Theorem ord1eln01 8431

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 4281	. . 3 ⊢ (1o ∈ 𝐴 → 𝐴 ≠ ∅)
2		1on 8417	. . . . . 6 ⊢ 1o ∈ On
3	2	onirri 6437	. . . . 5 ⊢ ¬ 1o ∈ 1o
4		eleq2 2825	. . . . 5 ⊢ (𝐴 = 1o → (1o ∈ 𝐴 ↔ 1o ∈ 1o))
5	3, 4	mtbiri 327	. . . 4 ⊢ (𝐴 = 1o → ¬ 1o ∈ 𝐴)
6	5	necon2ai 2961	. . 3 ⊢ (1o ∈ 𝐴 → 𝐴 ≠ 1o)
7	1, 6	jca 511	. 2 ⊢ (1o ∈ 𝐴 → (𝐴 ≠ ∅ ∧ 𝐴 ≠ 1o))
8		el1o 8430	. . . . . . 7 ⊢ (𝐴 ∈ 1o ↔ 𝐴 = ∅)
9	8	biimpi 216	. . . . . 6 ⊢ (𝐴 ∈ 1o → 𝐴 = ∅)
10	9	necon3ai 2957	. . . . 5 ⊢ (𝐴 ≠ ∅ → ¬ 𝐴 ∈ 1o)
11		nesym 2988	. . . . . 6 ⊢ (𝐴 ≠ 1o ↔ ¬ 1o = 𝐴)
12	11	biimpi 216	. . . . 5 ⊢ (𝐴 ≠ 1o → ¬ 1o = 𝐴)
13	10, 12	anim12ci 615	. . . 4 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ≠ 1o) → (¬ 1o = 𝐴 ∧ ¬ 𝐴 ∈ 1o))
14		pm4.56 991	. . . 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 6436	. . . 4 ⊢ Ord 1o
17		ordtri2 6358	. . . 4 ⊢ ((Ord 1o ∧ Ord 𝐴) → (1o ∈ 𝐴 ↔ ¬ (1o = 𝐴 ∨ 𝐴 ∈ 1o)))
18	16, 17	mpan 691	. . 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 848   = wceq 1542   ∈ wcel 2114   ≠ wne 2932  ∅c0 4273  Ord word 6322  1_oc1o 8398

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-tr 5193  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-ord 6326  df-on 6327  df-suc 6329  df-1o 8405

This theorem is referenced by:  1ellim  8433