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Theorem 1oequni2o 34648
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 8102 . . 3 2o = suc 1o
2 2on 8110 . . . 4 2o ∈ On
3 2on0 8112 . . . 4 2o ≠ ∅
4 2onn 8265 . . . . 5 2o ∈ ω
5 nnlim 7592 . . . . 5 (2o ∈ ω → ¬ Lim 2o)
64, 5ax-mp 5 . . . 4 ¬ Lim 2o
7 onsucuni3 34647 . . . 4 ((2o ∈ On ∧ 2o ≠ ∅ ∧ ¬ Lim 2o) → 2o = suc 2o)
82, 3, 6, 7mp3an 1457 . . 3 2o = suc 2o
91, 8eqtr3i 2846 . 2 suc 1o = suc 2o
10 1on 8108 . . 3 1o ∈ On
11 onuni 7507 . . . 4 (2o ∈ On → 2o ∈ On)
122, 11ax-mp 5 . . 3 2o ∈ On
13 suc11 6293 . . 3 ((1o ∈ On ∧ 2o ∈ On) → (suc 1o = suc 2o ↔ 1o = 2o))
1410, 12, 13mp2an 690 . 2 (suc 1o = suc 2o ↔ 1o = 2o)
159, 14mpbi 232 1 1o = 2o
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 208   = wceq 1533  wcel 2110  wne 3016  c0 4290   cuni 4837  Oncon0 6190  Lim wlim 6191  suc csuc 6192  ωcom 7579  1oc1o 8094  2oc2o 8095
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pr 5329  ax-un 7460
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-tp 4571  df-op 4573  df-uni 4838  df-br 5066  df-opab 5128  df-tr 5172  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-ord 6193  df-on 6194  df-lim 6195  df-suc 6196  df-om 7580  df-1o 8101  df-2o 8102
This theorem is referenced by:  finxpreclem4  34674
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