Theorem unimax 4968

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 4031	. . 3 ⊢ 𝐴 ⊆ 𝐴
2		sseq1 4034	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ⊆ 𝐴 ↔ 𝐴 ⊆ 𝐴))
3	2	elrab3 3709	. . 3 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴} ↔ 𝐴 ⊆ 𝐴))
4	1, 3	mpbiri 258	. 2 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴})
5		sseq1 4034	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 ⊆ 𝐴 ↔ 𝑦 ⊆ 𝐴))
6	5	elrab 3708	. . . 4 ⊢ (𝑦 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴} ↔ (𝑦 ∈ 𝐵 ∧ 𝑦 ⊆ 𝐴))
7	6	simprbi 496	. . 3 ⊢ (𝑦 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴} → 𝑦 ⊆ 𝐴)
8	7	rgen 3069	. 2 ⊢ ∀𝑦 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴}𝑦 ⊆ 𝐴
9		ssunieq 4967	. . 3 ⊢ ((𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴} ∧ ∀𝑦 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴}𝑦 ⊆ 𝐴) → 𝐴 = ∪ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴})
10	9	eqcomd 2746	. 2 ⊢ ((𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴} ∧ ∀𝑦 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴}𝑦 ⊆ 𝐴) → ∪ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴} = 𝐴)
11	4, 8, 10	sylancl 585	1 ⊢ (𝐴 ∈ 𝐵 → ∪ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴} = 𝐴)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2108  ∀wral 3067  {crab 3443   ⊆ wss 3976  ∪ cuni 4931

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rab 3444  df-v 3490  df-ss 3993  df-uni 4932

This theorem is referenced by:  lssuni  20960  chsupid  31444  shatomistici  32393  lssats  38968  lpssat  38969  lssatle  38971  lssat  38972  mrelatglbALT  48668  mreclat  48669  toplatmeet  48675