Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  dif20el Structured version   Visualization version   GIF version

Theorem dif20el 7630
 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 7627 . . 3 (𝐴 ∈ (On ∖ 2𝑜) ↔ (𝐴 ∈ On ∧ 1𝑜𝐴))
21simprbi 479 . 2 (𝐴 ∈ (On ∖ 2𝑜) → 1𝑜𝐴)
3 0lt1o 7629 . . 3 ∅ ∈ 1𝑜
4 eldifi 3765 . . . 4 (𝐴 ∈ (On ∖ 2𝑜) → 𝐴 ∈ On)
5 ontr1 5809 . . . 4 (𝐴 ∈ On → ((∅ ∈ 1𝑜 ∧ 1𝑜𝐴) → ∅ ∈ 𝐴))
64, 5syl 17 . . 3 (𝐴 ∈ (On ∖ 2𝑜) → ((∅ ∈ 1𝑜 ∧ 1𝑜𝐴) → ∅ ∈ 𝐴))
73, 6mpani 712 . 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 383   ∈ wcel 2030   ∖ cdif 3604  ∅c0 3948  Oncon0 5761  1𝑜c1o 7598  2𝑜c2o 7599 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936  ax-un 6991 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-tr 4786  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-ord 5764  df-on 5765  df-suc 5767  df-1o 7605  df-2o 7606 This theorem is referenced by:  oeordi  7712  oeworde  7718  oelimcl  7725  oeeulem  7726  oeeui  7727
 Copyright terms: Public domain W3C validator