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Theorem unon 7610
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 4823 . . . 4 (𝑥 On ↔ ∃𝑦 ∈ On 𝑥𝑦)
2 onelon 6238 . . . . 5 ((𝑦 ∈ On ∧ 𝑥𝑦) → 𝑥 ∈ On)
32rexlimiva 3200 . . . 4 (∃𝑦 ∈ On 𝑥𝑦𝑥 ∈ On)
41, 3sylbi 220 . . 3 (𝑥 On → 𝑥 ∈ On)
5 vex 3412 . . . . 5 𝑥 ∈ V
65sucid 6292 . . . 4 𝑥 ∈ suc 𝑥
7 suceloni 7592 . . . 4 (𝑥 ∈ On → suc 𝑥 ∈ On)
8 elunii 4824 . . . 4 ((𝑥 ∈ suc 𝑥 ∧ suc 𝑥 ∈ On) → 𝑥 On)
96, 7, 8sylancr 590 . . 3 (𝑥 ∈ On → 𝑥 On)
104, 9impbii 212 . 2 (𝑥 On ↔ 𝑥 ∈ On)
1110eqriv 2734 1 On = On
Colors of variables: wff setvar class
Syntax hints:   = wceq 1543  wcel 2110  wrex 3062   cuni 4819  Oncon0 6213  suc csuc 6215
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-11 2158  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322  ax-un 7523
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-tr 5162  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-we 5511  df-ord 6216  df-on 6217  df-suc 6219
This theorem is referenced by:  ordunisuc  7611  limon  7615  orduninsuc  7622  ordtoplem  34361  ordcmp  34373
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