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Theorem onm 4184
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 4175 . . 3 ∅ ∈ On
2 0ex 3925 . . . 4 ∅ ∈ V
3 eleq1 2145 . . . 4 (𝑥 = ∅ → (𝑥 ∈ On ↔ ∅ ∈ On))
42, 3ceqsexv 2647 . . 3 (∃𝑥(𝑥 = ∅ ∧ 𝑥 ∈ On) ↔ ∅ ∈ On)
51, 4mpbir 144 . 2 𝑥(𝑥 = ∅ ∧ 𝑥 ∈ On)
6 exsimpr 1550 . 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 102   = wceq 1285  wex 1422  wcel 1434  c0 3267  Oncon0 4146
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-nul 3924
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2612  df-dif 2984  df-in 2988  df-ss 2995  df-nul 3268  df-pw 3402  df-uni 3622  df-tr 3896  df-iord 4149  df-on 4151
This theorem is referenced by: (None)
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