Theorem dif20el 8537

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

Step	Hyp	Ref	Expression
1		ondif2 8534	. . 3 ⊢ (𝐴 ∈ (On ∖ 2o) ↔ (𝐴 ∈ On ∧ 1o ∈ 𝐴))
2	1	simprbi 495	. 2 ⊢ (𝐴 ∈ (On ∖ 2o) → 1o ∈ 𝐴)
3		0lt1o 8536	. . 3 ⊢ ∅ ∈ 1o
4		eldifi 4126	. . . 4 ⊢ (𝐴 ∈ (On ∖ 2o) → 𝐴 ∈ On)
5		ontr1 6424	. . . 4 ⊢ (𝐴 ∈ On → ((∅ ∈ 1o ∧ 1o ∈ 𝐴) → ∅ ∈ 𝐴))
6	4, 5	syl 17	. . 3 ⊢ (𝐴 ∈ (On ∖ 2o) → ((∅ ∈ 1o ∧ 1o ∈ 𝐴) → ∅ ∈ 𝐴))
7	3, 6	mpani 694	. 2 ⊢ (𝐴 ∈ (On ∖ 2o) → (1o ∈ 𝐴 → ∅ ∈ 𝐴))
8	2, 7	mpd 15	1 ⊢ (𝐴 ∈ (On ∖ 2o) → ∅ ∈ 𝐴)

Syntax hints:   → wi 4   ∧ wa 394   ∈ wcel 2099   ∖ cdif 3944  ∅c0 4325  Oncon0 6378  1_oc1o 8491  2_oc2o 8492

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697  ax-sep 5306  ax-nul 5313  ax-pr 5435

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4916  df-br 5156  df-opab 5218  df-tr 5273  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5639  df-we 5641  df-ord 6381  df-on 6382  df-suc 6384  df-1o 8498  df-2o 8499

This theorem is referenced by:  oeordi  8619  oeworde  8625  oelimcl  8632  oeeulem  8633  oeeui  8634  cantnfresb  43008