ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ordon Structured version   GIF version

Theorem ordon 4178
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 4085 . 2 Tr On
2 df-on 4071 . . . . 5 On = {x ∣ Ord x}
32abeq2i 2145 . . . 4 (x On ↔ Ord x)
4 ordtr 4081 . . . 4 (Ord x → Tr x)
53, 4sylbi 114 . . 3 (x On → Tr x)
65rgen 2368 . 2 x On Tr x
7 dford3 4070 . 2 (Ord On ↔ (Tr On x On Tr x))
81, 6, 7mpbir2an 848 1 Ord On
Colors of variables: wff set class
Syntax hints:   wcel 1390  wral 2300  Tr wtr 3845  Ord word 4065  Oncon0 4066
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-in 2918  df-ss 2925  df-uni 3572  df-tr 3846  df-iord 4069  df-on 4071
This theorem is referenced by:  ssorduni  4179  limon  4204  onprc  4230
  Copyright terms: Public domain W3C validator