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Theorem unon 4264
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 3611 . . . 4 (𝑥 On ↔ ∃𝑦 ∈ On 𝑥𝑦)
2 onelon 4148 . . . . 5 ((𝑦 ∈ On ∧ 𝑥𝑦) → 𝑥 ∈ On)
32rexlimiva 2445 . . . 4 (∃𝑦 ∈ On 𝑥𝑦𝑥 ∈ On)
41, 3sylbi 118 . . 3 (𝑥 On → 𝑥 ∈ On)
5 vex 2577 . . . . 5 𝑥 ∈ V
65sucid 4181 . . . 4 𝑥 ∈ suc 𝑥
7 suceloni 4254 . . . 4 (𝑥 ∈ On → suc 𝑥 ∈ On)
8 elunii 3612 . . . 4 ((𝑥 ∈ suc 𝑥 ∧ suc 𝑥 ∈ On) → 𝑥 On)
96, 7, 8sylancr 399 . . 3 (𝑥 ∈ On → 𝑥 On)
104, 9impbii 121 . 2 (𝑥 On ↔ 𝑥 ∈ On)
1110eqriv 2053 1 On = On
Colors of variables: wff set class
Syntax hints:   = wceq 1259  wcel 1409  wrex 2324   cuni 3607  Oncon0 4127  suc csuc 4129
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971  ax-un 4197
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-uni 3608  df-tr 3882  df-iord 4130  df-on 4132  df-suc 4135
This theorem is referenced by:  limon  4266  onintonm  4270
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