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Theorem dif20el 8117
 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 8114 . . 3 (𝐴 ∈ (On ∖ 2o) ↔ (𝐴 ∈ On ∧ 1o𝐴))
21simprbi 500 . 2 (𝐴 ∈ (On ∖ 2o) → 1o𝐴)
3 0lt1o 8116 . . 3 ∅ ∈ 1o
4 eldifi 4078 . . . 4 (𝐴 ∈ (On ∖ 2o) → 𝐴 ∈ On)
5 ontr1 6215 . . . 4 (𝐴 ∈ On → ((∅ ∈ 1o ∧ 1o𝐴) → ∅ ∈ 𝐴))
64, 5syl 17 . . 3 (𝐴 ∈ (On ∖ 2o) → ((∅ ∈ 1o ∧ 1o𝐴) → ∅ ∈ 𝐴))
73, 6mpani 695 . 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 399   ∈ wcel 2114   ∖ cdif 3905  ∅c0 4265  Oncon0 6169  1oc1o 8082  2oc2o 8083 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pr 5307  ax-un 7446 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-tp 4544  df-op 4546  df-uni 4814  df-br 5043  df-opab 5105  df-tr 5149  df-eprel 5442  df-po 5451  df-so 5452  df-fr 5491  df-we 5493  df-ord 6172  df-on 6173  df-suc 6175  df-1o 8089  df-2o 8090 This theorem is referenced by:  oeordi  8200  oeworde  8206  oelimcl  8213  oeeulem  8214  oeeui  8215
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