Theorem orddif0suc 43230

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 6419	. . . . . . . . 9 ⊢ ((Ord 𝐵 ∧ 𝐴 ∈ 𝐵) → 𝐴 ∈ On)
3	2	ancoms 458	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝐵 ∧ Ord 𝐵) → 𝐴 ∈ On)
4		ordeldifsucon 43221	. . . . . . . 8 ⊢ ((Ord 𝐵 ∧ 𝐴 ∈ On) → (𝑐 ∈ (𝐵 ∖ suc 𝐴) ↔ (𝑐 ∈ 𝐵 ∧ 𝐴 ∈ 𝑐)))
5	1, 3, 4	syl2anc 583	. . . . . . 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 6419	. . . . . . . . . 10 ⊢ ((Ord 𝐵 ∧ 𝑐 ∈ 𝐵) → 𝑐 ∈ On)
8	7	ad2ant2l 745	. . . . . . . . 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 1919	. . 3 ⊢ ((𝐴 ∈ 𝐵 ∧ Ord 𝐵) → (∀𝑐 ¬ 𝑐 ∈ (𝐵 ∖ suc 𝐴) ↔ ∀𝑐(𝑐 ∈ On → ¬ (𝐴 ∈ 𝑐 ∧ 𝑐 ∈ 𝐵))))
16		eq0 4373	. . 3 ⊢ ((𝐵 ∖ suc 𝐴) = ∅ ↔ ∀𝑐 ¬ 𝑐 ∈ (𝐵 ∖ suc 𝐴))
17		df-ral 3068	. . 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 43229	. 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 1535   = wceq 1537   ∈ wcel 2108  ∀wral 3067   ∖ cdif 3973  ∅c0 4352  Ord word 6394  Oncon0 6395  suc csuc 6397

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-tr 5284  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-ord 6398  df-on 6399  df-suc 6401

This theorem is referenced by: (None)