Theorem ord1eln01 8497

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 4329	. . 3 ⊢ (1o ∈ 𝐴 → 𝐴 ≠ ∅)
2		1on 8479	. . . . . 6 ⊢ 1o ∈ On
3	2	onirri 6471	. . . . 5 ⊢ ¬ 1o ∈ 1o
4		eleq2 2816	. . . . 5 ⊢ (𝐴 = 1o → (1o ∈ 𝐴 ↔ 1o ∈ 1o))
5	3, 4	mtbiri 327	. . . 4 ⊢ (𝐴 = 1o → ¬ 1o ∈ 𝐴)
6	5	necon2ai 2964	. . 3 ⊢ (1o ∈ 𝐴 → 𝐴 ≠ 1o)
7	1, 6	jca 511	. 2 ⊢ (1o ∈ 𝐴 → (𝐴 ≠ ∅ ∧ 𝐴 ≠ 1o))
8		el1o 8496	. . . . . . 7 ⊢ (𝐴 ∈ 1o ↔ 𝐴 = ∅)
9	8	biimpi 215	. . . . . 6 ⊢ (𝐴 ∈ 1o → 𝐴 = ∅)
10	9	necon3ai 2959	. . . . 5 ⊢ (𝐴 ≠ ∅ → ¬ 𝐴 ∈ 1o)
11		nesym 2991	. . . . . 6 ⊢ (𝐴 ≠ 1o ↔ ¬ 1o = 𝐴)
12	11	biimpi 215	. . . . 5 ⊢ (𝐴 ≠ 1o → ¬ 1o = 𝐴)
13	10, 12	anim12ci 613	. . . 4 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ≠ 1o) → (¬ 1o = 𝐴 ∧ ¬ 𝐴 ∈ 1o))
14		pm4.56 985	. . . 4 ⊢ ((¬ 1o = 𝐴 ∧ ¬ 𝐴 ∈ 1o) ↔ ¬ (1o = 𝐴 ∨ 𝐴 ∈ 1o))
15	13, 14	sylib 217	. . 3 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ≠ 1o) → ¬ (1o = 𝐴 ∨ 𝐴 ∈ 1o))
16	2	onordi 6469	. . . 4 ⊢ Ord 1o
17		ordtri2 6393	. . . 4 ⊢ ((Ord 1o ∧ Ord 𝐴) → (1o ∈ 𝐴 ↔ ¬ (1o = 𝐴 ∨ 𝐴 ∈ 1o)))
18	16, 17	mpan 687	. . 3 ⊢ (Ord 𝐴 → (1o ∈ 𝐴 ↔ ¬ (1o = 𝐴 ∨ 𝐴 ∈ 1o)))
19	15, 18	imbitrrid 245	. 2 ⊢ (Ord 𝐴 → ((𝐴 ≠ ∅ ∧ 𝐴 ≠ 1o) → 1o ∈ 𝐴))
20	7, 19	impbid2 225	1 ⊢ (Ord 𝐴 → (1o ∈ 𝐴 ↔ (𝐴 ≠ ∅ ∧ 𝐴 ≠ 1o)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   ∨ wo 844   = wceq 1533   ∈ wcel 2098   ≠ wne 2934  ∅c0 4317  Ord word 6357  1_oc1o 8460

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-tr 5259  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-ord 6361  df-on 6362  df-suc 6364  df-1o 8467

This theorem is referenced by:  1ellim  8499