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Theorem unon 7813
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 4904 . . . 4 (𝑥 On ↔ ∃𝑦 ∈ On 𝑥𝑦)
2 onelon 6380 . . . . 5 ((𝑦 ∈ On ∧ 𝑥𝑦) → 𝑥 ∈ On)
32rexlimiva 3139 . . . 4 (∃𝑦 ∈ On 𝑥𝑦𝑥 ∈ On)
41, 3sylbi 216 . . 3 (𝑥 On → 𝑥 ∈ On)
5 vex 3470 . . . . 5 𝑥 ∈ V
65sucid 6437 . . . 4 𝑥 ∈ suc 𝑥
7 onsuc 7793 . . . 4 (𝑥 ∈ On → suc 𝑥 ∈ On)
8 elunii 4905 . . . 4 ((𝑥 ∈ suc 𝑥 ∧ suc 𝑥 ∈ On) → 𝑥 On)
96, 7, 8sylancr 586 . . 3 (𝑥 ∈ On → 𝑥 On)
104, 9impbii 208 . 2 (𝑥 On ↔ 𝑥 ∈ On)
1110eqriv 2721 1 On = On
Colors of variables: wff setvar class
Syntax hints:   = wceq 1533  wcel 2098  wrex 3062   cuni 4900  Oncon0 6355  suc csuc 6357
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3960  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-tr 5257  df-eprel 5571  df-po 5579  df-so 5580  df-fr 5622  df-we 5624  df-ord 6358  df-on 6359  df-suc 6361
This theorem is referenced by:  ordunisuc  7814  limon  7818  orduninsuc  7826  ordtoplem  35811  ordcmp  35823  onsupnmax  42491
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