Theorem ondif2 8510

Description: Two ways to say that 𝐴 is an ordinal greater than one. (Contributed by Mario Carneiro, 21-May-2015.)

Assertion

Ref	Expression
ondif2	⊢ (𝐴 ∈ (On ∖ 2o) ↔ (𝐴 ∈ On ∧ 1o ∈ 𝐴))

Proof of Theorem ondif2

Step	Hyp	Ref	Expression
1		eldif 3909	. 2 ⊢ (𝐴 ∈ (On ∖ 2o) ↔ (𝐴 ∈ On ∧ ¬ 𝐴 ∈ 2o))
2		1on 8489	. . . . 5 ⊢ 1o ∈ On
3		ontri1 6397	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 1o ∈ On) → (𝐴 ⊆ 1o ↔ ¬ 1o ∈ 𝐴))
4		onsssuc 6455	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 1o ∈ On) → (𝐴 ⊆ 1o ↔ 𝐴 ∈ suc 1o))
5		df-2o 8477	. . . . . . . 8 ⊢ 2o = suc 1o
6	5	eleq2i 2853	. . . . . . 7 ⊢ (𝐴 ∈ 2o ↔ 𝐴 ∈ suc 1o)
7	4, 6	bitr4di 292	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 1o ∈ On) → (𝐴 ⊆ 1o ↔ 𝐴 ∈ 2o))
8	3, 7	bitr3d 284	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 1o ∈ On) → (¬ 1o ∈ 𝐴 ↔ 𝐴 ∈ 2o))
9	2, 8	mpan2 704	. . . 4 ⊢ (𝐴 ∈ On → (¬ 1o ∈ 𝐴 ↔ 𝐴 ∈ 2o))
10	9	con1bid 358	. . 3 ⊢ (𝐴 ∈ On → (¬ 𝐴 ∈ 2o ↔ 1o ∈ 𝐴))
11	10	pm5.32i 585	. 2 ⊢ ((𝐴 ∈ On ∧ ¬ 𝐴 ∈ 2o) ↔ (𝐴 ∈ On ∧ 1o ∈ 𝐴))
12	1, 11	bitri 278	1 ⊢ (𝐴 ∈ (On ∖ 2o) ↔ (𝐴 ∈ On ∧ 1o ∈ 𝐴))

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209   ∧ wa 401   ∈ wcel 2145   ∖ cdif 3896   ⊆ wss 3899  Oncon0 6362  suc csuc 6364  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:  dif20el  8513  oeordi  8596  oewordi  8600  oaabs2  8658  omabs  8660  cnfcom3clem  9706  infxpenc2lem1  10098  onexoegt  44245  oege2  44308  rp-oelim2  44309  oeord2lim  44310  oeord2i  44311  oeord2com  44312  nnoeomeqom  44313  oenord1  44317  cantnftermord  44321  cantnfresb  44325  cantnf2  44326  omabs2  44333  omcl2  44334