Theorem orddif0suc 43510

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 6341	. . . . . . . . 9 ⊢ ((Ord 𝐵 ∧ 𝐴 ∈ 𝐵) → 𝐴 ∈ On)
3	2	ancoms 458	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝐵 ∧ Ord 𝐵) → 𝐴 ∈ On)
4		ordeldifsucon 43501	. . . . . . . 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 6341	. . . . . . . . . 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 1921	. . 3 ⊢ ((𝐴 ∈ 𝐵 ∧ Ord 𝐵) → (∀𝑐 ¬ 𝑐 ∈ (𝐵 ∖ suc 𝐴) ↔ ∀𝑐(𝑐 ∈ On → ¬ (𝐴 ∈ 𝑐 ∧ 𝑐 ∈ 𝐵))))
16		eq0 4302	. . 3 ⊢ ((𝐵 ∖ suc 𝐴) = ∅ ↔ ∀𝑐 ¬ 𝑐 ∈ (𝐵 ∖ suc 𝐴))
17		df-ral 3052	. . 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 43509	. 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 1539   = wceq 1541   ∈ wcel 2113  ∀wral 3051   ∖ cdif 3898  ∅c0 4285  Ord word 6316  Oncon0 6317  suc csuc 6319

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-tr 5206  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-ord 6320  df-on 6321  df-suc 6323

This theorem is referenced by: (None)