Theorem intmin 4925

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 3457	. . . . 5 ⊢ 𝑦 ∈ V
2	1	elintrab 4917	. . . 4 ⊢ (𝑦 ∈ ∩ {𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥} ↔ ∀𝑥 ∈ 𝐵 (𝐴 ⊆ 𝑥 → 𝑦 ∈ 𝑥))
3		ssid 3958	. . . . 5 ⊢ 𝐴 ⊆ 𝐴
4		sseq2 3962	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝐴 ⊆ 𝑥 ↔ 𝐴 ⊆ 𝐴))
5		eleq2 2850	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑦 ∈ 𝑥 ↔ 𝑦 ∈ 𝐴))
6	4, 5	imbi12d 346	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝐴 ⊆ 𝑥 → 𝑦 ∈ 𝑥) ↔ (𝐴 ⊆ 𝐴 → 𝑦 ∈ 𝐴)))
7	6	rspcv 3577	. . . . 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 244	. . 3 ⊢ (𝐴 ∈ 𝐵 → (𝑦 ∈ ∩ {𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥} → 𝑦 ∈ 𝐴))
10	9	ssrdv 3942	. 2 ⊢ (𝐴 ∈ 𝐵 → ∩ {𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥} ⊆ 𝐴)
11		ssintub 4923	. . 3 ⊢ 𝐴 ⊆ ∩ {𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥}
12	11	a1i 11	. 2 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ⊆ ∩ {𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥})
13	10, 12	eqssd 3953	1 ⊢ (𝐴 ∈ 𝐵 → ∩ {𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥} = 𝐴)

Syntax hints:   → wi 4   = wceq 1559   ∈ wcel 2141  ∀wral 3075  {crab 3413   ⊆ wss 3904  ∩ cint 4904

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1562  df-ex 1799  df-nf 1803  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-ss 3921  df-int 4905

This theorem is referenced by:  intmin2  4932  ordintdif  6393  uniordint  7780  onsucmin  7797  naddrid  8649  naddasslem1  8660  naddasslem2  8661  rankonidlem  9783  rankval4  9822  harsucnn  9953  mrcid  17628  lspid  21029  aspid  21906  cldcls  23082  spanid  31496  chsupid  31561  fldgenidfld  33465  rankval4b  35360  igenidl2  38528  pclidN  40484  diaocN  41713  onuniintrab  43767  topclat  49583  toplatlub  49585  toplatjoin  49587