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Theorem dif20el 8543
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 8540 . . 3 (𝐴 ∈ (On ∖ 2o) ↔ (𝐴 ∈ On ∧ 1o𝐴))
21simprbi 496 . 2 (𝐴 ∈ (On ∖ 2o) → 1o𝐴)
3 0lt1o 8542 . . 3 ∅ ∈ 1o
4 eldifi 4131 . . . 4 (𝐴 ∈ (On ∖ 2o) → 𝐴 ∈ On)
5 ontr1 6430 . . . 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 2108  cdif 3948  c0 4333  Oncon0 6384  1oc1o 8499  2oc2o 8500
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-tr 5260  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-ord 6387  df-on 6388  df-suc 6390  df-1o 8506  df-2o 8507
This theorem is referenced by:  oeordi  8625  oeworde  8631  oelimcl  8638  oeeulem  8639  oeeui  8640  cantnfresb  43337
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