Theorem unimax 4920

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 3981	. . 3 ⊢ 𝐴 ⊆ 𝐴
2		sseq1 3984	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ⊆ 𝐴 ↔ 𝐴 ⊆ 𝐴))
3	2	elrab3 3672	. . 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 3984	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 ⊆ 𝐴 ↔ 𝑦 ⊆ 𝐴))
6	5	elrab 3671	. . . 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 3053	. 2 ⊢ ∀𝑦 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴}𝑦 ⊆ 𝐴
9		ssunieq 4919	. . 3 ⊢ ((𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴} ∧ ∀𝑦 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴}𝑦 ⊆ 𝐴) → 𝐴 = ∪ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴})
10	9	eqcomd 2741	. 2 ⊢ ((𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴} ∧ ∀𝑦 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴}𝑦 ⊆ 𝐴) → ∪ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴} = 𝐴)
11	4, 8, 10	sylancl 586	1 ⊢ (𝐴 ∈ 𝐵 → ∪ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴} = 𝐴)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ∀wral 3051  {crab 3415   ⊆ wss 3926  ∪ cuni 4883

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-ral 3052  df-rab 3416  df-v 3461  df-ss 3943  df-uni 4884

This theorem is referenced by:  lssuni  20896  chsupid  31393  shatomistici  32342  lssats  39030  lpssat  39031  lssatle  39033  lssat  39034  mrelatglbALT  48970  mreclat  48971  toplatmeet  48977