Theorem ordnexbtwnsuc 42506

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 6376	. . . . . 6 ⊢ ((Ord 𝐵 ∧ 𝐴 ∈ 𝐵) → Ord 𝐴)
2		ordnbtwn 6447	. . . . . . . . . 10 ⊢ (Ord 𝐴 → ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ suc 𝐴))
3	2	pm2.21d 121	. . . . . . . . 9 ⊢ (Ord 𝐴 → ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ suc 𝐴) → ∃𝑐 ∈ On (𝐴 ∈ 𝑐 ∧ 𝑐 ∈ 𝐵)))
4	3	expd 415	. . . . . . . 8 ⊢ (Ord 𝐴 → (𝐴 ∈ 𝐵 → (𝐵 ∈ suc 𝐴 → ∃𝑐 ∈ On (𝐴 ∈ 𝑐 ∧ 𝑐 ∈ 𝐵))))
5	4	com12 32	. . . . . . 7 ⊢ (𝐴 ∈ 𝐵 → (Ord 𝐴 → (𝐵 ∈ suc 𝐴 → ∃𝑐 ∈ On (𝐴 ∈ 𝑐 ∧ 𝑐 ∈ 𝐵))))
6	5	adantl 481	. . . . . 6 ⊢ ((Ord 𝐵 ∧ 𝐴 ∈ 𝐵) → (Ord 𝐴 → (𝐵 ∈ suc 𝐴 → ∃𝑐 ∈ On (𝐴 ∈ 𝑐 ∧ 𝑐 ∈ 𝐵))))
7	1, 6	mpd 15	. . . . 5 ⊢ ((Ord 𝐵 ∧ 𝐴 ∈ 𝐵) → (𝐵 ∈ suc 𝐴 → ∃𝑐 ∈ On (𝐴 ∈ 𝑐 ∧ 𝑐 ∈ 𝐵)))
8		sucidg 6435	. . . . . . 7 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ suc 𝐴)
9	8	adantl 481	. . . . . 6 ⊢ ((Ord 𝐵 ∧ 𝐴 ∈ 𝐵) → 𝐴 ∈ suc 𝐴)
10		ordelon 6378	. . . . . . . 8 ⊢ ((Ord 𝐵 ∧ 𝐴 ∈ 𝐵) → 𝐴 ∈ On)
11		onsuc 7792	. . . . . . . 8 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ On)
12	10, 11	syl 17	. . . . . . 7 ⊢ ((Ord 𝐵 ∧ 𝐴 ∈ 𝐵) → suc 𝐴 ∈ On)
13		eleq2 2814	. . . . . . . . 9 ⊢ (𝑐 = suc 𝐴 → (𝐴 ∈ 𝑐 ↔ 𝐴 ∈ suc 𝐴))
14		eleq1 2813	. . . . . . . . 9 ⊢ (𝑐 = suc 𝐴 → (𝑐 ∈ 𝐵 ↔ suc 𝐴 ∈ 𝐵))
15	13, 14	anbi12d 630	. . . . . . . 8 ⊢ (𝑐 = suc 𝐴 → ((𝐴 ∈ 𝑐 ∧ 𝑐 ∈ 𝐵) ↔ (𝐴 ∈ suc 𝐴 ∧ suc 𝐴 ∈ 𝐵)))
16	15	adantl 481	. . . . . . 7 ⊢ (((Ord 𝐵 ∧ 𝐴 ∈ 𝐵) ∧ 𝑐 = suc 𝐴) → ((𝐴 ∈ 𝑐 ∧ 𝑐 ∈ 𝐵) ↔ (𝐴 ∈ suc 𝐴 ∧ suc 𝐴 ∈ 𝐵)))
17	12, 16	rspcedv 3597	. . . . . 6 ⊢ ((Ord 𝐵 ∧ 𝐴 ∈ 𝐵) → ((𝐴 ∈ suc 𝐴 ∧ suc 𝐴 ∈ 𝐵) → ∃𝑐 ∈ On (𝐴 ∈ 𝑐 ∧ 𝑐 ∈ 𝐵)))
18	9, 17	mpand 692	. . . . 5 ⊢ ((Ord 𝐵 ∧ 𝐴 ∈ 𝐵) → (suc 𝐴 ∈ 𝐵 → ∃𝑐 ∈ On (𝐴 ∈ 𝑐 ∧ 𝑐 ∈ 𝐵)))
19	7, 18	jaod 856	. . . 4 ⊢ ((Ord 𝐵 ∧ 𝐴 ∈ 𝐵) → ((𝐵 ∈ suc 𝐴 ∨ suc 𝐴 ∈ 𝐵) → ∃𝑐 ∈ On (𝐴 ∈ 𝑐 ∧ 𝑐 ∈ 𝐵)))
20		ralnex 3064	. . . . 5 ⊢ (∀𝑐 ∈ On ¬ (𝐴 ∈ 𝑐 ∧ 𝑐 ∈ 𝐵) ↔ ¬ ∃𝑐 ∈ On (𝐴 ∈ 𝑐 ∧ 𝑐 ∈ 𝐵))
21	20	biimpi 215	. . . 4 ⊢ (∀𝑐 ∈ On ¬ (𝐴 ∈ 𝑐 ∧ 𝑐 ∈ 𝐵) → ¬ ∃𝑐 ∈ On (𝐴 ∈ 𝑐 ∧ 𝑐 ∈ 𝐵))
22	19, 21	nsyli 157	. . 3 ⊢ ((Ord 𝐵 ∧ 𝐴 ∈ 𝐵) → (∀𝑐 ∈ On ¬ (𝐴 ∈ 𝑐 ∧ 𝑐 ∈ 𝐵) → ¬ (𝐵 ∈ suc 𝐴 ∨ suc 𝐴 ∈ 𝐵)))
23		ordsuci 7789	. . . . 5 ⊢ (Ord 𝐴 → Ord suc 𝐴)
24	1, 23	syl 17	. . . 4 ⊢ ((Ord 𝐵 ∧ 𝐴 ∈ 𝐵) → Ord suc 𝐴)
25		ordtri3 6390	. . . 4 ⊢ ((Ord 𝐵 ∧ Ord suc 𝐴) → (𝐵 = suc 𝐴 ↔ ¬ (𝐵 ∈ suc 𝐴 ∨ suc 𝐴 ∈ 𝐵)))
26	24, 25	syldan 590	. . 3 ⊢ ((Ord 𝐵 ∧ 𝐴 ∈ 𝐵) → (𝐵 = suc 𝐴 ↔ ¬ (𝐵 ∈ suc 𝐴 ∨ suc 𝐴 ∈ 𝐵)))
27	22, 26	sylibrd 259	. 2 ⊢ ((Ord 𝐵 ∧ 𝐴 ∈ 𝐵) → (∀𝑐 ∈ On ¬ (𝐴 ∈ 𝑐 ∧ 𝑐 ∈ 𝐵) → 𝐵 = suc 𝐴))
28	27	ancoms 458	1 ⊢ ((𝐴 ∈ 𝐵 ∧ Ord 𝐵) → (∀𝑐 ∈ On ¬ (𝐴 ∈ 𝑐 ∧ 𝑐 ∈ 𝐵) → 𝐵 = suc 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   ∨ wo 844   = wceq 1533   ∈ wcel 2098  ∀wral 3053  ∃wrex 3062  Ord word 6353  Oncon0 6354  suc csuc 6356

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417  ax-un 7718

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3959  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-br 5139  df-opab 5201  df-tr 5256  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-we 5623  df-ord 6357  df-on 6358  df-suc 6360

This theorem is referenced by:  orddif0suc  42507