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Theorem unon 4528
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 3828 . . . 4 (𝑥 On ↔ ∃𝑦 ∈ On 𝑥𝑦)
2 onelon 4402 . . . . 5 ((𝑦 ∈ On ∧ 𝑥𝑦) → 𝑥 ∈ On)
32rexlimiva 2602 . . . 4 (∃𝑦 ∈ On 𝑥𝑦𝑥 ∈ On)
41, 3sylbi 121 . . 3 (𝑥 On → 𝑥 ∈ On)
5 vex 2755 . . . . 5 𝑥 ∈ V
65sucid 4435 . . . 4 𝑥 ∈ suc 𝑥
7 onsuc 4518 . . . 4 (𝑥 ∈ On → suc 𝑥 ∈ On)
8 elunii 3829 . . . 4 ((𝑥 ∈ suc 𝑥 ∧ suc 𝑥 ∈ On) → 𝑥 On)
96, 7, 8sylancr 414 . . 3 (𝑥 ∈ On → 𝑥 On)
104, 9impbii 126 . 2 (𝑥 On ↔ 𝑥 ∈ On)
1110eqriv 2186 1 On = On
Colors of variables: wff set class
Syntax hints:   = wceq 1364  wcel 2160  wrex 2469   cuni 3824  Oncon0 4381  suc csuc 4383
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-uni 3825  df-tr 4117  df-iord 4384  df-on 4386  df-suc 4389
This theorem is referenced by:  limon  4530  onintonm  4534  tfri1dALT  6376  rdgon  6411
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