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Theorem unon 7771
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 4874 . . . 4 (𝑥 On ↔ ∃𝑦 ∈ On 𝑥𝑦)
2 onelon 6347 . . . . 5 ((𝑦 ∈ On ∧ 𝑥𝑦) → 𝑥 ∈ On)
32rexlimiva 3145 . . . 4 (∃𝑦 ∈ On 𝑥𝑦𝑥 ∈ On)
41, 3sylbi 216 . . 3 (𝑥 On → 𝑥 ∈ On)
5 vex 3452 . . . . 5 𝑥 ∈ V
65sucid 6404 . . . 4 𝑥 ∈ suc 𝑥
7 onsuc 7751 . . . 4 (𝑥 ∈ On → suc 𝑥 ∈ On)
8 elunii 4875 . . . 4 ((𝑥 ∈ suc 𝑥 ∧ suc 𝑥 ∈ On) → 𝑥 On)
96, 7, 8sylancr 588 . . 3 (𝑥 ∈ On → 𝑥 On)
104, 9impbii 208 . 2 (𝑥 On ↔ 𝑥 ∈ On)
1110eqriv 2734 1 On = On
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542  wcel 2107  wrex 3074   cuni 4870  Oncon0 6322  suc csuc 6324
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-tr 5228  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-ord 6325  df-on 6326  df-suc 6328
This theorem is referenced by:  ordunisuc  7772  limon  7776  orduninsuc  7784  ordtoplem  34936  ordcmp  34948  onsupnmax  41591
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