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Theorem ondif2 7854
Description: Two ways to say that 𝐴 is an ordinal greater than one. (Contributed by Mario Carneiro, 21-May-2015.)
Assertion
Ref Expression
ondif2 (𝐴 ∈ (On ∖ 2o) ↔ (𝐴 ∈ On ∧ 1o𝐴))

Proof of Theorem ondif2
StepHypRef Expression
1 eldif 3808 . 2 (𝐴 ∈ (On ∖ 2o) ↔ (𝐴 ∈ On ∧ ¬ 𝐴 ∈ 2o))
2 1on 7838 . . . . 5 1o ∈ On
3 ontri1 6001 . . . . . 6 ((𝐴 ∈ On ∧ 1o ∈ On) → (𝐴 ⊆ 1o ↔ ¬ 1o𝐴))
4 onsssuc 6054 . . . . . . 7 ((𝐴 ∈ On ∧ 1o ∈ On) → (𝐴 ⊆ 1o𝐴 ∈ suc 1o))
5 df-2o 7832 . . . . . . . 8 2o = suc 1o
65eleq2i 2898 . . . . . . 7 (𝐴 ∈ 2o𝐴 ∈ suc 1o)
74, 6syl6bbr 281 . . . . . 6 ((𝐴 ∈ On ∧ 1o ∈ On) → (𝐴 ⊆ 1o𝐴 ∈ 2o))
83, 7bitr3d 273 . . . . 5 ((𝐴 ∈ On ∧ 1o ∈ On) → (¬ 1o𝐴𝐴 ∈ 2o))
92, 8mpan2 682 . . . 4 (𝐴 ∈ On → (¬ 1o𝐴𝐴 ∈ 2o))
109con1bid 347 . . 3 (𝐴 ∈ On → (¬ 𝐴 ∈ 2o ↔ 1o𝐴))
1110pm5.32i 570 . 2 ((𝐴 ∈ On ∧ ¬ 𝐴 ∈ 2o) ↔ (𝐴 ∈ On ∧ 1o𝐴))
121, 11bitri 267 1 (𝐴 ∈ (On ∖ 2o) ↔ (𝐴 ∈ On ∧ 1o𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 198  wa 386  wcel 2164  cdif 3795  wss 3798  Oncon0 5967  suc csuc 5969  1oc1o 7824  2oc2o 7825
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pr 5129  ax-un 7214
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3or 1112  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-tp 4404  df-op 4406  df-uni 4661  df-br 4876  df-opab 4938  df-tr 4978  df-eprel 5257  df-po 5265  df-so 5266  df-fr 5305  df-we 5307  df-ord 5970  df-on 5971  df-suc 5973  df-1o 7831  df-2o 7832
This theorem is referenced by:  dif20el  7857  oeordi  7939  oewordi  7943  oaabs2  7997  omabs  7999  cnfcom3clem  8886  infxpenc2lem1  9162
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