Theorem unisseq 1946

Description: A set is a union of its subsets.

Assertion

Ref	Expression
unisseq	⊢ (A ∈ B → A = ∪{x ∈ B∣x ⊆ A})

Distinct variable group(s):   x,A   x,B

Proof of Theorem unisseq

Step	Hyp	Ref	Expression
1		ssid 1519	. . . 4 ⊢ A ⊆ A
2	1	jctr 239	. . 3 ⊢ (A ∈ B → (A ∈ B ∧ A ⊆ A))
3		sseq1 1521	. . . 4 ⊢ (x = A → (x ⊆ A ↔ A ⊆ A))
4	3	elrab 1422	. . 3 ⊢ (A ∈ {x ∈ B∣x ⊆ A} ↔ (A ∈ B ∧ A ⊆ A))
5	2, 4	sylibr 175	. 2 ⊢ (A ∈ B → A ∈ {x ∈ B∣x ⊆ A})
6		sseq1 1521	. . . . . 6 ⊢ (x = y → (x ⊆ A ↔ y ⊆ A))
7	6	elrab 1422	. . . . 5 ⊢ (y ∈ {x ∈ B∣x ⊆ A} ↔ (y ∈ B ∧ y ⊆ A))
8	7	pm3.27bd 263	. . . 4 ⊢ (y ∈ {x ∈ B∣x ⊆ A} → y ⊆ A)
9	8	rgen 1247	. . 3 ⊢ ∀y ∈ {x ∈ B∣x ⊆ A}y ⊆ A
10		ssunieq 1945	. . 3 ⊢ ((A ∈ {x ∈ B∣x ⊆ A} ∧ ∀y ∈ {x ∈ B∣x ⊆ A}y ⊆ A) → A = ∪{x ∈ B∣x ⊆ A})
11	9, 10	mpan2 519	. 2 ⊢ (A ∈ {x ∈ B∣x ⊆ A} → A = ∪{x ∈ B∣x ⊆ A})
12	5, 11	syl 12	1 ⊢ (A ∈ B → A = ∪{x ∈ B∣x ⊆ A})

Syntax hints:   → wi 2   ∧ wa 196   = wceq 1091   ∈ wcel 1092  ∀wral 1201  {crab 1204   ⊆ wss 1487  ∪cuni 1919

This theorem is referenced by:  chsupid 5312

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-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rab 1208  df-v 1349  df-in 1491  df-ss 1492  df-uni 1920

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