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Theorem faosnf0.11b 41773
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 𝐴𝐴 ≠ ∅) → ∃𝑥 ∈ On 𝐴 = suc 𝑥)
Distinct variable group:   𝑥,𝐴

Proof of Theorem faosnf0.11b
StepHypRef Expression
1 3ancomb 1100 . . 3 ((Ord 𝐴 ∧ ¬ Lim 𝐴𝐴 ≠ ∅) ↔ (Ord 𝐴𝐴 ≠ ∅ ∧ ¬ Lim 𝐴))
2 df-3an 1090 . . 3 ((Ord 𝐴𝐴 ≠ ∅ ∧ ¬ Lim 𝐴) ↔ ((Ord 𝐴𝐴 ≠ ∅) ∧ ¬ Lim 𝐴))
3 df-ne 2945 . . . . . . . 8 (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅)
43anbi2i 624 . . . . . . 7 ((Ord 𝐴𝐴 ≠ ∅) ↔ (Ord 𝐴 ∧ ¬ 𝐴 = ∅))
54imbi1i 350 . . . . . 6 (((Ord 𝐴𝐴 ≠ ∅) → ∃𝑥 ∈ On 𝐴 = suc 𝑥) ↔ ((Ord 𝐴 ∧ ¬ 𝐴 = ∅) → ∃𝑥 ∈ On 𝐴 = suc 𝑥))
6 pm5.6 1001 . . . . . 6 (((Ord 𝐴 ∧ ¬ 𝐴 = ∅) → ∃𝑥 ∈ On 𝐴 = suc 𝑥) ↔ (Ord 𝐴 → (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥)))
7 iman 403 . . . . . 6 ((Ord 𝐴 → (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥)) ↔ ¬ (Ord 𝐴 ∧ ¬ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥)))
85, 6, 73bitrri 298 . . . . 5 (¬ (Ord 𝐴 ∧ ¬ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥)) ↔ ((Ord 𝐴𝐴 ≠ ∅) → ∃𝑥 ∈ On 𝐴 = suc 𝑥))
9 dflim3 7788 . . . . 5 (Lim 𝐴 ↔ (Ord 𝐴 ∧ ¬ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥)))
108, 9xchnxbir 333 . . . 4 (¬ Lim 𝐴 ↔ ((Ord 𝐴𝐴 ≠ ∅) → ∃𝑥 ∈ On 𝐴 = suc 𝑥))
1110anbi2i 624 . . 3 (((Ord 𝐴𝐴 ≠ ∅) ∧ ¬ Lim 𝐴) ↔ ((Ord 𝐴𝐴 ≠ ∅) ∧ ((Ord 𝐴𝐴 ≠ ∅) → ∃𝑥 ∈ On 𝐴 = suc 𝑥)))
121, 2, 113bitri 297 . 2 ((Ord 𝐴 ∧ ¬ Lim 𝐴𝐴 ≠ ∅) ↔ ((Ord 𝐴𝐴 ≠ ∅) ∧ ((Ord 𝐴𝐴 ≠ ∅) → ∃𝑥 ∈ On 𝐴 = suc 𝑥)))
13 pm3.35 802 . 2 (((Ord 𝐴𝐴 ≠ ∅) ∧ ((Ord 𝐴𝐴 ≠ ∅) → ∃𝑥 ∈ On 𝐴 = suc 𝑥)) → ∃𝑥 ∈ On 𝐴 = suc 𝑥)
1412, 13sylbi 216 1 ((Ord 𝐴 ∧ ¬ Lim 𝐴𝐴 ≠ ∅) → ∃𝑥 ∈ On 𝐴 = suc 𝑥)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 397  wo 846  w3a 1088   = wceq 1542  wne 2944  wrex 3074  c0 4287  Ord word 6321  Oncon0 6322  Lim wlim 6323  suc csuc 6324
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-tr 5228  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328
This theorem is referenced by: (None)
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