Theorem onsucuni 7767

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 7719	. 2 ⊢ (𝐴 ⊆ On → Ord ∪ 𝐴)
2		ssid 3960	. . 3 ⊢ ∪ 𝐴 ⊆ ∪ 𝐴
3		ordunisssuc 6419	. . 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 3905  ∪ cuni 4861  Ord word 6310  Oncon0 6311  suc csuc 6313

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 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pr 5374

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 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-br 5096  df-opab 5158  df-tr 5203  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-we 5578  df-ord 6314  df-on 6315  df-suc 6317

This theorem is referenced by:  ordsucuni  7768  cofon1  8597  cofon2  8598  naddcllem  8601  tz9.12lem3  9704  onssnum  9953  dfac12lem2  10058  ackbij1lem16  10147  cfslb2n  10181  hsmexlem1  10339  noeta2  27713  etasslt2  27743  zs12bday  28379  cantnfub2  43295  onsucunifi  43343