Theorem onsucuni 7772

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 7726	. 2 ⊢ (𝐴 ⊆ On → Ord ∪ 𝐴)
2		ssid 3939	. . 3 ⊢ ∪ 𝐴 ⊆ ∪ 𝐴
3		ordunisssuc 6422	. . 3 ⊢ ((𝐴 ⊆ On ∧ Ord ∪ 𝐴) → (∪ 𝐴 ⊆ ∪ 𝐴 ↔ 𝐴 ⊆ suc ∪ 𝐴))
4	2, 3	mpbii 235	. 2 ⊢ ((𝐴 ⊆ On ∧ Ord ∪ 𝐴) → 𝐴 ⊆ suc ∪ 𝐴)
5	1, 4	mpdan 694	1 ⊢ (𝐴 ⊆ On → 𝐴 ⊆ suc ∪ 𝐴)

Syntax hints:   → wi 4   ∧ wa 397   ⊆ wss 3885  ∪ cuni 4841  Ord word 6313  Oncon0 6314  suc csuc 6316

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713  ax-sep 5221  ax-pr 5365

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3or 1094  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-ne 2937  df-ral 3056  df-rex 3066  df-rab 3394  df-v 3435  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-pss 3905  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-br 5076  df-opab 5138  df-tr 5183  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-we 5576  df-ord 6317  df-on 6318  df-suc 6320

This theorem is referenced by:  ordsucuni  7773  cofon1  8602  cofon2  8603  naddcllem  8606  tz9.12lem3  9708  onssnum  9957  dfac12lem2  10062  ackbij1lem16  10151  cfslb2n  10185  hsmexlem1  10343  noeta2  27775  etaslts2  27808  cantnfub2  43782  onsucunifi  43830