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

Theorem onm 4104
Description: The class of all ordinal numbers is inhabited. (Contributed by Jim Kingdon, 6-Mar-2019.)
Assertion
Ref Expression
onm x x On

Proof of Theorem onm
StepHypRef Expression
1 0elon 4095 . . 3 On
2 0ex 3875 . . . 4 V
3 eleq1 2097 . . . 4 (x = ∅ → (x On ↔ ∅ On))
42, 3ceqsexv 2587 . . 3 (x(x = ∅ x On) ↔ ∅ On)
51, 4mpbir 134 . 2 x(x = ∅ x On)
6 exsimpr 1506 . 2 (x(x = ∅ x On) → x x On)
75, 6ax-mp 7 1 x x On
Colors of variables: wff set class
Syntax hints:   wa 97   = wceq 1242  wex 1378   wcel 1390  c0 3218  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-in1 544  ax-in2 545  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  ax-nul 3874
This theorem depends on definitions:  df-bi 110  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-dif 2914  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-uni 3572  df-tr 3846  df-iord 4069  df-on 4071
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator