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Theorem dif20el 8541
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 8538 . . 3 (𝐴 ∈ (On ∖ 2o) ↔ (𝐴 ∈ On ∧ 1o𝐴))
21simprbi 496 . 2 (𝐴 ∈ (On ∖ 2o) → 1o𝐴)
3 0lt1o 8540 . . 3 ∅ ∈ 1o
4 eldifi 4140 . . . 4 (𝐴 ∈ (On ∖ 2o) → 𝐴 ∈ On)
5 ontr1 6431 . . . 4 (𝐴 ∈ On → ((∅ ∈ 1o ∧ 1o𝐴) → ∅ ∈ 𝐴))
64, 5syl 17 . . 3 (𝐴 ∈ (On ∖ 2o) → ((∅ ∈ 1o ∧ 1o𝐴) → ∅ ∈ 𝐴))
73, 6mpani 696 . 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 395  wcel 2105  cdif 3959  c0 4338  Oncon0 6385  1oc1o 8497  2oc2o 8498
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-tr 5265  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-ord 6388  df-on 6389  df-suc 6391  df-1o 8504  df-2o 8505
This theorem is referenced by:  oeordi  8623  oeworde  8629  oelimcl  8636  oeeulem  8637  oeeui  8638  cantnfresb  43313
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