Theorem ordunidif 6314

Description: The union of an ordinal stays the same if a subset equal to one of its elements is removed. (Contributed by NM, 10-Dec-2004.)

Assertion

Ref	Expression
ordunidif	⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → ∪ (𝐴 ∖ 𝐵) = ∪ 𝐴)

Proof of Theorem ordunidif

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ordelon 6290	. . . . . . . 8 ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝐵 ∈ On)
2		onelss 6308	. . . . . . . 8 ⊢ (𝐵 ∈ On → (𝑥 ∈ 𝐵 → 𝑥 ⊆ 𝐵))
3	1, 2	syl 17	. . . . . . 7 ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝑥 ∈ 𝐵 → 𝑥 ⊆ 𝐵))
4		eloni 6276	. . . . . . . . . . 11 ⊢ (𝐵 ∈ On → Ord 𝐵)
5		ordirr 6284	. . . . . . . . . . 11 ⊢ (Ord 𝐵 → ¬ 𝐵 ∈ 𝐵)
6	4, 5	syl 17	. . . . . . . . . 10 ⊢ (𝐵 ∈ On → ¬ 𝐵 ∈ 𝐵)
7		eldif 3897	. . . . . . . . . . 11 ⊢ (𝐵 ∈ (𝐴 ∖ 𝐵) ↔ (𝐵 ∈ 𝐴 ∧ ¬ 𝐵 ∈ 𝐵))
8	7	simplbi2 501	. . . . . . . . . 10 ⊢ (𝐵 ∈ 𝐴 → (¬ 𝐵 ∈ 𝐵 → 𝐵 ∈ (𝐴 ∖ 𝐵)))
9	6, 8	syl5 34	. . . . . . . . 9 ⊢ (𝐵 ∈ 𝐴 → (𝐵 ∈ On → 𝐵 ∈ (𝐴 ∖ 𝐵)))
10	9	adantl 482	. . . . . . . 8 ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝐵 ∈ On → 𝐵 ∈ (𝐴 ∖ 𝐵)))
11	1, 10	mpd 15	. . . . . . 7 ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝐵 ∈ (𝐴 ∖ 𝐵))
12	3, 11	jctild 526	. . . . . 6 ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝑥 ∈ 𝐵 → (𝐵 ∈ (𝐴 ∖ 𝐵) ∧ 𝑥 ⊆ 𝐵)))
13	12	adantr 481	. . . . 5 ⊢ (((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ 𝐵 → (𝐵 ∈ (𝐴 ∖ 𝐵) ∧ 𝑥 ⊆ 𝐵)))
14		sseq2 3947	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝑥 ⊆ 𝑦 ↔ 𝑥 ⊆ 𝐵))
15	14	rspcev 3561	. . . . 5 ⊢ ((𝐵 ∈ (𝐴 ∖ 𝐵) ∧ 𝑥 ⊆ 𝐵) → ∃𝑦 ∈ (𝐴 ∖ 𝐵)𝑥 ⊆ 𝑦)
16	13, 15	syl6 35	. . . 4 ⊢ (((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ 𝐵 → ∃𝑦 ∈ (𝐴 ∖ 𝐵)𝑥 ⊆ 𝑦))
17		eldif 3897	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵))
18	17	biimpri 227	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) → 𝑥 ∈ (𝐴 ∖ 𝐵))
19		ssid 3943	. . . . . . . 8 ⊢ 𝑥 ⊆ 𝑥
20	18, 19	jctir 521	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) → (𝑥 ∈ (𝐴 ∖ 𝐵) ∧ 𝑥 ⊆ 𝑥))
21	20	ex 413	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 → (¬ 𝑥 ∈ 𝐵 → (𝑥 ∈ (𝐴 ∖ 𝐵) ∧ 𝑥 ⊆ 𝑥)))
22		sseq2 3947	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (𝑥 ⊆ 𝑦 ↔ 𝑥 ⊆ 𝑥))
23	22	rspcev 3561	. . . . . 6 ⊢ ((𝑥 ∈ (𝐴 ∖ 𝐵) ∧ 𝑥 ⊆ 𝑥) → ∃𝑦 ∈ (𝐴 ∖ 𝐵)𝑥 ⊆ 𝑦)
24	21, 23	syl6 35	. . . . 5 ⊢ (𝑥 ∈ 𝐴 → (¬ 𝑥 ∈ 𝐵 → ∃𝑦 ∈ (𝐴 ∖ 𝐵)𝑥 ⊆ 𝑦))
25	24	adantl 482	. . . 4 ⊢ (((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑥 ∈ 𝐴) → (¬ 𝑥 ∈ 𝐵 → ∃𝑦 ∈ (𝐴 ∖ 𝐵)𝑥 ⊆ 𝑦))
26	16, 25	pm2.61d 179	. . 3 ⊢ (((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ (𝐴 ∖ 𝐵)𝑥 ⊆ 𝑦)
27	26	ralrimiva 3103	. 2 ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ (𝐴 ∖ 𝐵)𝑥 ⊆ 𝑦)
28		unidif 4875	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ (𝐴 ∖ 𝐵)𝑥 ⊆ 𝑦 → ∪ (𝐴 ∖ 𝐵) = ∪ 𝐴)
29	27, 28	syl 17	1 ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → ∪ (𝐴 ∖ 𝐵) = ∪ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ∀wral 3064  ∃wrex 3065   ∖ cdif 3884   ⊆ wss 3887  ∪ cuni 4839  Ord word 6265  Oncon0 6266

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-11 2154  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-tr 5192  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-ord 6269  df-on 6270

This theorem is referenced by: (None)