Theorem ord1eln01 8423

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 4293	. . 3 ⊢ (1o ∈ 𝐴 → 𝐴 ≠ ∅)
2		1on 8409	. . . . . 6 ⊢ 1o ∈ On
3	2	onirri 6431	. . . . 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 8422	. . . . . . 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 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 6430	. . . 4 ⊢ Ord 1o
17		ordtri2 6352	. . . 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 1541   ∈ wcel 2113   ≠ wne 2932  ∅c0 4285  Ord word 6316  1_oc1o 8390

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-tr 5206  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-ord 6320  df-on 6321  df-suc 6323  df-1o 8397

This theorem is referenced by:  1ellim  8425