Theorem dif20el 7472

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 7469	. . 3 ⊢ (𝐴 ∈ (On ∖ 2𝑜) ↔ (𝐴 ∈ On ∧ 1𝑜 ∈ 𝐴))
2	1	simprbi 479	. 2 ⊢ (𝐴 ∈ (On ∖ 2𝑜) → 1𝑜 ∈ 𝐴)
3		0lt1o 7471	. . 3 ⊢ ∅ ∈ 1𝑜
4		eldifi 3694	. . . 4 ⊢ (𝐴 ∈ (On ∖ 2𝑜) → 𝐴 ∈ On)
5		ontr1 5688	. . . 4 ⊢ (𝐴 ∈ On → ((∅ ∈ 1𝑜 ∧ 1𝑜 ∈ 𝐴) → ∅ ∈ 𝐴))
6	4, 5	syl 17	. . 3 ⊢ (𝐴 ∈ (On ∖ 2𝑜) → ((∅ ∈ 1𝑜 ∧ 1𝑜 ∈ 𝐴) → ∅ ∈ 𝐴))
7	3, 6	mpani 708	. 2 ⊢ (𝐴 ∈ (On ∖ 2𝑜) → (1𝑜 ∈ 𝐴 → ∅ ∈ 𝐴))
8	2, 7	mpd 15	1 ⊢ (𝐴 ∈ (On ∖ 2𝑜) → ∅ ∈ 𝐴)

Syntax hints:   → wi 4   ∧ wa 383   ∈ wcel 1977   ∖ cdif 3537  ∅c0 3874  Oncon0 5640  1_𝑜c1o 7440  2_𝑜c2o 7441

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-tr 4681  df-eprel 4949  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-ord 5643  df-on 5644  df-suc 5646  df-1o 7447  df-2o 7448

This theorem is referenced by:  oeordi  7554  oeworde  7560  oelimcl  7567  oeeulem  7568  oeeui  7569