MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  onsucuni2 Structured version   Visualization version   GIF version

Theorem onsucuni2 7855
Description: A successor ordinal is the successor of its union. (Contributed by NM, 10-Dec-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Assertion
Ref Expression
onsucuni2 ((𝐴 ∈ On ∧ 𝐴 = suc 𝐵) → suc 𝐴 = 𝐴)

Proof of Theorem onsucuni2
StepHypRef Expression
1 eleq1 2828 . . . . . 6 (𝐴 = suc 𝐵 → (𝐴 ∈ On ↔ suc 𝐵 ∈ On))
21biimpac 478 . . . . 5 ((𝐴 ∈ On ∧ 𝐴 = suc 𝐵) → suc 𝐵 ∈ On)
3 eloni 6393 . . . . 5 (suc 𝐵 ∈ On → Ord suc 𝐵)
4 ordsuc 7834 . . . . . . . 8 (Ord 𝐵 ↔ Ord suc 𝐵)
5 ordunisuc 7853 . . . . . . . 8 (Ord 𝐵 suc 𝐵 = 𝐵)
64, 5sylbir 235 . . . . . . 7 (Ord suc 𝐵 suc 𝐵 = 𝐵)
7 suceq 6449 . . . . . . 7 ( suc 𝐵 = 𝐵 → suc suc 𝐵 = suc 𝐵)
86, 7syl 17 . . . . . 6 (Ord suc 𝐵 → suc suc 𝐵 = suc 𝐵)
9 ordunisuc 7853 . . . . . 6 (Ord suc 𝐵 suc suc 𝐵 = suc 𝐵)
108, 9eqtr4d 2779 . . . . 5 (Ord suc 𝐵 → suc suc 𝐵 = suc suc 𝐵)
112, 3, 103syl 18 . . . 4 ((𝐴 ∈ On ∧ 𝐴 = suc 𝐵) → suc suc 𝐵 = suc suc 𝐵)
12 unieq 4917 . . . . . 6 (𝐴 = suc 𝐵 𝐴 = suc 𝐵)
13 suceq 6449 . . . . . 6 ( 𝐴 = suc 𝐵 → suc 𝐴 = suc suc 𝐵)
1412, 13syl 17 . . . . 5 (𝐴 = suc 𝐵 → suc 𝐴 = suc suc 𝐵)
15 suceq 6449 . . . . . 6 (𝐴 = suc 𝐵 → suc 𝐴 = suc suc 𝐵)
1615unieqd 4919 . . . . 5 (𝐴 = suc 𝐵 suc 𝐴 = suc suc 𝐵)
1714, 16eqeq12d 2752 . . . 4 (𝐴 = suc 𝐵 → (suc 𝐴 = suc 𝐴 ↔ suc suc 𝐵 = suc suc 𝐵))
1811, 17imbitrrid 246 . . 3 (𝐴 = suc 𝐵 → ((𝐴 ∈ On ∧ 𝐴 = suc 𝐵) → suc 𝐴 = suc 𝐴))
1918anabsi7 671 . 2 ((𝐴 ∈ On ∧ 𝐴 = suc 𝐵) → suc 𝐴 = suc 𝐴)
20 eloni 6393 . . . 4 (𝐴 ∈ On → Ord 𝐴)
21 ordunisuc 7853 . . . 4 (Ord 𝐴 suc 𝐴 = 𝐴)
2220, 21syl 17 . . 3 (𝐴 ∈ On → suc 𝐴 = 𝐴)
2322adantr 480 . 2 ((𝐴 ∈ On ∧ 𝐴 = suc 𝐵) → suc 𝐴 = 𝐴)
2419, 23eqtrd 2776 1 ((𝐴 ∈ On ∧ 𝐴 = suc 𝐵) → suc 𝐴 = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1539  wcel 2107   cuni 4906  Ord word 6382  Oncon0 6383  suc csuc 6385
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-tr 5259  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-ord 6386  df-on 6387  df-suc 6389
This theorem is referenced by:  rankxplim3  9922  rankxpsuc  9923  onsucf1lem  43287
  Copyright terms: Public domain W3C validator