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Theorem unon 7758
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 4867 . . . 4 (𝑥 On ↔ ∃𝑦 ∈ On 𝑥𝑦)
2 onelon 6340 . . . . 5 ((𝑦 ∈ On ∧ 𝑥𝑦) → 𝑥 ∈ On)
32rexlimiva 3142 . . . 4 (∃𝑦 ∈ On 𝑥𝑦𝑥 ∈ On)
41, 3sylbi 216 . . 3 (𝑥 On → 𝑥 ∈ On)
5 vex 3447 . . . . 5 𝑥 ∈ V
65sucid 6397 . . . 4 𝑥 ∈ suc 𝑥
7 onsuc 7738 . . . 4 (𝑥 ∈ On → suc 𝑥 ∈ On)
8 elunii 4868 . . . 4 ((𝑥 ∈ suc 𝑥 ∧ suc 𝑥 ∈ On) → 𝑥 On)
96, 7, 8sylancr 587 . . 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 1541  wcel 2106  wrex 3071   cuni 4863  Oncon0 6315  suc csuc 6317
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2708  ax-sep 5254  ax-nul 5261  ax-pr 5382  ax-un 7664
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-br 5104  df-opab 5166  df-tr 5221  df-eprel 5535  df-po 5543  df-so 5544  df-fr 5586  df-we 5588  df-ord 6318  df-on 6319  df-suc 6321
This theorem is referenced by:  ordunisuc  7759  limon  7763  orduninsuc  7771  ordtoplem  34838  ordcmp  34850  onsupnmax  41464
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