Theorem ordnexbtwnsuc 44035

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	⊢ ((𝐴 ∈ 𝐵 ∧ Ord 𝐵) → (∀𝑐 ∈ On ¬ (𝐴 ∈ 𝑐 ∧ 𝑐 ∈ 𝐵) → 𝐵 = suc 𝐴))

Distinct variable groups:   𝐴,𝑐   𝐵,𝑐

Proof of Theorem ordnexbtwnsuc

Step	Hyp	Ref	Expression
1		ordelord 6389	. . . . . 6 ⊢ ((Ord 𝐵 ∧ 𝐴 ∈ 𝐵) → Ord 𝐴)
2		ordnbtwn 6463	. . . . . . . . . 10 ⊢ (Ord 𝐴 → ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ suc 𝐴))
3	2	pm2.21d 122	. . . . . . . . 9 ⊢ (Ord 𝐴 → ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ suc 𝐴) → ∃𝑐 ∈ On (𝐴 ∈ 𝑐 ∧ 𝑐 ∈ 𝐵)))
4	3	expd 421	. . . . . . . 8 ⊢ (Ord 𝐴 → (𝐴 ∈ 𝐵 → (𝐵 ∈ suc 𝐴 → ∃𝑐 ∈ On (𝐴 ∈ 𝑐 ∧ 𝑐 ∈ 𝐵))))
5	4	com12 33	. . . . . . 7 ⊢ (𝐴 ∈ 𝐵 → (Ord 𝐴 → (𝐵 ∈ suc 𝐴 → ∃𝑐 ∈ On (𝐴 ∈ 𝑐 ∧ 𝑐 ∈ 𝐵))))
6	5	adantl 487	. . . . . 6 ⊢ ((Ord 𝐵 ∧ 𝐴 ∈ 𝐵) → (Ord 𝐴 → (𝐵 ∈ suc 𝐴 → ∃𝑐 ∈ On (𝐴 ∈ 𝑐 ∧ 𝑐 ∈ 𝐵))))
7	1, 6	mpd 16	. . . . 5 ⊢ ((Ord 𝐵 ∧ 𝐴 ∈ 𝐵) → (𝐵 ∈ suc 𝐴 → ∃𝑐 ∈ On (𝐴 ∈ 𝑐 ∧ 𝑐 ∈ 𝐵)))
8		sucidg 6451	. . . . . . 7 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ suc 𝐴)
9	8	adantl 487	. . . . . 6 ⊢ ((Ord 𝐵 ∧ 𝐴 ∈ 𝐵) → 𝐴 ∈ suc 𝐴)
10		ordelon 6391	. . . . . . . 8 ⊢ ((Ord 𝐵 ∧ 𝐴 ∈ 𝐵) → 𝐴 ∈ On)
11		onsuc 7818	. . . . . . . 8 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ On)
12	10, 11	syl 18	. . . . . . 7 ⊢ ((Ord 𝐵 ∧ 𝐴 ∈ 𝐵) → suc 𝐴 ∈ On)
13		eleq2 2855	. . . . . . . . 9 ⊢ (𝑐 = suc 𝐴 → (𝐴 ∈ 𝑐 ↔ 𝐴 ∈ suc 𝐴))
14		eleq1 2854	. . . . . . . . 9 ⊢ (𝑐 = suc 𝐴 → (𝑐 ∈ 𝐵 ↔ suc 𝐴 ∈ 𝐵))
15	13, 14	anbi12d 644	. . . . . . . 8 ⊢ (𝑐 = suc 𝐴 → ((𝐴 ∈ 𝑐 ∧ 𝑐 ∈ 𝐵) ↔ (𝐴 ∈ suc 𝐴 ∧ suc 𝐴 ∈ 𝐵)))
16	15	adantl 487	. . . . . . 7 ⊢ (((Ord 𝐵 ∧ 𝐴 ∈ 𝐵) ∧ 𝑐 = suc 𝐴) → ((𝐴 ∈ 𝑐 ∧ 𝑐 ∈ 𝐵) ↔ (𝐴 ∈ suc 𝐴 ∧ suc 𝐴 ∈ 𝐵)))
17	12, 16	rspcedv 3577	. . . . . 6 ⊢ ((Ord 𝐵 ∧ 𝐴 ∈ 𝐵) → ((𝐴 ∈ suc 𝐴 ∧ suc 𝐴 ∈ 𝐵) → ∃𝑐 ∈ On (𝐴 ∈ 𝑐 ∧ 𝑐 ∈ 𝐵)))
18	9, 17	mpand 708	. . . . 5 ⊢ ((Ord 𝐵 ∧ 𝐴 ∈ 𝐵) → (suc 𝐴 ∈ 𝐵 → ∃𝑐 ∈ On (𝐴 ∈ 𝑐 ∧ 𝑐 ∈ 𝐵)))
19	7, 18	jaod 873	. . . 4 ⊢ ((Ord 𝐵 ∧ 𝐴 ∈ 𝐵) → ((𝐵 ∈ suc 𝐴 ∨ suc 𝐴 ∈ 𝐵) → ∃𝑐 ∈ On (𝐴 ∈ 𝑐 ∧ 𝑐 ∈ 𝐵)))
20		ralnex 3094	. . . . 5 ⊢ (∀𝑐 ∈ On ¬ (𝐴 ∈ 𝑐 ∧ 𝑐 ∈ 𝐵) ↔ ¬ ∃𝑐 ∈ On (𝐴 ∈ 𝑐 ∧ 𝑐 ∈ 𝐵))
21	20	biimpi 219	. . . 4 ⊢ (∀𝑐 ∈ On ¬ (𝐴 ∈ 𝑐 ∧ 𝑐 ∈ 𝐵) → ¬ ∃𝑐 ∈ On (𝐴 ∈ 𝑐 ∧ 𝑐 ∈ 𝐵))
22	19, 21	nsyli 158	. . 3 ⊢ ((Ord 𝐵 ∧ 𝐴 ∈ 𝐵) → (∀𝑐 ∈ On ¬ (𝐴 ∈ 𝑐 ∧ 𝑐 ∈ 𝐵) → ¬ (𝐵 ∈ suc 𝐴 ∨ suc 𝐴 ∈ 𝐵)))
23		ordsuci 7816	. . . . 5 ⊢ (Ord 𝐴 → Ord suc 𝐴)
24	1, 23	syl 18	. . . 4 ⊢ ((Ord 𝐵 ∧ 𝐴 ∈ 𝐵) → Ord suc 𝐴)
25		ordtri3 6404	. . . 4 ⊢ ((Ord 𝐵 ∧ Ord suc 𝐴) → (𝐵 = suc 𝐴 ↔ ¬ (𝐵 ∈ suc 𝐴 ∨ suc 𝐴 ∈ 𝐵)))
26	24, 25	syldan 603	. . 3 ⊢ ((Ord 𝐵 ∧ 𝐴 ∈ 𝐵) → (𝐵 = suc 𝐴 ↔ ¬ (𝐵 ∈ suc 𝐴 ∨ suc 𝐴 ∈ 𝐵)))
27	22, 26	sylibrd 262	. 2 ⊢ ((Ord 𝐵 ∧ 𝐴 ∈ 𝐵) → (∀𝑐 ∈ On ¬ (𝐴 ∈ 𝑐 ∧ 𝑐 ∈ 𝐵) → 𝐵 = suc 𝐴))
28	27	ancoms 464	1 ⊢ ((𝐴 ∈ 𝐵 ∧ Ord 𝐵) → (∀𝑐 ∈ On ¬ (𝐴 ∈ 𝑐 ∧ 𝑐 ∈ 𝐵) → 𝐵 = suc 𝐴))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2146  ∀wral 3082  ∃wrex 3092  Ord word 6366  Oncon0 6367  suc csuc 6369

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738  ax-sep 5262  ax-pr 5409  ax-un 7745

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-ne 2962  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3928  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-opab 5179  df-tr 5224  df-eprel 5566  df-po 5574  df-so 5575  df-fr 5619  df-we 5621  df-ord 6370  df-on 6371  df-suc 6373

This theorem is used by:  orddif0suc  44036