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Theorem ordon 4293
Description: The class of all ordinal numbers is ordinal. Proposition 7.12 of [TakeutiZaring] p. 38, but without using the Axiom of Regularity. (Contributed by NM, 17-May-1994.)
Assertion
Ref Expression
ordon Ord On

Proof of Theorem ordon
StepHypRef Expression
1 tron 4200 . 2 Tr On
2 df-on 4186 . . . . 5 On = {𝑥 ∣ Ord 𝑥}
32abeq2i 2198 . . . 4 (𝑥 ∈ On ↔ Ord 𝑥)
4 ordtr 4196 . . . 4 (Ord 𝑥 → Tr 𝑥)
53, 4sylbi 119 . . 3 (𝑥 ∈ On → Tr 𝑥)
65rgen 2428 . 2 𝑥 ∈ On Tr 𝑥
7 dford3 4185 . 2 (Ord On ↔ (Tr On ∧ ∀𝑥 ∈ On Tr 𝑥))
81, 6, 7mpbir2an 888 1 Ord On
Colors of variables: wff set class
Syntax hints:  wcel 1438  wral 2359  Tr wtr 3928  Ord word 4180  Oncon0 4181
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-in 3003  df-ss 3010  df-uni 3649  df-tr 3929  df-iord 4184  df-on 4186
This theorem is referenced by:  ssorduni  4294  limon  4320  onprc  4358  tfri1dALT  6098  rdgon  6133
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