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Theorem unon 7529
 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 4805 . . . 4 (𝑥 On ↔ ∃𝑦 ∈ On 𝑥𝑦)
2 onelon 6185 . . . . 5 ((𝑦 ∈ On ∧ 𝑥𝑦) → 𝑥 ∈ On)
32rexlimiva 3240 . . . 4 (∃𝑦 ∈ On 𝑥𝑦𝑥 ∈ On)
41, 3sylbi 220 . . 3 (𝑥 On → 𝑥 ∈ On)
5 vex 3444 . . . . 5 𝑥 ∈ V
65sucid 6239 . . . 4 𝑥 ∈ suc 𝑥
7 suceloni 7511 . . . 4 (𝑥 ∈ On → suc 𝑥 ∈ On)
8 elunii 4806 . . . 4 ((𝑥 ∈ suc 𝑥 ∧ suc 𝑥 ∈ On) → 𝑥 On)
96, 7, 8sylancr 590 . . 3 (𝑥 ∈ On → 𝑥 On)
104, 9impbii 212 . 2 (𝑥 On ↔ 𝑥 ∈ On)
1110eqriv 2795 1 On = On
 Colors of variables: wff setvar class Syntax hints:   = wceq 1538   ∈ wcel 2111  ∃wrex 3107  ∪ cuni 4801  Oncon0 6160  suc csuc 6162 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pr 5296  ax-un 7444 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-br 5032  df-opab 5094  df-tr 5138  df-eprel 5431  df-po 5439  df-so 5440  df-fr 5479  df-we 5481  df-ord 6163  df-on 6164  df-suc 6166 This theorem is referenced by:  ordunisuc  7530  limon  7534  orduninsuc  7541  ordtoplem  33911  ordcmp  33923
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