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Theorem onm 4138
 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 4129 . . 3 ∅ ∈ On
2 0ex 3884 . . . 4 ∅ ∈ V
3 eleq1 2100 . . . 4 (𝑥 = ∅ → (𝑥 ∈ On ↔ ∅ ∈ On))
42, 3ceqsexv 2593 . . 3 (∃𝑥(𝑥 = ∅ ∧ 𝑥 ∈ On) ↔ ∅ ∈ On)
51, 4mpbir 134 . 2 𝑥(𝑥 = ∅ ∧ 𝑥 ∈ On)
6 exsimpr 1509 . 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 1243  ∃wex 1381   ∈ wcel 1393  ∅c0 3224  Oncon0 4100 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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-nul 3883 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-dif 2920  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-uni 3581  df-tr 3855  df-iord 4103  df-on 4105 This theorem is referenced by: (None)
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