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Theorem ordnexbtwnsuc 43280
Description: For any distinct pair of ordinals, if there is no ordinal between the lesser and the greater, the greater is the successor of the lesser. Lemma 1.16 of [Schloeder] p. 2. (Contributed by RP, 16-Jan-2025.)
Assertion
Ref Expression
ordnexbtwnsuc ((𝐴𝐵 ∧ Ord 𝐵) → (∀𝑐 ∈ On ¬ (𝐴𝑐𝑐𝐵) → 𝐵 = suc 𝐴))
Distinct variable groups:   𝐴,𝑐   𝐵,𝑐

Proof of Theorem ordnexbtwnsuc
StepHypRef Expression
1 ordelord 6406 . . . . . 6 ((Ord 𝐵𝐴𝐵) → Ord 𝐴)
2 ordnbtwn 6477 . . . . . . . . . 10 (Ord 𝐴 → ¬ (𝐴𝐵𝐵 ∈ suc 𝐴))
32pm2.21d 121 . . . . . . . . 9 (Ord 𝐴 → ((𝐴𝐵𝐵 ∈ suc 𝐴) → ∃𝑐 ∈ On (𝐴𝑐𝑐𝐵)))
43expd 415 . . . . . . . 8 (Ord 𝐴 → (𝐴𝐵 → (𝐵 ∈ suc 𝐴 → ∃𝑐 ∈ On (𝐴𝑐𝑐𝐵))))
54com12 32 . . . . . . 7 (𝐴𝐵 → (Ord 𝐴 → (𝐵 ∈ suc 𝐴 → ∃𝑐 ∈ On (𝐴𝑐𝑐𝐵))))
65adantl 481 . . . . . 6 ((Ord 𝐵𝐴𝐵) → (Ord 𝐴 → (𝐵 ∈ suc 𝐴 → ∃𝑐 ∈ On (𝐴𝑐𝑐𝐵))))
71, 6mpd 15 . . . . 5 ((Ord 𝐵𝐴𝐵) → (𝐵 ∈ suc 𝐴 → ∃𝑐 ∈ On (𝐴𝑐𝑐𝐵)))
8 sucidg 6465 . . . . . . 7 (𝐴𝐵𝐴 ∈ suc 𝐴)
98adantl 481 . . . . . 6 ((Ord 𝐵𝐴𝐵) → 𝐴 ∈ suc 𝐴)
10 ordelon 6408 . . . . . . . 8 ((Ord 𝐵𝐴𝐵) → 𝐴 ∈ On)
11 onsuc 7831 . . . . . . . 8 (𝐴 ∈ On → suc 𝐴 ∈ On)
1210, 11syl 17 . . . . . . 7 ((Ord 𝐵𝐴𝐵) → suc 𝐴 ∈ On)
13 eleq2 2830 . . . . . . . . 9 (𝑐 = suc 𝐴 → (𝐴𝑐𝐴 ∈ suc 𝐴))
14 eleq1 2829 . . . . . . . . 9 (𝑐 = suc 𝐴 → (𝑐𝐵 ↔ suc 𝐴𝐵))
1513, 14anbi12d 632 . . . . . . . 8 (𝑐 = suc 𝐴 → ((𝐴𝑐𝑐𝐵) ↔ (𝐴 ∈ suc 𝐴 ∧ suc 𝐴𝐵)))
1615adantl 481 . . . . . . 7 (((Ord 𝐵𝐴𝐵) ∧ 𝑐 = suc 𝐴) → ((𝐴𝑐𝑐𝐵) ↔ (𝐴 ∈ suc 𝐴 ∧ suc 𝐴𝐵)))
1712, 16rspcedv 3615 . . . . . 6 ((Ord 𝐵𝐴𝐵) → ((𝐴 ∈ suc 𝐴 ∧ suc 𝐴𝐵) → ∃𝑐 ∈ On (𝐴𝑐𝑐𝐵)))
189, 17mpand 695 . . . . 5 ((Ord 𝐵𝐴𝐵) → (suc 𝐴𝐵 → ∃𝑐 ∈ On (𝐴𝑐𝑐𝐵)))
197, 18jaod 860 . . . 4 ((Ord 𝐵𝐴𝐵) → ((𝐵 ∈ suc 𝐴 ∨ suc 𝐴𝐵) → ∃𝑐 ∈ On (𝐴𝑐𝑐𝐵)))
20 ralnex 3072 . . . . 5 (∀𝑐 ∈ On ¬ (𝐴𝑐𝑐𝐵) ↔ ¬ ∃𝑐 ∈ On (𝐴𝑐𝑐𝐵))
2120biimpi 216 . . . 4 (∀𝑐 ∈ On ¬ (𝐴𝑐𝑐𝐵) → ¬ ∃𝑐 ∈ On (𝐴𝑐𝑐𝐵))
2219, 21nsyli 157 . . 3 ((Ord 𝐵𝐴𝐵) → (∀𝑐 ∈ On ¬ (𝐴𝑐𝑐𝐵) → ¬ (𝐵 ∈ suc 𝐴 ∨ suc 𝐴𝐵)))
23 ordsuci 7828 . . . . 5 (Ord 𝐴 → Ord suc 𝐴)
241, 23syl 17 . . . 4 ((Ord 𝐵𝐴𝐵) → Ord suc 𝐴)
25 ordtri3 6420 . . . 4 ((Ord 𝐵 ∧ Ord suc 𝐴) → (𝐵 = suc 𝐴 ↔ ¬ (𝐵 ∈ suc 𝐴 ∨ suc 𝐴𝐵)))
2624, 25syldan 591 . . 3 ((Ord 𝐵𝐴𝐵) → (𝐵 = suc 𝐴 ↔ ¬ (𝐵 ∈ suc 𝐴 ∨ suc 𝐴𝐵)))
2722, 26sylibrd 259 . 2 ((Ord 𝐵𝐴𝐵) → (∀𝑐 ∈ On ¬ (𝐴𝑐𝑐𝐵) → 𝐵 = suc 𝐴))
2827ancoms 458 1 ((𝐴𝐵 ∧ Ord 𝐵) → (∀𝑐 ∈ On ¬ (𝐴𝑐𝑐𝐵) → 𝐵 = suc 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wo 848   = wceq 1540  wcel 2108  wral 3061  wrex 3070  Ord word 6383  Oncon0 6384  suc csuc 6386
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-tr 5260  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-ord 6387  df-on 6388  df-suc 6390
This theorem is referenced by:  orddif0suc  43281
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