Theorem ord1eln01 8516

Description: An ordinal that is not 0 or 1 contains 1. (Contributed by BTernaryTau, 1-Dec-2024.)

Assertion

Ref	Expression
ord1eln01	⊢ (Ord 𝐴 → (1o ∈ 𝐴 ↔ (𝐴 ≠ ∅ ∧ 𝐴 ≠ 1o)))

Proof of Theorem ord1eln01

Step	Hyp	Ref	Expression
1		ne0i 4321	. . 3 ⊢ (1o ∈ 𝐴 → 𝐴 ≠ ∅)
2		1on 8500	. . . . . 6 ⊢ 1o ∈ On
3	2	onirri 6477	. . . . 5 ⊢ ¬ 1o ∈ 1o
4		eleq2 2822	. . . . 5 ⊢ (𝐴 = 1o → (1o ∈ 𝐴 ↔ 1o ∈ 1o))
5	3, 4	mtbiri 327	. . . 4 ⊢ (𝐴 = 1o → ¬ 1o ∈ 𝐴)
6	5	necon2ai 2960	. . 3 ⊢ (1o ∈ 𝐴 → 𝐴 ≠ 1o)
7	1, 6	jca 511	. 2 ⊢ (1o ∈ 𝐴 → (𝐴 ≠ ∅ ∧ 𝐴 ≠ 1o))
8		el1o 8515	. . . . . . 7 ⊢ (𝐴 ∈ 1o ↔ 𝐴 = ∅)
9	8	biimpi 216	. . . . . 6 ⊢ (𝐴 ∈ 1o → 𝐴 = ∅)
10	9	necon3ai 2956	. . . . 5 ⊢ (𝐴 ≠ ∅ → ¬ 𝐴 ∈ 1o)
11		nesym 2987	. . . . . 6 ⊢ (𝐴 ≠ 1o ↔ ¬ 1o = 𝐴)
12	11	biimpi 216	. . . . 5 ⊢ (𝐴 ≠ 1o → ¬ 1o = 𝐴)
13	10, 12	anim12ci 614	. . . 4 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ≠ 1o) → (¬ 1o = 𝐴 ∧ ¬ 𝐴 ∈ 1o))
14		pm4.56 990	. . . 4 ⊢ ((¬ 1o = 𝐴 ∧ ¬ 𝐴 ∈ 1o) ↔ ¬ (1o = 𝐴 ∨ 𝐴 ∈ 1o))
15	13, 14	sylib 218	. . 3 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ≠ 1o) → ¬ (1o = 𝐴 ∨ 𝐴 ∈ 1o))
16	2	onordi 6475	. . . 4 ⊢ Ord 1o
17		ordtri2 6398	. . . 4 ⊢ ((Ord 1o ∧ Ord 𝐴) → (1o ∈ 𝐴 ↔ ¬ (1o = 𝐴 ∨ 𝐴 ∈ 1o)))
18	16, 17	mpan 690	. . 3 ⊢ (Ord 𝐴 → (1o ∈ 𝐴 ↔ ¬ (1o = 𝐴 ∨ 𝐴 ∈ 1o)))
19	15, 18	imbitrrid 246	. 2 ⊢ (Ord 𝐴 → ((𝐴 ≠ ∅ ∧ 𝐴 ≠ 1o) → 1o ∈ 𝐴))
20	7, 19	impbid2 226	1 ⊢ (Ord 𝐴 → (1o ∈ 𝐴 ↔ (𝐴 ≠ ∅ ∧ 𝐴 ≠ 1o)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1539   ∈ wcel 2107   ≠ wne 2931  ∅c0 4313  Ord word 6362  1_oc1o 8481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2706  ax-sep 5276  ax-nul 5286  ax-pr 5412

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-ne 2932  df-ral 3051  df-rex 3060  df-rab 3420  df-v 3465  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4888  df-br 5124  df-opab 5186  df-tr 5240  df-eprel 5564  df-po 5572  df-so 5573  df-fr 5617  df-we 5619  df-ord 6366  df-on 6367  df-suc 6369  df-1o 8488

This theorem is referenced by:  1ellim  8518