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Theorem ordnexbtwnsuc 43256
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 6407 . . . . . 6 ((Ord 𝐵𝐴𝐵) → Ord 𝐴)
2 ordnbtwn 6478 . . . . . . . . . 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 6466 . . . . . . 7 (𝐴𝐵𝐴 ∈ suc 𝐴)
98adantl 481 . . . . . 6 ((Ord 𝐵𝐴𝐵) → 𝐴 ∈ suc 𝐴)
10 ordelon 6409 . . . . . . . 8 ((Ord 𝐵𝐴𝐵) → 𝐴 ∈ On)
11 onsuc 7830 . . . . . . . 8 (𝐴 ∈ On → suc 𝐴 ∈ On)
1210, 11syl 17 . . . . . . 7 ((Ord 𝐵𝐴𝐵) → suc 𝐴 ∈ On)
13 eleq2 2827 . . . . . . . . 9 (𝑐 = suc 𝐴 → (𝐴𝑐𝐴 ∈ suc 𝐴))
14 eleq1 2826 . . . . . . . . 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 3614 . . . . . 6 ((Ord 𝐵𝐴𝐵) → ((𝐴 ∈ suc 𝐴 ∧ suc 𝐴𝐵) → ∃𝑐 ∈ On (𝐴𝑐𝑐𝐵)))
189, 17mpand 695 . . . . 5 ((Ord 𝐵𝐴𝐵) → (suc 𝐴𝐵 → ∃𝑐 ∈ On (𝐴𝑐𝑐𝐵)))
197, 18jaod 859 . . . 4 ((Ord 𝐵𝐴𝐵) → ((𝐵 ∈ suc 𝐴 ∨ suc 𝐴𝐵) → ∃𝑐 ∈ On (𝐴𝑐𝑐𝐵)))
20 ralnex 3069 . . . . 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 7827 . . . . 5 (Ord 𝐴 → Ord suc 𝐴)
241, 23syl 17 . . . 4 ((Ord 𝐵𝐴𝐵) → Ord suc 𝐴)
25 ordtri3 6421 . . . 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 847   = wceq 1536  wcel 2105  wral 3058  wrex 3067  Ord word 6384  Oncon0 6385  suc csuc 6387
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-tr 5265  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-ord 6388  df-on 6389  df-suc 6391
This theorem is referenced by:  orddif0suc  43257
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