Theorem onsucuni 7822

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

Step	Hyp	Ref	Expression
1		ssorduni 7773	. 2 ⊢ (𝐴 ⊆ On → Ord ∪ 𝐴)
2		ssid 3981	. . 3 ⊢ ∪ 𝐴 ⊆ ∪ 𝐴
3		ordunisssuc 6460	. . 3 ⊢ ((𝐴 ⊆ On ∧ Ord ∪ 𝐴) → (∪ 𝐴 ⊆ ∪ 𝐴 ↔ 𝐴 ⊆ suc ∪ 𝐴))
4	2, 3	mpbii 233	. 2 ⊢ ((𝐴 ⊆ On ∧ Ord ∪ 𝐴) → 𝐴 ⊆ suc ∪ 𝐴)
5	1, 4	mpdan 687	1 ⊢ (𝐴 ⊆ On → 𝐴 ⊆ suc ∪ 𝐴)

Syntax hints:   → wi 4   ∧ wa 395   ⊆ wss 3926  ∪ cuni 4883  Ord word 6351  Oncon0 6352  suc csuc 6354

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-tr 5230  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-we 5608  df-ord 6355  df-on 6356  df-suc 6358

This theorem is referenced by:  ordsucuni  7823  cofon1  8684  cofon2  8685  naddcllem  8688  tz9.12lem3  9803  onssnum  10054  dfac12lem2  10159  ackbij1lem16  10248  cfslb2n  10282  hsmexlem1  10440  noeta2  27748  etasslt2  27778  zs12bday  28395  cantnfub2  43346  onsucunifi  43394