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Theorem ordnexbtwnsuc 41788
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 6375 . . . . . 6 ((Ord 𝐵𝐴𝐵) → Ord 𝐴)
2 ordnbtwn 6446 . . . . . . . . . 10 (Ord 𝐴 → ¬ (𝐴𝐵𝐵 ∈ suc 𝐴))
32pm2.21d 121 . . . . . . . . 9 (Ord 𝐴 → ((𝐴𝐵𝐵 ∈ suc 𝐴) → ∃𝑐 ∈ On (𝐴𝑐𝑐𝐵)))
43expd 416 . . . . . . . 8 (Ord 𝐴 → (𝐴𝐵 → (𝐵 ∈ suc 𝐴 → ∃𝑐 ∈ On (𝐴𝑐𝑐𝐵))))
54com12 32 . . . . . . 7 (𝐴𝐵 → (Ord 𝐴 → (𝐵 ∈ suc 𝐴 → ∃𝑐 ∈ On (𝐴𝑐𝑐𝐵))))
65adantl 482 . . . . . 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 6434 . . . . . . 7 (𝐴𝐵𝐴 ∈ suc 𝐴)
98adantl 482 . . . . . 6 ((Ord 𝐵𝐴𝐵) → 𝐴 ∈ suc 𝐴)
10 ordelon 6377 . . . . . . . 8 ((Ord 𝐵𝐴𝐵) → 𝐴 ∈ On)
11 onsuc 7782 . . . . . . . 8 (𝐴 ∈ On → suc 𝐴 ∈ On)
1210, 11syl 17 . . . . . . 7 ((Ord 𝐵𝐴𝐵) → suc 𝐴 ∈ On)
13 eleq2 2821 . . . . . . . . 9 (𝑐 = suc 𝐴 → (𝐴𝑐𝐴 ∈ suc 𝐴))
14 eleq1 2820 . . . . . . . . 9 (𝑐 = suc 𝐴 → (𝑐𝐵 ↔ suc 𝐴𝐵))
1513, 14anbi12d 631 . . . . . . . 8 (𝑐 = suc 𝐴 → ((𝐴𝑐𝑐𝐵) ↔ (𝐴 ∈ suc 𝐴 ∧ suc 𝐴𝐵)))
1615adantl 482 . . . . . . 7 (((Ord 𝐵𝐴𝐵) ∧ 𝑐 = suc 𝐴) → ((𝐴𝑐𝑐𝐵) ↔ (𝐴 ∈ suc 𝐴 ∧ suc 𝐴𝐵)))
1712, 16rspcedv 3602 . . . . . 6 ((Ord 𝐵𝐴𝐵) → ((𝐴 ∈ suc 𝐴 ∧ suc 𝐴𝐵) → ∃𝑐 ∈ On (𝐴𝑐𝑐𝐵)))
189, 17mpand 693 . . . . 5 ((Ord 𝐵𝐴𝐵) → (suc 𝐴𝐵 → ∃𝑐 ∈ On (𝐴𝑐𝑐𝐵)))
197, 18jaod 857 . . . 4 ((Ord 𝐵𝐴𝐵) → ((𝐵 ∈ suc 𝐴 ∨ suc 𝐴𝐵) → ∃𝑐 ∈ On (𝐴𝑐𝑐𝐵)))
20 ralnex 3071 . . . . 5 (∀𝑐 ∈ On ¬ (𝐴𝑐𝑐𝐵) ↔ ¬ ∃𝑐 ∈ On (𝐴𝑐𝑐𝐵))
2120biimpi 215 . . . 4 (∀𝑐 ∈ On ¬ (𝐴𝑐𝑐𝐵) → ¬ ∃𝑐 ∈ On (𝐴𝑐𝑐𝐵))
2219, 21nsyli 157 . . 3 ((Ord 𝐵𝐴𝐵) → (∀𝑐 ∈ On ¬ (𝐴𝑐𝑐𝐵) → ¬ (𝐵 ∈ suc 𝐴 ∨ suc 𝐴𝐵)))
23 ordsuci 7779 . . . . 5 (Ord 𝐴 → Ord suc 𝐴)
241, 23syl 17 . . . 4 ((Ord 𝐵𝐴𝐵) → Ord suc 𝐴)
25 ordtri3 6389 . . . 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 258 . 2 ((Ord 𝐵𝐴𝐵) → (∀𝑐 ∈ On ¬ (𝐴𝑐𝑐𝐵) → 𝐵 = suc 𝐴))
2827ancoms 459 1 ((𝐴𝐵 ∧ Ord 𝐵) → (∀𝑐 ∈ On ¬ (𝐴𝑐𝑐𝐵) → 𝐵 = suc 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  wo 845   = wceq 1541  wcel 2106  wral 3060  wrex 3069  Ord word 6352  Oncon0 6353  suc csuc 6355
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7708
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  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 6356  df-on 6357  df-suc 6359
This theorem is referenced by:  orddif0suc  41789
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