Theorem ssmin 4932

Description: Subclass of the minimum value of class of supersets. (Contributed by NM, 10-Aug-2006.)

Assertion

Ref	Expression
ssmin	⊢ 𝐴 ⊆ ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ 𝜑)}

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem ssmin

Step	Hyp	Ref	Expression
1		ssintab 4930	. 2 ⊢ (𝐴 ⊆ ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ 𝜑)} ↔ ∀𝑥((𝐴 ⊆ 𝑥 ∧ 𝜑) → 𝐴 ⊆ 𝑥))
2		simpl 488	. 2 ⊢ ((𝐴 ⊆ 𝑥 ∧ 𝜑) → 𝐴 ⊆ 𝑥)
3	1, 2	mpgbir 1832	1 ⊢ 𝐴 ⊆ ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ 𝜑)}

This proof depends on syntax axioms:   → wi 4   ∧ wa 401  {cab 2741   ⊆ wss 3900  ∩ cint 4912

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735

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 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-v 3456  df-ss 3901  df-int 4913

This theorem is used by:  tcid  9775  trclfvlb  15204  trclun  15210  tz9.1regs  35943  tz9.1tco  37486  dfttc3gw  37526  dmtrcl  44888  rntrcl  44889  dfrtrcl5  44890