Theorem ordunidif 6299

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 6275	. . . . . . . 8 ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝐵 ∈ On)
2		onelss 6293	. . . . . . . 8 ⊢ (𝐵 ∈ On → (𝑥 ∈ 𝐵 → 𝑥 ⊆ 𝐵))
3	1, 2	syl 17	. . . . . . 7 ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝑥 ∈ 𝐵 → 𝑥 ⊆ 𝐵))
4		eloni 6261	. . . . . . . . . . 11 ⊢ (𝐵 ∈ On → Ord 𝐵)
5		ordirr 6269	. . . . . . . . . . 11 ⊢ (Ord 𝐵 → ¬ 𝐵 ∈ 𝐵)
6	4, 5	syl 17	. . . . . . . . . 10 ⊢ (𝐵 ∈ On → ¬ 𝐵 ∈ 𝐵)
7		eldif 3893	. . . . . . . . . . 11 ⊢ (𝐵 ∈ (𝐴 ∖ 𝐵) ↔ (𝐵 ∈ 𝐴 ∧ ¬ 𝐵 ∈ 𝐵))
8	7	simplbi2 500	. . . . . . . . . 10 ⊢ (𝐵 ∈ 𝐴 → (¬ 𝐵 ∈ 𝐵 → 𝐵 ∈ (𝐴 ∖ 𝐵)))
9	6, 8	syl5 34	. . . . . . . . 9 ⊢ (𝐵 ∈ 𝐴 → (𝐵 ∈ On → 𝐵 ∈ (𝐴 ∖ 𝐵)))
10	9	adantl 481	. . . . . . . 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 525	. . . . . 6 ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝑥 ∈ 𝐵 → (𝐵 ∈ (𝐴 ∖ 𝐵) ∧ 𝑥 ⊆ 𝐵)))
13	12	adantr 480	. . . . 5 ⊢ (((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ 𝐵 → (𝐵 ∈ (𝐴 ∖ 𝐵) ∧ 𝑥 ⊆ 𝐵)))
14		sseq2 3943	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝑥 ⊆ 𝑦 ↔ 𝑥 ⊆ 𝐵))
15	14	rspcev 3552	. . . . 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 3893	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵))
18	17	biimpri 227	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) → 𝑥 ∈ (𝐴 ∖ 𝐵))
19		ssid 3939	. . . . . . . 8 ⊢ 𝑥 ⊆ 𝑥
20	18, 19	jctir 520	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) → (𝑥 ∈ (𝐴 ∖ 𝐵) ∧ 𝑥 ⊆ 𝑥))
21	20	ex 412	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 → (¬ 𝑥 ∈ 𝐵 → (𝑥 ∈ (𝐴 ∖ 𝐵) ∧ 𝑥 ⊆ 𝑥)))
22		sseq2 3943	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (𝑥 ⊆ 𝑦 ↔ 𝑥 ⊆ 𝑥))
23	22	rspcev 3552	. . . . . 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 481	. . . 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 3107	. 2 ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ (𝐴 ∖ 𝐵)𝑥 ⊆ 𝑦)
28		unidif 4872	. 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 395   = wceq 1539   ∈ wcel 2108  ∀wral 3063  ∃wrex 3064   ∖ cdif 3880   ⊆ wss 3883  ∪ cuni 4836  Ord word 6250  Oncon0 6251

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-11 2156  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-tr 5188  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-ord 6254  df-on 6255

This theorem is referenced by: (None)