Theorem intmin 4932

Description: Any member of a class is the smallest of those members that include it. (Contributed by NM, 13-Aug-2002.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Assertion

Ref	Expression
intmin	⊢ (𝐴 ∈ 𝐵 → ∩ {𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥} = 𝐴)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem intmin

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 3451	. . . . 5 ⊢ 𝑦 ∈ V
2	1	elintrab 4924	. . . 4 ⊢ (𝑦 ∈ ∩ {𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥} ↔ ∀𝑥 ∈ 𝐵 (𝐴 ⊆ 𝑥 → 𝑦 ∈ 𝑥))
3		ssid 3969	. . . . 5 ⊢ 𝐴 ⊆ 𝐴
4		sseq2 3973	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝐴 ⊆ 𝑥 ↔ 𝐴 ⊆ 𝐴))
5		eleq2 2817	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑦 ∈ 𝑥 ↔ 𝑦 ∈ 𝐴))
6	4, 5	imbi12d 344	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝐴 ⊆ 𝑥 → 𝑦 ∈ 𝑥) ↔ (𝐴 ⊆ 𝐴 → 𝑦 ∈ 𝐴)))
7	6	rspcv 3584	. . . . 5 ⊢ (𝐴 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 (𝐴 ⊆ 𝑥 → 𝑦 ∈ 𝑥) → (𝐴 ⊆ 𝐴 → 𝑦 ∈ 𝐴)))
8	3, 7	mpii 46	. . . 4 ⊢ (𝐴 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 (𝐴 ⊆ 𝑥 → 𝑦 ∈ 𝑥) → 𝑦 ∈ 𝐴))
9	2, 8	biimtrid 242	. . 3 ⊢ (𝐴 ∈ 𝐵 → (𝑦 ∈ ∩ {𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥} → 𝑦 ∈ 𝐴))
10	9	ssrdv 3952	. 2 ⊢ (𝐴 ∈ 𝐵 → ∩ {𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥} ⊆ 𝐴)
11		ssintub 4930	. . 3 ⊢ 𝐴 ⊆ ∩ {𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥}
12	11	a1i 11	. 2 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ⊆ ∩ {𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥})
13	10, 12	eqssd 3964	1 ⊢ (𝐴 ∈ 𝐵 → ∩ {𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥} = 𝐴)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  ∀wral 3044  {crab 3405   ⊆ wss 3914  ∩ cint 4910

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-ss 3931  df-int 4911

This theorem is referenced by:  intmin2  4939  ordintdif  6383  uniordint  7777  onsucmin  7796  naddrid  8647  naddasslem1  8658  naddasslem2  8659  rankonidlem  9781  rankval4  9820  harsucnn  9951  mrcid  17574  lspid  20888  aspid  21784  cldcls  22929  spanid  31276  chsupid  31341  fldgenidfld  33267  igenidl2  38059  pclidN  39890  diaocN  41119  onuniintrab  43215  topclat  48986  toplatlub  48988  toplatjoin  48990