Theorem dif20el 8513

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 8510	. . 3 ⊢ (𝐴 ∈ (On ∖ 2o) ↔ (𝐴 ∈ On ∧ 1o ∈ 𝐴))
2	1	simprbi 503	. 2 ⊢ (𝐴 ∈ (On ∖ 2o) → 1o ∈ 𝐴)
3		0lt1o 8512	. . 3 ⊢ ∅ ∈ 1o
4		eldifi 4078	. . . 4 ⊢ (𝐴 ∈ (On ∖ 2o) → 𝐴 ∈ On)
5		ontr1 6410	. . . 4 ⊢ (𝐴 ∈ On → ((∅ ∈ 1o ∧ 1o ∈ 𝐴) → ∅ ∈ 𝐴))
6	4, 5	syl 18	. . 3 ⊢ (𝐴 ∈ (On ∖ 2o) → ((∅ ∈ 1o ∧ 1o ∈ 𝐴) → ∅ ∈ 𝐴))
7	3, 6	mpani 709	. 2 ⊢ (𝐴 ∈ (On ∖ 2o) → (1o ∈ 𝐴 → ∅ ∈ 𝐴))
8	2, 7	mpd 16	1 ⊢ (𝐴 ∈ (On ∖ 2o) → ∅ ∈ 𝐴)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∈ wcel 2145   ∖ cdif 3896  ∅c0 4279  Oncon0 6362  1_oc1o 8469  2_oc2o 8470

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-tr 5213  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-ord 6365  df-on 6366  df-suc 6368  df-1o 8476  df-2o 8477

This theorem is used by:  oeordi  8596  oeworde  8602  oelimcl  8609  oeeulem  8610  oeeui  8611  cantnfresb  44325