Theorem orddif0suc 43625

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 6349	. . . . . . . . 9 ⊢ ((Ord 𝐵 ∧ 𝐴 ∈ 𝐵) → 𝐴 ∈ On)
3	2	ancoms 458	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝐵 ∧ Ord 𝐵) → 𝐴 ∈ On)
4		ordeldifsucon 43616	. . . . . . . 8 ⊢ ((Ord 𝐵 ∧ 𝐴 ∈ On) → (𝑐 ∈ (𝐵 ∖ suc 𝐴) ↔ (𝑐 ∈ 𝐵 ∧ 𝐴 ∈ 𝑐)))
5	1, 3, 4	syl2anc 585	. . . . . . 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 6349	. . . . . . . . . 10 ⊢ ((Ord 𝐵 ∧ 𝑐 ∈ 𝐵) → 𝑐 ∈ On)
8	7	ad2ant2l 747	. . . . . . . . 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 1922	. . 3 ⊢ ((𝐴 ∈ 𝐵 ∧ Ord 𝐵) → (∀𝑐 ¬ 𝑐 ∈ (𝐵 ∖ suc 𝐴) ↔ ∀𝑐(𝑐 ∈ On → ¬ (𝐴 ∈ 𝑐 ∧ 𝑐 ∈ 𝐵))))
16		eq0 4304	. . 3 ⊢ ((𝐵 ∖ suc 𝐴) = ∅ ↔ ∀𝑐 ¬ 𝑐 ∈ (𝐵 ∖ suc 𝐴))
17		df-ral 3053	. . 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 43624	. 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 1540   = wceq 1542   ∈ wcel 2114  ∀wral 3052   ∖ cdif 3900  ∅c0 4287  Ord word 6324  Oncon0 6325  suc csuc 6327

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5243  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-tr 5208  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-ord 6328  df-on 6329  df-suc 6331

This theorem is referenced by: (None)