Theorem faosnf0.11b 43871

Description: 𝐵 is called a non-limit ordinal if it is not a limit ordinal. (Contributed by RP, 27-Sep-2023.)

Alling, Norman L. "Fundamentals of Analysis Over Surreal Numbers Fields." The Rocky Mountain Journal of Mathematics 19, no. 3 (1989): 565-73. http://www.jstor.org/stable/44237243.

Assertion

Ref	Expression
faosnf0.11b	⊢ ((Ord 𝐴 ∧ ¬ Lim 𝐴 ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ On 𝐴 = suc 𝑥)

Distinct variable group:   𝑥,𝐴

Proof of Theorem faosnf0.11b

Step	Hyp	Ref	Expression
1		3ancomb 1104	. . 3 ⊢ ((Ord 𝐴 ∧ ¬ Lim 𝐴 ∧ 𝐴 ≠ ∅) ↔ (Ord 𝐴 ∧ 𝐴 ≠ ∅ ∧ ¬ Lim 𝐴))
2		df-3an 1094	. . 3 ⊢ ((Ord 𝐴 ∧ 𝐴 ≠ ∅ ∧ ¬ Lim 𝐴) ↔ ((Ord 𝐴 ∧ 𝐴 ≠ ∅) ∧ ¬ Lim 𝐴))
3		df-ne 2935	. . . . . . . 8 ⊢ (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅)
4	3	anbi2i 629	. . . . . . 7 ⊢ ((Ord 𝐴 ∧ 𝐴 ≠ ∅) ↔ (Ord 𝐴 ∧ ¬ 𝐴 = ∅))
5	4	imbi1i 350	. . . . . 6 ⊢ (((Ord 𝐴 ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ On 𝐴 = suc 𝑥) ↔ ((Ord 𝐴 ∧ ¬ 𝐴 = ∅) → ∃𝑥 ∈ On 𝐴 = suc 𝑥))
6		pm5.6 1009	. . . . . 6 ⊢ (((Ord 𝐴 ∧ ¬ 𝐴 = ∅) → ∃𝑥 ∈ On 𝐴 = suc 𝑥) ↔ (Ord 𝐴 → (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥)))
7		iman 402	. . . . . 6 ⊢ ((Ord 𝐴 → (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥)) ↔ ¬ (Ord 𝐴 ∧ ¬ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥)))
8	5, 6, 7	3bitrri 299	. . . . 5 ⊢ (¬ (Ord 𝐴 ∧ ¬ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥)) ↔ ((Ord 𝐴 ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ On 𝐴 = suc 𝑥))
9		dflim3 7787	. . . . 5 ⊢ (Lim 𝐴 ↔ (Ord 𝐴 ∧ ¬ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥)))
10	8, 9	xchnxbir 334	. . . 4 ⊢ (¬ Lim 𝐴 ↔ ((Ord 𝐴 ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ On 𝐴 = suc 𝑥))
11	10	anbi2i 629	. . 3 ⊢ (((Ord 𝐴 ∧ 𝐴 ≠ ∅) ∧ ¬ Lim 𝐴) ↔ ((Ord 𝐴 ∧ 𝐴 ≠ ∅) ∧ ((Ord 𝐴 ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ On 𝐴 = suc 𝑥)))
12	1, 2, 11	3bitri 298	. 2 ⊢ ((Ord 𝐴 ∧ ¬ Lim 𝐴 ∧ 𝐴 ≠ ∅) ↔ ((Ord 𝐴 ∧ 𝐴 ≠ ∅) ∧ ((Ord 𝐴 ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ On 𝐴 = suc 𝑥)))
13		pm3.35 808	. 2 ⊢ (((Ord 𝐴 ∧ 𝐴 ≠ ∅) ∧ ((Ord 𝐴 ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ On 𝐴 = suc 𝑥)) → ∃𝑥 ∈ On 𝐴 = suc 𝑥)
14	12, 13	sylbi 218	1 ⊢ ((Ord 𝐴 ∧ ¬ Lim 𝐴 ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ On 𝐴 = suc 𝑥)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   ∨ wo 853   ∧ w3a 1092   = wceq 1547   ≠ wne 2934  ∃wrex 3063  ∅c0 4261  Ord word 6309  Oncon0 6310  Lim wlim 6311  suc csuc 6312

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-opab 5135  df-tr 5180  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-we 5573  df-ord 6313  df-on 6314  df-lim 6315  df-suc 6316

This theorem is referenced by: (None)