Theorem unimax 3926

Description: Any member of a class is the largest of those members that it includes. (Contributed by NM, 13-Aug-2002.)

Assertion

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

Distinct variable groups:   x,A   x,B

Proof of Theorem unimax

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssid 3291	. . 3 ⊢ A ⊆ A
2		sseq1 3293	. . . 4 ⊢ (x = A → (x ⊆ A ↔ A ⊆ A))
3	2	elrab3 2996	. . 3 ⊢ (A ∈ B → (A ∈ {x ∈ B ∣ x ⊆ A} ↔ A ⊆ A))
4	1, 3	mpbiri 224	. 2 ⊢ (A ∈ B → A ∈ {x ∈ B ∣ x ⊆ A})
5		sseq1 3293	. . . . 5 ⊢ (x = y → (x ⊆ A ↔ y ⊆ A))
6	5	elrab 2995	. . . 4 ⊢ (y ∈ {x ∈ B ∣ x ⊆ A} ↔ (y ∈ B ∧ y ⊆ A))
7	6	simprbi 450	. . 3 ⊢ (y ∈ {x ∈ B ∣ x ⊆ A} → y ⊆ A)
8	7	rgen 2680	. 2 ⊢ ∀y ∈ {x ∈ B ∣ x ⊆ A}y ⊆ A
9		ssunieq 3925	. . 3 ⊢ ((A ∈ {x ∈ B ∣ x ⊆ A} ∧ ∀y ∈ {x ∈ B ∣ x ⊆ A}y ⊆ A) → A = ∪{x ∈ B ∣ x ⊆ A})
10	9	eqcomd 2358	. 2 ⊢ ((A ∈ {x ∈ B ∣ x ⊆ A} ∧ ∀y ∈ {x ∈ B ∣ x ⊆ A}y ⊆ A) → ∪{x ∈ B ∣ x ⊆ A} = A)
11	4, 8, 10	sylancl 643	1 ⊢ (A ∈ B → ∪{x ∈ B ∣ x ⊆ A} = A)

Syntax hints:   → wi 4   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∀wral 2615  {crab 2619   ⊆ wss 3258  ∪cuni 3892

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ral 2620  df-rab 2624  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260  df-uni 3893

This theorem is referenced by: (None)