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Theorem 1oequni2o 34531
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 8092 . . 3 2o = suc 1o
2 2on 8100 . . . 4 2o ∈ On
3 2on0 8102 . . . 4 2o ≠ ∅
4 2onn 8255 . . . . 5 2o ∈ ω
5 nnlim 7582 . . . . 5 (2o ∈ ω → ¬ Lim 2o)
64, 5ax-mp 5 . . . 4 ¬ Lim 2o
7 onsucuni3 34530 . . . 4 ((2o ∈ On ∧ 2o ≠ ∅ ∧ ¬ Lim 2o) → 2o = suc 2o)
82, 3, 6, 7mp3an 1452 . . 3 2o = suc 2o
91, 8eqtr3i 2843 . 2 suc 1o = suc 2o
10 1on 8098 . . 3 1o ∈ On
11 onuni 7497 . . . 4 (2o ∈ On → 2o ∈ On)
122, 11ax-mp 5 . . 3 2o ∈ On
13 suc11 6287 . . 3 ((1o ∈ On ∧ 2o ∈ On) → (suc 1o = suc 2o ↔ 1o = 2o))
1410, 12, 13mp2an 688 . 2 (suc 1o = suc 2o ↔ 1o = 2o)
159, 14mpbi 231 1 1o = 2o
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 207   = wceq 1528  wcel 2105  wne 3013  c0 4288   cuni 4830  Oncon0 6184  Lim wlim 6185  suc csuc 6186  ωcom 7569  1oc1o 8084  2oc2o 8085
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-tr 5164  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-om 7570  df-1o 8091  df-2o 8092
This theorem is referenced by:  finxpreclem4  34557
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