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Theorem 1oequni2o 32887
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 7521 . . 3 2𝑜 = suc 1𝑜
2 2on 7528 . . . 4 2𝑜 ∈ On
3 2on0 7529 . . . 4 2𝑜 ≠ ∅
4 2onn 7680 . . . . 5 2𝑜 ∈ ω
5 nnlim 7040 . . . . 5 (2𝑜 ∈ ω → ¬ Lim 2𝑜)
64, 5ax-mp 5 . . . 4 ¬ Lim 2𝑜
7 onsucuni3 32886 . . . 4 ((2𝑜 ∈ On ∧ 2𝑜 ≠ ∅ ∧ ¬ Lim 2𝑜) → 2𝑜 = suc 2𝑜)
82, 3, 6, 7mp3an 1421 . . 3 2𝑜 = suc 2𝑜
91, 8eqtr3i 2645 . 2 suc 1𝑜 = suc 2𝑜
10 suc11reg 8476 . 2 (suc 1𝑜 = suc 2𝑜 ↔ 1𝑜 = 2𝑜)
119, 10mpbi 220 1 1𝑜 = 2𝑜
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1480  wcel 1987  wne 2790  c0 3897   cuni 4409  Oncon0 5692  Lim wlim 5693  suc csuc 5694  ωcom 7027  1𝑜c1o 7513  2𝑜c2o 7514
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pr 4877  ax-un 6914  ax-reg 8457
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-tr 4723  df-eprel 4995  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-om 7028  df-1o 7520  df-2o 7521
This theorem is referenced by:  finxpreclem4  32902
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