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Theorem dif20el 7545
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 7542 . . 3 (𝐴 ∈ (On ∖ 2𝑜) ↔ (𝐴 ∈ On ∧ 1𝑜𝐴))
21simprbi 480 . 2 (𝐴 ∈ (On ∖ 2𝑜) → 1𝑜𝐴)
3 0lt1o 7544 . . 3 ∅ ∈ 1𝑜
4 eldifi 3716 . . . 4 (𝐴 ∈ (On ∖ 2𝑜) → 𝐴 ∈ On)
5 ontr1 5740 . . . 4 (𝐴 ∈ On → ((∅ ∈ 1𝑜 ∧ 1𝑜𝐴) → ∅ ∈ 𝐴))
64, 5syl 17 . . 3 (𝐴 ∈ (On ∖ 2𝑜) → ((∅ ∈ 1𝑜 ∧ 1𝑜𝐴) → ∅ ∈ 𝐴))
73, 6mpani 711 . 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 384  wcel 1987  cdif 3557  c0 3897  Oncon0 5692  1𝑜c1o 7513  2𝑜c2o 7514
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-tr 4723  df-eprel 4995  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-ord 5695  df-on 5696  df-suc 5698  df-1o 7520  df-2o 7521
This theorem is referenced by:  oeordi  7627  oeworde  7633  oelimcl  7640  oeeulem  7641  oeeui  7642
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