Theorem ordunidif 2260

Description: The union of an ordinal stays the same if a subset equal to one of its elements is removed.

Assertion

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

Proof of Theorem ordunidif

Step	Hyp	Ref	Expression
1		ordelon 2222	. . . . . . . . . 10 ⊢ ((Ord A ∧ B ∈ A) → B ∈ On)
2		eldif 1496	. . . . . . . . . . . . . 14 ⊢ (B ∈ (A ∖ B) ↔ (B ∈ A ∧ ¬ B ∈ B))
3	2	biimpr 134	. . . . . . . . . . . . 13 ⊢ ((B ∈ A ∧ ¬ B ∈ B) → B ∈ (A ∖ B))
4	3	exp 291	. . . . . . . . . . . 12 ⊢ (B ∈ A → (¬ B ∈ B → B ∈ (A ∖ B)))
5		eloni 2209	. . . . . . . . . . . . 13 ⊢ (B ∈ On → Ord B)
6		ordeirr 2217	. . . . . . . . . . . . 13 ⊢ (Ord B → ¬ B ∈ B)
7	5, 6	syl 12	. . . . . . . . . . . 12 ⊢ (B ∈ On → ¬ B ∈ B)
8	4, 7	syl5 22	. . . . . . . . . . 11 ⊢ (B ∈ A → (B ∈ On → B ∈ (A ∖ B)))
9	8	adantl 305	. . . . . . . . . 10 ⊢ ((Ord A ∧ B ∈ A) → (B ∈ On → B ∈ (A ∖ B)))
10	1, 9	mpd 46	. . . . . . . . 9 ⊢ ((Ord A ∧ B ∈ A) → B ∈ (A ∖ B))
11	10	a1d 14	. . . . . . . 8 ⊢ ((Ord A ∧ B ∈ A) → (x ∈ B → B ∈ (A ∖ B)))
12		onelsst 2255	. . . . . . . . 9 ⊢ (B ∈ On → (x ∈ B → x ⊆ B))
13	1, 12	syl 12	. . . . . . . 8 ⊢ ((Ord A ∧ B ∈ A) → (x ∈ B → x ⊆ B))
14	11, 13	jcad 455	. . . . . . 7 ⊢ ((Ord A ∧ B ∈ A) → (x ∈ B → (B ∈ (A ∖ B) ∧ x ⊆ B)))
15	14	adantr 306	. . . . . 6 ⊢ (((Ord A ∧ B ∈ A) ∧ x ∈ A) → (x ∈ B → (B ∈ (A ∖ B) ∧ x ⊆ B)))
16		sseq2 1522	. . . . . . 7 ⊢ (y = B → (x ⊆ y ↔ x ⊆ B))
17	16	rcla4ev 1403	. . . . . 6 ⊢ ((B ∈ (A ∖ B) ∧ x ⊆ B) → ∃y ∈ (A ∖ B)x ⊆ y)
18	15, 17	syl6 23	. . . . 5 ⊢ (((Ord A ∧ B ∈ A) ∧ x ∈ A) → (x ∈ B → ∃y ∈ (A ∖ B)x ⊆ y))
19		eldif 1496	. . . . . . . . . 10 ⊢ (x ∈ (A ∖ B) ↔ (x ∈ A ∧ ¬ x ∈ B))
20	19	biimpr 134	. . . . . . . . 9 ⊢ ((x ∈ A ∧ ¬ x ∈ B) → x ∈ (A ∖ B))
21		ssid 1519	. . . . . . . . 9 ⊢ x ⊆ x
22	20, 21	jctir 241	. . . . . . . 8 ⊢ ((x ∈ A ∧ ¬ x ∈ B) → (x ∈ (A ∖ B) ∧ x ⊆ x))
23	22	exp 291	. . . . . . 7 ⊢ (x ∈ A → (¬ x ∈ B → (x ∈ (A ∖ B) ∧ x ⊆ x)))
24		sseq2 1522	. . . . . . . 8 ⊢ (y = x → (x ⊆ y ↔ x ⊆ x))
25	24	rcla4ev 1403	. . . . . . 7 ⊢ ((x ∈ (A ∖ B) ∧ x ⊆ x) → ∃y ∈ (A ∖ B)x ⊆ y)
26	23, 25	syl6 23	. . . . . 6 ⊢ (x ∈ A → (¬ x ∈ B → ∃y ∈ (A ∖ B)x ⊆ y))
27	26	adantl 305	. . . . 5 ⊢ (((Ord A ∧ B ∈ A) ∧ x ∈ A) → (¬ x ∈ B → ∃y ∈ (A ∖ B)x ⊆ y))
28	18, 27	pm2.61d 112	. . . 4 ⊢ (((Ord A ∧ B ∈ A) ∧ x ∈ A) → ∃y ∈ (A ∖ B)x ⊆ y)
29	28	exp 291	. . 3 ⊢ ((Ord A ∧ B ∈ A) → (x ∈ A → ∃y ∈ (A ∖ B)x ⊆ y))
30	29	r19.21aiv 1259	. 2 ⊢ ((Ord A ∧ B ∈ A) → ∀x ∈ A ∃y ∈ (A ∖ B)x ⊆ y)
31		unidif 1943	. 2 ⊢ (∀x ∈ A ∃y ∈ (A ∖ B)x ⊆ y → ∪(A ∖ B) = ∪A)
32	30, 31	syl 12	1 ⊢ ((Ord A ∧ B ∈ A) → ∪(A ∖ B) = ∪A)

Syntax hints:  ¬ wn 1   → wi 2   ∧ wa 196   = wceq 1091   ∈ wcel 1092  ∀wral 1201  ∃wrex 1202   ∖ cdif 1484   ⊆ wss 1487  ∪cuni 1919  Ord word 2198  Oncon0 2199

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3an 583  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203

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