Theorem ordunisuc 7867

Description: An ordinal class is equal to the union of its successor. (Contributed by NM, 10-Dec-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Assertion

Ref	Expression
ordunisuc	⊢ (Ord 𝐴 → ∪ suc 𝐴 = 𝐴)

Proof of Theorem ordunisuc

Step	Hyp	Ref	Expression
1		ordeleqon 7820	. 2 ⊢ (Ord 𝐴 ↔ (𝐴 ∈ On ∨ 𝐴 = On))
2		onunisuc 6493	. . 3 ⊢ (𝐴 ∈ On → ∪ suc 𝐴 = 𝐴)
3		sucon 7841	. . . . . 6 ⊢ suc On = On
4	3	unieqi 4882	. . . . 5 ⊢ ∪ suc On = ∪ On
5		unon 7866	. . . . 5 ⊢ ∪ On = On
6	4, 5	eqtri 2786	. . . 4 ⊢ ∪ suc On = On
7		suceq 6449	. . . . 5 ⊢ (𝐴 = On → suc 𝐴 = suc On)
8	7	unieqd 4883	. . . 4 ⊢ (𝐴 = On → ∪ suc 𝐴 = ∪ suc On)
9		id 23	. . . 4 ⊢ (𝐴 = On → 𝐴 = On)
10	6, 8, 9	3eqtr4a 2824	. . 3 ⊢ (𝐴 = On → ∪ suc 𝐴 = 𝐴)
11	2, 10	jaoi 871	. 2 ⊢ ((𝐴 ∈ On ∨ 𝐴 = On) → ∪ suc 𝐴 = 𝐴)
12	1, 11	sylbi 220	1 ⊢ (Ord 𝐴 → ∪ suc 𝐴 = 𝐴)

This proof depends on syntax axioms:   → wi 4   ∨ wo 861   = wceq 1570   ∈ wcel 2147  ∪ 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  ax-un 7775

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:  orduniss2  7868  onsucuni2  7869  nlimsucg  7877  tz7.44-2  8443  rnttrcl  9750  ttrclselem2  9754  ttukeylem7  10620  tsksuc  10874  nnuni  36670  dfrdg2  36735  ontgsucval  37398  onsuctopon  37400  limsucncmpi  37411  finxpsuclem  38512