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Theorem unon 7073
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 4472 . . . 4 (𝑥 On ↔ ∃𝑦 ∈ On 𝑥𝑦)
2 onelon 5786 . . . . 5 ((𝑦 ∈ On ∧ 𝑥𝑦) → 𝑥 ∈ On)
32rexlimiva 3057 . . . 4 (∃𝑦 ∈ On 𝑥𝑦𝑥 ∈ On)
41, 3sylbi 207 . . 3 (𝑥 On → 𝑥 ∈ On)
5 vex 3234 . . . . 5 𝑥 ∈ V
65sucid 5842 . . . 4 𝑥 ∈ suc 𝑥
7 suceloni 7055 . . . 4 (𝑥 ∈ On → suc 𝑥 ∈ On)
8 elunii 4473 . . . 4 ((𝑥 ∈ suc 𝑥 ∧ suc 𝑥 ∈ On) → 𝑥 On)
96, 7, 8sylancr 696 . . 3 (𝑥 ∈ On → 𝑥 On)
104, 9impbii 199 . 2 (𝑥 On ↔ 𝑥 ∈ On)
1110eqriv 2648 1 On = On
Colors of variables: wff setvar class
Syntax hints:   = wceq 1523  wcel 2030  wrex 2942   cuni 4468  Oncon0 5761  suc csuc 5763
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-tr 4786  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-ord 5764  df-on 5765  df-suc 5767
This theorem is referenced by:  ordunisuc  7074  limon  7078  orduninsuc  7085  ordtoplem  32559  ordcmp  32571
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