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Theorem unon 7265
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 4632 . . . 4 (𝑥 On ↔ ∃𝑦 ∈ On 𝑥𝑦)
2 onelon 5966 . . . . 5 ((𝑦 ∈ On ∧ 𝑥𝑦) → 𝑥 ∈ On)
32rexlimiva 3209 . . . 4 (∃𝑦 ∈ On 𝑥𝑦𝑥 ∈ On)
41, 3sylbi 209 . . 3 (𝑥 On → 𝑥 ∈ On)
5 vex 3388 . . . . 5 𝑥 ∈ V
65sucid 6020 . . . 4 𝑥 ∈ suc 𝑥
7 suceloni 7247 . . . 4 (𝑥 ∈ On → suc 𝑥 ∈ On)
8 elunii 4633 . . . 4 ((𝑥 ∈ suc 𝑥 ∧ suc 𝑥 ∈ On) → 𝑥 On)
96, 7, 8sylancr 582 . . 3 (𝑥 ∈ On → 𝑥 On)
104, 9impbii 201 . 2 (𝑥 On ↔ 𝑥 ∈ On)
1110eqriv 2796 1 On = On
Colors of variables: wff setvar class
Syntax hints:   = wceq 1653  wcel 2157  wrex 3090   cuni 4628  Oncon0 5941  suc csuc 5943
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097  ax-un 7183
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-tp 4373  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-tr 4946  df-eprel 5225  df-po 5233  df-so 5234  df-fr 5271  df-we 5273  df-ord 5944  df-on 5945  df-suc 5947
This theorem is referenced by:  ordunisuc  7266  limon  7270  orduninsuc  7277  ordtoplem  32942  ordcmp  32954
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