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Theorem dif20el 7869
Description: An ordinal greater than one is greater than zero. (Contributed by Mario Carneiro, 25-May-2015.)
Assertion
Ref Expression
dif20el (𝐴 ∈ (On ∖ 2o) → ∅ ∈ 𝐴)

Proof of Theorem dif20el
StepHypRef Expression
1 ondif2 7866 . . 3 (𝐴 ∈ (On ∖ 2o) ↔ (𝐴 ∈ On ∧ 1o𝐴))
21simprbi 492 . 2 (𝐴 ∈ (On ∖ 2o) → 1o𝐴)
3 0lt1o 7868 . . 3 ∅ ∈ 1o
4 eldifi 3954 . . . 4 (𝐴 ∈ (On ∖ 2o) → 𝐴 ∈ On)
5 ontr1 6022 . . . 4 (𝐴 ∈ On → ((∅ ∈ 1o ∧ 1o𝐴) → ∅ ∈ 𝐴))
64, 5syl 17 . . 3 (𝐴 ∈ (On ∖ 2o) → ((∅ ∈ 1o ∧ 1o𝐴) → ∅ ∈ 𝐴))
73, 6mpani 686 . 2 (𝐴 ∈ (On ∖ 2o) → (1o𝐴 → ∅ ∈ 𝐴))
82, 7mpd 15 1 (𝐴 ∈ (On ∖ 2o) → ∅ ∈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386  wcel 2106  cdif 3788  c0 4140  Oncon0 5976  1oc1o 7836  2oc2o 7837
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-8 2108  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-sep 5017  ax-nul 5025  ax-pr 5138  ax-un 7226
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ne 2969  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3399  df-sbc 3652  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-pss 3807  df-nul 4141  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-tp 4402  df-op 4404  df-uni 4672  df-br 4887  df-opab 4949  df-tr 4988  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-we 5316  df-ord 5979  df-on 5980  df-suc 5982  df-1o 7843  df-2o 7844
This theorem is referenced by:  oeordi  7951  oeworde  7957  oelimcl  7964  oeeulem  7965  oeeui  7966
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