Theorem ssintub 4925

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 4923	. 2 ⊢ (𝐴 ⊆ ∩ {𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥} ↔ ∀𝑦 ∈ {𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥}𝐴 ⊆ 𝑦)
2		sseq2 3935	. . . 4 ⊢ (𝑥 = 𝑦 → (𝐴 ⊆ 𝑥 ↔ 𝐴 ⊆ 𝑦))
3	2	elrab 3644	. . 3 ⊢ (𝑦 ∈ {𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥} ↔ (𝑦 ∈ 𝐵 ∧ 𝐴 ⊆ 𝑦))
4	3	simprbi 503	. 2 ⊢ (𝑦 ∈ {𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥} → 𝐴 ⊆ 𝑦)
5	1, 4	mprgbir 3083	1 ⊢ 𝐴 ⊆ ∩ {𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥}

This proof depends on syntax axioms:   ∈ wcel 2144  {crab 3412   ⊆ wss 3896  ∩ cint 4906

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-11 2193  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rab 3413  df-v 3452  df-ss 3897  df-int 4907

This theorem is used by:  intmin  4927  cofon2  8690  naddunif  8711  wuncid  10835  mrcssid  17800  rgspnssid  20875  lspssid  21269  lbsextlem3  21447  aspssid  22194  sscls  23383  filufint  24248  spanss2  31956  shsval2i  31998  ococin  32019  chsupsn  32024  fldgenssid  33925  sssigagen  34828  dynkin  34850  igenss  39133  pclssidN  41089  dochocss  42560  intubeu  50234