Theorem ordunidif 6415

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 6388	. . . . . . . 8 ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝐵 ∈ On)
2		onelss 6407	. . . . . . . 8 ⊢ (𝐵 ∈ On → (𝑥 ∈ 𝐵 → 𝑥 ⊆ 𝐵))
3	1, 2	syl 18	. . . . . . 7 ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝑥 ∈ 𝐵 → 𝑥 ⊆ 𝐵))
4		eloni 6374	. . . . . . . . . . 11 ⊢ (𝐵 ∈ On → Ord 𝐵)
5		ordirr 6382	. . . . . . . . . . 11 ⊢ (Ord 𝐵 → ¬ 𝐵 ∈ 𝐵)
6	4, 5	syl 18	. . . . . . . . . 10 ⊢ (𝐵 ∈ On → ¬ 𝐵 ∈ 𝐵)
7		eldif 3916	. . . . . . . . . . 11 ⊢ (𝐵 ∈ (𝐴 ∖ 𝐵) ↔ (𝐵 ∈ 𝐴 ∧ ¬ 𝐵 ∈ 𝐵))
8	7	simplbi2 506	. . . . . . . . . 10 ⊢ (𝐵 ∈ 𝐴 → (¬ 𝐵 ∈ 𝐵 → 𝐵 ∈ (𝐴 ∖ 𝐵)))
9	6, 8	syl5 35	. . . . . . . . 9 ⊢ (𝐵 ∈ 𝐴 → (𝐵 ∈ On → 𝐵 ∈ (𝐴 ∖ 𝐵)))
10	9	adantl 487	. . . . . . . 8 ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝐵 ∈ On → 𝐵 ∈ (𝐴 ∖ 𝐵)))
11	1, 10	mpd 16	. . . . . . 7 ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝐵 ∈ (𝐴 ∖ 𝐵))
12	3, 11	jctild 535	. . . . . 6 ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝑥 ∈ 𝐵 → (𝐵 ∈ (𝐴 ∖ 𝐵) ∧ 𝑥 ⊆ 𝐵)))
13	12	adantr 486	. . . . 5 ⊢ (((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ 𝐵 → (𝐵 ∈ (𝐴 ∖ 𝐵) ∧ 𝑥 ⊆ 𝐵)))
14		sseq2 3964	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝑥 ⊆ 𝑦 ↔ 𝑥 ⊆ 𝐵))
15	14	rspcev 3583	. . . . 5 ⊢ ((𝐵 ∈ (𝐴 ∖ 𝐵) ∧ 𝑥 ⊆ 𝐵) → ∃𝑦 ∈ (𝐴 ∖ 𝐵)𝑥 ⊆ 𝑦)
16	13, 15	syl6 36	. . . 4 ⊢ (((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ 𝐵 → ∃𝑦 ∈ (𝐴 ∖ 𝐵)𝑥 ⊆ 𝑦))
17		eldif 3916	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵))
18	17	biimpri 231	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) → 𝑥 ∈ (𝐴 ∖ 𝐵))
19		ssid 3960	. . . . . . . 8 ⊢ 𝑥 ⊆ 𝑥
20	18, 19	jctir 530	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) → (𝑥 ∈ (𝐴 ∖ 𝐵) ∧ 𝑥 ⊆ 𝑥))
21	20	ex 418	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 → (¬ 𝑥 ∈ 𝐵 → (𝑥 ∈ (𝐴 ∖ 𝐵) ∧ 𝑥 ⊆ 𝑥)))
22		sseq2 3964	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (𝑥 ⊆ 𝑦 ↔ 𝑥 ⊆ 𝑥))
23	22	rspcev 3583	. . . . . 6 ⊢ ((𝑥 ∈ (𝐴 ∖ 𝐵) ∧ 𝑥 ⊆ 𝑥) → ∃𝑦 ∈ (𝐴 ∖ 𝐵)𝑥 ⊆ 𝑦)
24	21, 23	syl6 36	. . . . 5 ⊢ (𝑥 ∈ 𝐴 → (¬ 𝑥 ∈ 𝐵 → ∃𝑦 ∈ (𝐴 ∖ 𝐵)𝑥 ⊆ 𝑦))
25	24	adantl 487	. . . 4 ⊢ (((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑥 ∈ 𝐴) → (¬ 𝑥 ∈ 𝐵 → ∃𝑦 ∈ (𝐴 ∖ 𝐵)𝑥 ⊆ 𝑦))
26	16, 25	pm2.61d 181	. . 3 ⊢ (((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ (𝐴 ∖ 𝐵)𝑥 ⊆ 𝑦)
27	26	ralrimiva 3159	. 2 ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ (𝐴 ∖ 𝐵)𝑥 ⊆ 𝑦)
28		unidif 4910	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ (𝐴 ∖ 𝐵)𝑥 ⊆ 𝑦 → ∪ (𝐴 ∖ 𝐵) = ∪ 𝐴)
29	27, 28	syl 18	1 ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → ∪ (𝐴 ∖ 𝐵) = ∪ 𝐴)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2146  ∀wral 3081  ∃wrex 3091   ∖ cdif 3903   ⊆ wss 3906  ∪ cuni 4874  Ord word 6363  Oncon0 6364

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737  ax-sep 5259  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-tr 5221  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-ord 6367  df-on 6368

This theorem is used by: (None)