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Theorem ordon 4372
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 4274 . 2 Tr On
2 df-on 4260 . . . . 5 On = {𝑥 ∣ Ord 𝑥}
32abeq2i 2228 . . . 4 (𝑥 ∈ On ↔ Ord 𝑥)
4 ordtr 4270 . . . 4 (Ord 𝑥 → Tr 𝑥)
53, 4sylbi 120 . . 3 (𝑥 ∈ On → Tr 𝑥)
65rgen 2462 . 2 𝑥 ∈ On Tr 𝑥
7 dford3 4259 . 2 (Ord On ↔ (Tr On ∧ ∀𝑥 ∈ On Tr 𝑥))
81, 6, 7mpbir2an 911 1 Ord On
Colors of variables: wff set class
Syntax hints:  wcel 1465  wral 2393  Tr wtr 3996  Ord word 4254  Oncon0 4255
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-in 3047  df-ss 3054  df-uni 3707  df-tr 3997  df-iord 4258  df-on 4260
This theorem is referenced by:  ssorduni  4373  limon  4399  onprc  4437  tfri1dALT  6216  rdgon  6251
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