Theorem orddif0suc 43281

Description: For any distinct pair of ordinals, if the set difference between the greater and the successor of the lesser is empty, the greater is the successor of the lesser. Lemma 1.16 of [Schloeder] p. 2. (Contributed by RP, 17-Jan-2025.)

Assertion

Ref	Expression
orddif0suc	⊢ ((𝐴 ∈ 𝐵 ∧ Ord 𝐵) → ((𝐵 ∖ suc 𝐴) = ∅ → 𝐵 = suc 𝐴))

Proof of Theorem orddif0suc

Dummy variable 𝑐 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 484	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝐵 ∧ Ord 𝐵) → Ord 𝐵)
2		ordelon 6408	. . . . . . . . 9 ⊢ ((Ord 𝐵 ∧ 𝐴 ∈ 𝐵) → 𝐴 ∈ On)
3	2	ancoms 458	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝐵 ∧ Ord 𝐵) → 𝐴 ∈ On)
4		ordeldifsucon 43272	. . . . . . . 8 ⊢ ((Ord 𝐵 ∧ 𝐴 ∈ On) → (𝑐 ∈ (𝐵 ∖ suc 𝐴) ↔ (𝑐 ∈ 𝐵 ∧ 𝐴 ∈ 𝑐)))
5	1, 3, 4	syl2anc 584	. . . . . . 7 ⊢ ((𝐴 ∈ 𝐵 ∧ Ord 𝐵) → (𝑐 ∈ (𝐵 ∖ suc 𝐴) ↔ (𝑐 ∈ 𝐵 ∧ 𝐴 ∈ 𝑐)))
6	5	biancomd 463	. . . . . 6 ⊢ ((𝐴 ∈ 𝐵 ∧ Ord 𝐵) → (𝑐 ∈ (𝐵 ∖ suc 𝐴) ↔ (𝐴 ∈ 𝑐 ∧ 𝑐 ∈ 𝐵)))
7		ordelon 6408	. . . . . . . . . 10 ⊢ ((Ord 𝐵 ∧ 𝑐 ∈ 𝐵) → 𝑐 ∈ On)
8	7	ad2ant2l 746	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝐵 ∧ Ord 𝐵) ∧ (𝐴 ∈ 𝑐 ∧ 𝑐 ∈ 𝐵)) → 𝑐 ∈ On)
9	8	ex 412	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝐵 ∧ Ord 𝐵) → ((𝐴 ∈ 𝑐 ∧ 𝑐 ∈ 𝐵) → 𝑐 ∈ On))
10	9	pm4.71rd 562	. . . . . . 7 ⊢ ((𝐴 ∈ 𝐵 ∧ Ord 𝐵) → ((𝐴 ∈ 𝑐 ∧ 𝑐 ∈ 𝐵) ↔ (𝑐 ∈ On ∧ (𝐴 ∈ 𝑐 ∧ 𝑐 ∈ 𝐵))))
11		df-an 396	. . . . . . 7 ⊢ ((𝑐 ∈ On ∧ (𝐴 ∈ 𝑐 ∧ 𝑐 ∈ 𝐵)) ↔ ¬ (𝑐 ∈ On → ¬ (𝐴 ∈ 𝑐 ∧ 𝑐 ∈ 𝐵)))
12	10, 11	bitrdi 287	. . . . . 6 ⊢ ((𝐴 ∈ 𝐵 ∧ Ord 𝐵) → ((𝐴 ∈ 𝑐 ∧ 𝑐 ∈ 𝐵) ↔ ¬ (𝑐 ∈ On → ¬ (𝐴 ∈ 𝑐 ∧ 𝑐 ∈ 𝐵))))
13	6, 12	bitr2d 280	. . . . 5 ⊢ ((𝐴 ∈ 𝐵 ∧ Ord 𝐵) → (¬ (𝑐 ∈ On → ¬ (𝐴 ∈ 𝑐 ∧ 𝑐 ∈ 𝐵)) ↔ 𝑐 ∈ (𝐵 ∖ suc 𝐴)))
14	13	con1bid 355	. . . 4 ⊢ ((𝐴 ∈ 𝐵 ∧ Ord 𝐵) → (¬ 𝑐 ∈ (𝐵 ∖ suc 𝐴) ↔ (𝑐 ∈ On → ¬ (𝐴 ∈ 𝑐 ∧ 𝑐 ∈ 𝐵))))
15	14	albidv 1920	. . 3 ⊢ ((𝐴 ∈ 𝐵 ∧ Ord 𝐵) → (∀𝑐 ¬ 𝑐 ∈ (𝐵 ∖ suc 𝐴) ↔ ∀𝑐(𝑐 ∈ On → ¬ (𝐴 ∈ 𝑐 ∧ 𝑐 ∈ 𝐵))))
16		eq0 4350	. . 3 ⊢ ((𝐵 ∖ suc 𝐴) = ∅ ↔ ∀𝑐 ¬ 𝑐 ∈ (𝐵 ∖ suc 𝐴))
17		df-ral 3062	. . 3 ⊢ (∀𝑐 ∈ On ¬ (𝐴 ∈ 𝑐 ∧ 𝑐 ∈ 𝐵) ↔ ∀𝑐(𝑐 ∈ On → ¬ (𝐴 ∈ 𝑐 ∧ 𝑐 ∈ 𝐵)))
18	15, 16, 17	3bitr4g 314	. 2 ⊢ ((𝐴 ∈ 𝐵 ∧ Ord 𝐵) → ((𝐵 ∖ suc 𝐴) = ∅ ↔ ∀𝑐 ∈ On ¬ (𝐴 ∈ 𝑐 ∧ 𝑐 ∈ 𝐵)))
19		ordnexbtwnsuc 43280	. 2 ⊢ ((𝐴 ∈ 𝐵 ∧ Ord 𝐵) → (∀𝑐 ∈ On ¬ (𝐴 ∈ 𝑐 ∧ 𝑐 ∈ 𝐵) → 𝐵 = suc 𝐴))
20	18, 19	sylbid 240	1 ⊢ ((𝐴 ∈ 𝐵 ∧ Ord 𝐵) → ((𝐵 ∖ suc 𝐴) = ∅ → 𝐵 = suc 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1538   = wceq 1540   ∈ wcel 2108  ∀wral 3061   ∖ cdif 3948  ∅c0 4333  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: (None)