Theorem unimax 3932

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 3248	. . 3 ⊢ 𝐴 ⊆ 𝐴
2		sseq1 3251	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ⊆ 𝐴 ↔ 𝐴 ⊆ 𝐴))
3	2	elrab3 2964	. . 3 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴} ↔ 𝐴 ⊆ 𝐴))
4	1, 3	mpbiri 168	. 2 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴})
5		sseq1 3251	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 ⊆ 𝐴 ↔ 𝑦 ⊆ 𝐴))
6	5	elrab 2963	. . . 4 ⊢ (𝑦 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴} ↔ (𝑦 ∈ 𝐵 ∧ 𝑦 ⊆ 𝐴))
7	6	simprbi 275	. . 3 ⊢ (𝑦 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴} → 𝑦 ⊆ 𝐴)
8	7	rgen 2586	. 2 ⊢ ∀𝑦 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴}𝑦 ⊆ 𝐴
9		ssunieq 3931	. . 3 ⊢ ((𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴} ∧ ∀𝑦 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴}𝑦 ⊆ 𝐴) → 𝐴 = ∪ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴})
10	9	eqcomd 2237	. 2 ⊢ ((𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴} ∧ ∀𝑦 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴}𝑦 ⊆ 𝐴) → ∪ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴} = 𝐴)
11	4, 8, 10	sylancl 413	1 ⊢ (𝐴 ∈ 𝐵 → ∪ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴} = 𝐴)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1398   ∈ wcel 2202  ∀wral 2511  {crab 2515   ⊆ wss 3201  ∪ cuni 3898

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rab 2520  df-v 2805  df-in 3207  df-ss 3214  df-uni 3899

This theorem is referenced by:  onuniss2  4616  lssuni  14459