Theorem unimax 3687

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 3044	. . 3 ⊢ 𝐴 ⊆ 𝐴
2		sseq1 3047	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ⊆ 𝐴 ↔ 𝐴 ⊆ 𝐴))
3	2	elrab3 2772	. . 3 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴} ↔ 𝐴 ⊆ 𝐴))
4	1, 3	mpbiri 166	. 2 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴})
5		sseq1 3047	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 ⊆ 𝐴 ↔ 𝑦 ⊆ 𝐴))
6	5	elrab 2771	. . . 4 ⊢ (𝑦 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴} ↔ (𝑦 ∈ 𝐵 ∧ 𝑦 ⊆ 𝐴))
7	6	simprbi 269	. . 3 ⊢ (𝑦 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴} → 𝑦 ⊆ 𝐴)
8	7	rgen 2428	. 2 ⊢ ∀𝑦 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴}𝑦 ⊆ 𝐴
9		ssunieq 3686	. . 3 ⊢ ((𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴} ∧ ∀𝑦 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴}𝑦 ⊆ 𝐴) → 𝐴 = ∪ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴})
10	9	eqcomd 2093	. 2 ⊢ ((𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴} ∧ ∀𝑦 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴}𝑦 ⊆ 𝐴) → ∪ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴} = 𝐴)
11	4, 8, 10	sylancl 404	1 ⊢ (𝐴 ∈ 𝐵 → ∪ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴} = 𝐴)

Syntax hints:   → wi 4   ∧ wa 102   = wceq 1289   ∈ wcel 1438  ∀wral 2359  {crab 2363   ⊆ wss 2999  ∪ cuni 3653

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rab 2368  df-v 2621  df-in 3005  df-ss 3012  df-uni 3654

This theorem is referenced by:  onuniss2  4329