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Theorem unon 4365
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 3687 . . . 4 (𝑥 On ↔ ∃𝑦 ∈ On 𝑥𝑦)
2 onelon 4244 . . . . 5 ((𝑦 ∈ On ∧ 𝑥𝑦) → 𝑥 ∈ On)
32rexlimiva 2503 . . . 4 (∃𝑦 ∈ On 𝑥𝑦𝑥 ∈ On)
41, 3sylbi 120 . . 3 (𝑥 On → 𝑥 ∈ On)
5 vex 2644 . . . . 5 𝑥 ∈ V
65sucid 4277 . . . 4 𝑥 ∈ suc 𝑥
7 suceloni 4355 . . . 4 (𝑥 ∈ On → suc 𝑥 ∈ On)
8 elunii 3688 . . . 4 ((𝑥 ∈ suc 𝑥 ∧ suc 𝑥 ∈ On) → 𝑥 On)
96, 7, 8sylancr 408 . . 3 (𝑥 ∈ On → 𝑥 On)
104, 9impbii 125 . 2 (𝑥 On ↔ 𝑥 ∈ On)
1110eqriv 2097 1 On = On
Colors of variables: wff set class
Syntax hints:   = wceq 1299  wcel 1448  wrex 2376   cuni 3683  Oncon0 4223  suc csuc 4225
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069  ax-un 4293
This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-v 2643  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-uni 3684  df-tr 3967  df-iord 4226  df-on 4228  df-suc 4231
This theorem is referenced by:  limon  4367  onintonm  4371  tfri1dALT  6178  rdgon  6213
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