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Theorem unon 7852
Description: The class of all ordinal numbers is its own union. Exercise 11 of [TakeutiZaring] p. 40. (Contributed by NM, 12-Nov-2003.)
Assertion
Ref Expression
unon On = On

Proof of Theorem unon
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eluni2 4910 . . . 4 (𝑥 On ↔ ∃𝑦 ∈ On 𝑥𝑦)
2 onelon 6408 . . . . 5 ((𝑦 ∈ On ∧ 𝑥𝑦) → 𝑥 ∈ On)
32rexlimiva 3146 . . . 4 (∃𝑦 ∈ On 𝑥𝑦𝑥 ∈ On)
41, 3sylbi 217 . . 3 (𝑥 On → 𝑥 ∈ On)
5 vex 3483 . . . . 5 𝑥 ∈ V
65sucid 6465 . . . 4 𝑥 ∈ suc 𝑥
7 onsuc 7832 . . . 4 (𝑥 ∈ On → suc 𝑥 ∈ On)
8 elunii 4911 . . . 4 ((𝑥 ∈ suc 𝑥 ∧ suc 𝑥 ∈ On) → 𝑥 On)
96, 7, 8sylancr 587 . . 3 (𝑥 ∈ On → 𝑥 On)
104, 9impbii 209 . 2 (𝑥 On ↔ 𝑥 ∈ On)
1110eqriv 2733 1 On = On
Colors of variables: wff setvar class
Syntax hints:   = wceq 1539  wcel 2107  wrex 3069   cuni 4906  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:  ordunisuc  7853  limon  7857  orduninsuc  7865  ordtoplem  36437  ordcmp  36449  onsupnmax  43245
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