Theorem ssintub 4931

Description: Subclass of the least upper bound. (Contributed by NM, 8-Aug-2000.)

Assertion

Ref	Expression
ssintub	⊢ 𝐴 ⊆ ∩ {𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥}

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem ssintub

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssint 4929	. 2 ⊢ (𝐴 ⊆ ∩ {𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥} ↔ ∀𝑦 ∈ {𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥}𝐴 ⊆ 𝑦)
2		sseq2 3963	. . . 4 ⊢ (𝑥 = 𝑦 → (𝐴 ⊆ 𝑥 ↔ 𝐴 ⊆ 𝑦))
3	2	elrab 3650	. . 3 ⊢ (𝑦 ∈ {𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥} ↔ (𝑦 ∈ 𝐵 ∧ 𝐴 ⊆ 𝑦))
4	3	simprbi 502	. 2 ⊢ (𝑦 ∈ {𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥} → 𝐴 ⊆ 𝑦)
5	1, 4	mprgbir 3086	1 ⊢ 𝐴 ⊆ ∩ {𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥}

Syntax hints:   ∈ wcel 2143  {crab 3416   ⊆ wss 3905  ∩ cint 4912

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-11 2192  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1573  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rab 3417  df-v 3457  df-ss 3922  df-int 4913

This theorem is referenced by:  intmin  4933  cofon2  8655  naddunif  8676  wuncid  10723  mrcssid  17668  rgspnssid  20713  lspssid  21106  lbsextlem3  21284  aspssid  22027  sscls  23213  filufint  24077  spanss2  31697  shsval2i  31739  ococin  31760  chsupsn  31765  fldgenssid  33634  sssigagen  34535  dynkin  34557  igenss  38713  pclssidN  40669  dochocss  42140  intubeu  49762