Theorem dif20el 8446

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 8443	. . 3 ⊢ (𝐴 ∈ (On ∖ 2o) ↔ (𝐴 ∈ On ∧ 1o ∈ 𝐴))
2	1	simprbi 496	. 2 ⊢ (𝐴 ∈ (On ∖ 2o) → 1o ∈ 𝐴)
3		0lt1o 8445	. . 3 ⊢ ∅ ∈ 1o
4		eldifi 4090	. . . 4 ⊢ (𝐴 ∈ (On ∖ 2o) → 𝐴 ∈ On)
5		ontr1 6367	. . . 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 2109   ∖ cdif 3908  ∅c0 4292  Oncon0 6320  1_oc1o 8404  2_oc2o 8405

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-tr 5210  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-ord 6323  df-on 6324  df-suc 6326  df-1o 8411  df-2o 8412

This theorem is referenced by:  oeordi  8528  oeworde  8534  oelimcl  8541  oeeulem  8542  oeeui  8543  cantnfresb  43286