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Theorem ordon 4212
 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
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 tron 4119 . 2 Tr On
2 df-on 4105 . . . . 5 On = {𝑥 ∣ Ord 𝑥}
32abeq2i 2148 . . . 4 (𝑥 ∈ On ↔ Ord 𝑥)
4 ordtr 4115 . . . 4 (Ord 𝑥 → Tr 𝑥)
53, 4sylbi 114 . . 3 (𝑥 ∈ On → Tr 𝑥)
65rgen 2374 . 2 𝑥 ∈ On Tr 𝑥
7 dford3 4104 . 2 (Ord On ↔ (Tr On ∧ ∀𝑥 ∈ On Tr 𝑥))
81, 6, 7mpbir2an 849 1 Ord On
 Colors of variables: wff set class Syntax hints:   ∈ wcel 1393  ∀wral 2306  Tr wtr 3854  Ord word 4099  Oncon0 4100 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-in 2924  df-ss 2931  df-uni 3581  df-tr 3855  df-iord 4103  df-on 4105 This theorem is referenced by:  ssorduni  4213  limon  4239  onprc  4276
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