Theorem ssintub 3654

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 3652	. 2 ⊢ (𝐴 ⊆ ∩ {𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥} ↔ ∀𝑦 ∈ {𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥}𝐴 ⊆ 𝑦)
2		sseq2 3021	. . . 4 ⊢ (𝑥 = 𝑦 → (𝐴 ⊆ 𝑥 ↔ 𝐴 ⊆ 𝑦))
3	2	elrab 2749	. . 3 ⊢ (𝑦 ∈ {𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥} ↔ (𝑦 ∈ 𝐵 ∧ 𝐴 ⊆ 𝑦))
4	3	simprbi 269	. 2 ⊢ (𝑦 ∈ {𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥} → 𝐴 ⊆ 𝑦)
5	1, 4	mprgbir 2421	1 ⊢ 𝐴 ⊆ ∩ {𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥}

Syntax hints:   ∈ wcel 1433  {crab 2352   ⊆ wss 2973  ∩ cint 3636

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rab 2357  df-v 2603  df-in 2979  df-ss 2986  df-int 3637

This theorem is referenced by:  intmin  3656