc 6380
This theorem is referenced by:  ordsucuni  7838  cofon1  8699  cofon2  8700  naddcllem  8703  tz9.12lem3  9820  onssnum  10071  dfac12lem2  10175  ackbij1lem16  10266  cfslb2n  10299  hsmexlem1  10457  noeta2  27737  etasslt2  27767  cantnfub2  42782  onsucunifi  42830
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