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Theorem ordon 4481
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 4378 . 2 Tr On
2 df-on 4364 . . . . 5 On = {𝑥 ∣ Ord 𝑥}
32abeq2i 2288 . . . 4 (𝑥 ∈ On ↔ Ord 𝑥)
4 ordtr 4374 . . . 4 (Ord 𝑥 → Tr 𝑥)
53, 4sylbi 121 . . 3 (𝑥 ∈ On → Tr 𝑥)
65rgen 2530 . 2 𝑥 ∈ On Tr 𝑥
7 dford3 4363 . 2 (Ord On ↔ (Tr On ∧ ∀𝑥 ∈ On Tr 𝑥))
81, 6, 7mpbir2an 942 1 Ord On
Colors of variables: wff set class
Syntax hints:  wcel 2148  wral 2455  Tr wtr 4098  Ord word 4358  Oncon0 4359
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-in 3135  df-ss 3142  df-uni 3808  df-tr 4099  df-iord 4362  df-on 4364
This theorem is referenced by:  ssorduni  4482  limon  4508  onprc  4547  tfri1dALT  6345  rdgon  6380
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