Theorem dif20el 8415

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 8412	. . 3 ⊢ (𝐴 ∈ (On ∖ 2o) ↔ (𝐴 ∈ On ∧ 1o ∈ 𝐴))
2	1	simprbi 496	. 2 ⊢ (𝐴 ∈ (On ∖ 2o) → 1o ∈ 𝐴)
3		0lt1o 8414	. . 3 ⊢ ∅ ∈ 1o
4		eldifi 4079	. . . 4 ⊢ (𝐴 ∈ (On ∖ 2o) → 𝐴 ∈ On)
5		ontr1 6349	. . . 4 ⊢ (𝐴 ∈ On → ((∅ ∈ 1o ∧ 1o ∈ 𝐴) → ∅ ∈ 𝐴))
6	4, 5	syl 17	. . 3 ⊢ (𝐴 ∈ (On ∖ 2o) → ((∅ ∈ 1o ∧ 1o ∈ 𝐴) → ∅ ∈ 𝐴))
7	3, 6	mpani 696	. 2 ⊢ (𝐴 ∈ (On ∖ 2o) → (1o ∈ 𝐴 → ∅ ∈ 𝐴))
8	2, 7	mpd 15	1 ⊢ (𝐴 ∈ (On ∖ 2o) → ∅ ∈ 𝐴)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2110   ∖ cdif 3897  ∅c0 4281  Oncon0 6302  1_oc1o 8373  2_oc2o 8374

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-ext 2702  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2722  df-clel 2804  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3394  df-v 3436  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-pss 3920  df-nul 4282  df-if 4474  df-pw 4550  df-sn 4575  df-pr 4577  df-op 4581  df-uni 4858  df-br 5090  df-opab 5152  df-tr 5197  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-ord 6305  df-on 6306  df-suc 6308  df-1o 8380  df-2o 8381

This theorem is referenced by:  oeordi  8497  oeworde  8503  oelimcl  8510  oeeulem  8511  oeeui  8512  cantnfresb  43336