Theorem onsucuni 7780

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 7734	. 2 ⊢ (𝐴 ⊆ On → Ord ∪ 𝐴)
2		ssid 3958	. . 3 ⊢ ∪ 𝐴 ⊆ ∪ 𝐴
3		ordunisssuc 6433	. . 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 688	1 ⊢ (𝐴 ⊆ On → 𝐴 ⊆ suc ∪ 𝐴)

Syntax hints:   → wi 4   ∧ wa 395   ⊆ wss 3903  ∪ cuni 4865  Ord word 6324  Oncon0 6325  suc csuc 6327

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5243  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-tr 5208  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-ord 6328  df-on 6329  df-suc 6331

This theorem is referenced by:  ordsucuni  7781  cofon1  8610  cofon2  8611  naddcllem  8614  tz9.12lem3  9713  onssnum  9962  dfac12lem2  10067  ackbij1lem16  10156  cfslb2n  10190  hsmexlem1  10348  noeta2  27769  etaslts2  27802  cantnfub2  43673  onsucunifi  43721