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Theorem unon 7402
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 4749 . . . 4 (𝑥 On ↔ ∃𝑦 ∈ On 𝑥𝑦)
2 onelon 6091 . . . . 5 ((𝑦 ∈ On ∧ 𝑥𝑦) → 𝑥 ∈ On)
32rexlimiva 3244 . . . 4 (∃𝑦 ∈ On 𝑥𝑦𝑥 ∈ On)
41, 3sylbi 218 . . 3 (𝑥 On → 𝑥 ∈ On)
5 vex 3440 . . . . 5 𝑥 ∈ V
65sucid 6145 . . . 4 𝑥 ∈ suc 𝑥
7 suceloni 7384 . . . 4 (𝑥 ∈ On → suc 𝑥 ∈ On)
8 elunii 4750 . . . 4 ((𝑥 ∈ suc 𝑥 ∧ suc 𝑥 ∈ On) → 𝑥 On)
96, 7, 8sylancr 587 . . 3 (𝑥 ∈ On → 𝑥 On)
104, 9impbii 210 . 2 (𝑥 On ↔ 𝑥 ∈ On)
1110eqriv 2792 1 On = On
Colors of variables: wff setvar class
Syntax hints:   = wceq 1522  wcel 2081  wrex 3106   cuni 4745  Oncon0 6066  suc csuc 6068
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5094  ax-nul 5101  ax-pr 5221  ax-un 7319
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-rab 3114  df-v 3439  df-sbc 3707  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-pss 3876  df-nul 4212  df-if 4382  df-sn 4473  df-pr 4475  df-tp 4477  df-op 4479  df-uni 4746  df-br 4963  df-opab 5025  df-tr 5064  df-eprel 5353  df-po 5362  df-so 5363  df-fr 5402  df-we 5404  df-ord 6069  df-on 6070  df-suc 6072
This theorem is referenced by:  ordunisuc  7403  limon  7407  orduninsuc  7414  ordtoplem  33392  ordcmp  33404
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