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Theorem unon 7194
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 4590 . . . 4 (𝑥 On ↔ ∃𝑦 ∈ On 𝑥𝑦)
2 onelon 5907 . . . . 5 ((𝑦 ∈ On ∧ 𝑥𝑦) → 𝑥 ∈ On)
32rexlimiva 3164 . . . 4 (∃𝑦 ∈ On 𝑥𝑦𝑥 ∈ On)
41, 3sylbi 207 . . 3 (𝑥 On → 𝑥 ∈ On)
5 vex 3341 . . . . 5 𝑥 ∈ V
65sucid 5963 . . . 4 𝑥 ∈ suc 𝑥
7 suceloni 7176 . . . 4 (𝑥 ∈ On → suc 𝑥 ∈ On)
8 elunii 4591 . . . 4 ((𝑥 ∈ suc 𝑥 ∧ suc 𝑥 ∈ On) → 𝑥 On)
96, 7, 8sylancr 698 . . 3 (𝑥 ∈ On → 𝑥 On)
104, 9impbii 199 . 2 (𝑥 On ↔ 𝑥 ∈ On)
1110eqriv 2755 1 On = On
Colors of variables: wff setvar class
Syntax hints:   = wceq 1630  wcel 2137  wrex 3049   cuni 4586  Oncon0 5882  suc csuc 5884
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-8 2139  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738  ax-sep 4931  ax-nul 4939  ax-pr 5053  ax-un 7112
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-eu 2609  df-mo 2610  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ne 2931  df-ral 3053  df-rex 3054  df-rab 3057  df-v 3340  df-sbc 3575  df-dif 3716  df-un 3718  df-in 3720  df-ss 3727  df-pss 3729  df-nul 4057  df-if 4229  df-sn 4320  df-pr 4322  df-tp 4324  df-op 4326  df-uni 4587  df-br 4803  df-opab 4863  df-tr 4903  df-eprel 5177  df-po 5185  df-so 5186  df-fr 5223  df-we 5225  df-ord 5885  df-on 5886  df-suc 5888
This theorem is referenced by:  ordunisuc  7195  limon  7199  orduninsuc  7206  ordtoplem  32738  ordcmp  32750
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