Theorem onsucuni 7863

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 7817	. 2 ⊢ (𝐴 ⊆ On → Ord ∪ 𝐴)
2		ssid 3935	. . 3 ⊢ ∪ 𝐴 ⊆ ∪ 𝐴
3		ordunisssuc 6489	. . 3 ⊢ ((𝐴 ⊆ On ∧ Ord ∪ 𝐴) → (∪ 𝐴 ⊆ ∪ 𝐴 ↔ 𝐴 ⊆ suc ∪ 𝐴))
4	2, 3	mpbii 236	. 2 ⊢ ((𝐴 ⊆ On ∧ Ord ∪ 𝐴) → 𝐴 ⊆ suc ∪ 𝐴)
5	1, 4	mpdan 700	1 ⊢ (𝐴 ⊆ On → 𝐴 ⊆ suc ∪ 𝐴)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ⊆ wss 3900  ∪ cuni 4870  Ord word 6379  Oncon0 6380  suc csuc 6382

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5252  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-tr 5216  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-ord 6383  df-on 6384  df-suc 6386

This theorem is used by:  ordsucuni  7864  cofon1  8709  cofon2  8710  naddcllem  8713  tz9.12lem3  9823  onssnum  10146  dfac12lem2  10250  ackbij1lem16  10339  cfslb2n  10373  hsmexlem1  10531  noeta2  28177  etaslts2  28210  cantnfub2  44516  onsucunifi  44564