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Theorem ordon 4411
 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 4313 . 2 Tr On
2 df-on 4299 . . . . 5 On = {𝑥 ∣ Ord 𝑥}
32abeq2i 2251 . . . 4 (𝑥 ∈ On ↔ Ord 𝑥)
4 ordtr 4309 . . . 4 (Ord 𝑥 → Tr 𝑥)
53, 4sylbi 120 . . 3 (𝑥 ∈ On → Tr 𝑥)
65rgen 2489 . 2 𝑥 ∈ On Tr 𝑥
7 dford3 4298 . 2 (Ord On ↔ (Tr On ∧ ∀𝑥 ∈ On Tr 𝑥))
81, 6, 7mpbir2an 927 1 Ord On
 Colors of variables: wff set class Syntax hints:   ∈ wcel 1481  ∀wral 2417  Tr wtr 4035  Ord word 4293  Oncon0 4294 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2692  df-in 3083  df-ss 3090  df-uni 3746  df-tr 4036  df-iord 4297  df-on 4299 This theorem is referenced by:  ssorduni  4412  limon  4438  onprc  4476  tfri1dALT  6257  rdgon  6292
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