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

Step	Hyp	Ref	Expression
1		ondif2 7627	. . 3 ⊢ (𝐴 ∈ (On ∖ 2𝑜) ↔ (𝐴 ∈ On ∧ 1𝑜 ∈ 𝐴))
2	1	simprbi 479	. 2 ⊢ (𝐴 ∈ (On ∖ 2𝑜) → 1𝑜 ∈ 𝐴)
3		0lt1o 7629	. . 3 ⊢ ∅ ∈ 1𝑜
4		eldifi 3765	. . . 4 ⊢ (𝐴 ∈ (On ∖ 2𝑜) → 𝐴 ∈ On)
5		ontr1 5809	. . . 4 ⊢ (𝐴 ∈ On → ((∅ ∈ 1𝑜 ∧ 1𝑜 ∈ 𝐴) → ∅ ∈ 𝐴))
6	4, 5	syl 17	. . 3 ⊢ (𝐴 ∈ (On ∖ 2𝑜) → ((∅ ∈ 1𝑜 ∧ 1𝑜 ∈ 𝐴) → ∅ ∈ 𝐴))
7	3, 6	mpani 712	. 2 ⊢ (𝐴 ∈ (On ∖ 2𝑜) → (1𝑜 ∈ 𝐴 → ∅ ∈ 𝐴))
8	2, 7	mpd 15	1 ⊢ (𝐴 ∈ (On ∖ 2𝑜) → ∅ ∈ 𝐴)

Syntax hints:   → wi 4   ∧ wa 383   ∈ wcel 2030   ∖ cdif 3604  ∅c0 3948  Oncon0 5761  1_𝑜c1o 7598  2_𝑜c2o 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