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Theorem ordon 4240
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 4147 . 2 Tr On
2 df-on 4133 . . . . 5 On = {𝑥 ∣ Ord 𝑥}
32abeq2i 2164 . . . 4 (𝑥 ∈ On ↔ Ord 𝑥)
4 ordtr 4143 . . . 4 (Ord 𝑥 → Tr 𝑥)
53, 4sylbi 118 . . 3 (𝑥 ∈ On → Tr 𝑥)
65rgen 2391 . 2 𝑥 ∈ On Tr 𝑥
7 dford3 4132 . 2 (Ord On ↔ (Tr On ∧ ∀𝑥 ∈ On Tr 𝑥))
81, 6, 7mpbir2an 860 1 Ord On
Colors of variables: wff set class
Syntax hints:  wcel 1409  wral 2323  Tr wtr 3882  Ord word 4127  Oncon0 4128
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-in 2952  df-ss 2959  df-uni 3609  df-tr 3883  df-iord 4131  df-on 4133
This theorem is referenced by:  ssorduni  4241  limon  4267  onprc  4304
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