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Theorem 1oequni2o 37336
Description: The ordinal number 1o is the predecessor of the ordinal number 2o. (Contributed by ML, 19-Oct-2020.)
Assertion
Ref Expression
1oequni2o 1o = 2o

Proof of Theorem 1oequni2o
StepHypRef Expression
1 df-2o 8525 . . 3 2o = suc 1o
2 2on 8538 . . . 4 2o ∈ On
3 2on0 8540 . . . 4 2o ≠ ∅
4 2onn 8700 . . . . 5 2o ∈ ω
5 nnlim 7919 . . . . 5 (2o ∈ ω → ¬ Lim 2o)
64, 5ax-mp 5 . . . 4 ¬ Lim 2o
7 onsucuni3 37335 . . . 4 ((2o ∈ On ∧ 2o ≠ ∅ ∧ ¬ Lim 2o) → 2o = suc 2o)
82, 3, 6, 7mp3an 1461 . . 3 2o = suc 2o
91, 8eqtr3i 2770 . 2 suc 1o = suc 2o
10 1on 8536 . . 3 1o ∈ On
11 onuni 7826 . . . 4 (2o ∈ On → 2o ∈ On)
122, 11ax-mp 5 . . 3 2o ∈ On
13 suc11 6504 . . 3 ((1o ∈ On ∧ 2o ∈ On) → (suc 1o = suc 2o ↔ 1o = 2o))
1410, 12, 13mp2an 691 . 2 (suc 1o = suc 2o ↔ 1o = 2o)
159, 14mpbi 230 1 1o = 2o
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 206   = wceq 1537  wcel 2108  wne 2946  c0 4352   cuni 4931  Oncon0 6397  Lim wlim 6398  suc csuc 6399  ωcom 7905  1oc1o 8517  2oc2o 8518
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7772
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-tr 5284  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-ord 6400  df-on 6401  df-lim 6402  df-suc 6403  df-om 7906  df-1o 8524  df-2o 8525
This theorem is referenced by:  finxpreclem4  37362
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