Theorem intmin 4928

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 3455	. . . . 5 ⊢ 𝑦 ∈ V
2	1	elintrab 4920	. . . 4 ⊢ (𝑦 ∈ ∩ {𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥} ↔ ∀𝑥 ∈ 𝐵 (𝐴 ⊆ 𝑥 → 𝑦 ∈ 𝑥))
3		ssid 3953	. . . . 5 ⊢ 𝐴 ⊆ 𝐴
4		sseq2 3957	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝐴 ⊆ 𝑥 ↔ 𝐴 ⊆ 𝐴))
5		eleq2 2850	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑦 ∈ 𝑥 ↔ 𝑦 ∈ 𝐴))
6	4, 5	imbi12d 347	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝐴 ⊆ 𝑥 → 𝑦 ∈ 𝑥) ↔ (𝐴 ⊆ 𝐴 → 𝑦 ∈ 𝐴)))
7	6	rspcv 3573	. . . . 5 ⊢ (𝐴 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 (𝐴 ⊆ 𝑥 → 𝑦 ∈ 𝑥) → (𝐴 ⊆ 𝐴 → 𝑦 ∈ 𝐴)))
8	3, 7	mpii 47	. . . 4 ⊢ (𝐴 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 (𝐴 ⊆ 𝑥 → 𝑦 ∈ 𝑥) → 𝑦 ∈ 𝐴))
9	2, 8	biimtrid 245	. . 3 ⊢ (𝐴 ∈ 𝐵 → (𝑦 ∈ ∩ {𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥} → 𝑦 ∈ 𝐴))
10	9	ssrdv 3937	. 2 ⊢ (𝐴 ∈ 𝐵 → ∩ {𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥} ⊆ 𝐴)
11		ssintub 4926	. . 3 ⊢ 𝐴 ⊆ ∩ {𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥}
12	11	a1i 11	. 2 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ⊆ ∩ {𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥})
13	10, 12	eqssd 3948	1 ⊢ (𝐴 ∈ 𝐵 → ∩ {𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥} = 𝐴)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  ∀wral 3077  {crab 3413   ⊆ wss 3899  ∩ cint 4907

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-ss 3916  df-int 4908

This theorem is used by:  intmin2  4935  ordintdif  6407  uniordint  7804  onsucmin  7821  naddrid  8677  naddasslem1  8688  naddasslem2  8689  rankonidlem  9819  rankval4b  9861  rankval4  9865  harsucnn  10060  mrcid  17767  lspid  21237  aspid  22162  cldcls  23340  spanid  31931  chsupid  31996  fldgenidfld  33861  nmulrid  36916  igenidl2  38967  pclidN  40921  diaocN  42150  onuniintrab  44186  topclat  50050  toplatlub  50052  toplatjoin  50054