Theorem ssintab 4766

Description: Subclass of the intersection of a class abstraction. (Contributed by NM, 31-Jul-2006.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Assertion

Ref	Expression
ssintab	⊢ (𝐴 ⊆ ∩ {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝜑 → 𝐴 ⊆ 𝑥))

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem ssintab

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssint 4765	. 2 ⊢ (𝐴 ⊆ ∩ {𝑥 ∣ 𝜑} ↔ ∀𝑦 ∈ {𝑥 ∣ 𝜑}𝐴 ⊆ 𝑦)
2		sseq2 3884	. . 3 ⊢ (𝑦 = 𝑥 → (𝐴 ⊆ 𝑦 ↔ 𝐴 ⊆ 𝑥))
3	2	ralab2 3605	. 2 ⊢ (∀𝑦 ∈ {𝑥 ∣ 𝜑}𝐴 ⊆ 𝑦 ↔ ∀𝑥(𝜑 → 𝐴 ⊆ 𝑥))
4	1, 3	bitri 267	1 ⊢ (𝐴 ⊆ ∩ {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝜑 → 𝐴 ⊆ 𝑥))

Syntax hints:   → wi 4   ↔ wb 198  ∀wal 1505  {cab 2759  ∀wral 3089   ⊆ wss 3830  ∩ cint 4749

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-ext 2751

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-clab 2760  df-cleq 2772  df-clel 2847  df-nfc 2919  df-ral 3094  df-v 3418  df-in 3837  df-ss 3844  df-int 4750

This theorem is referenced by:  ssmin  4768  ssintrab  4772  intmin4  4778  dffi2  8682  rankval3b  9049  sstskm  10062  dfuzi  11886  cycsubg  18091  ssmclslem  32329  mptrcllem  39333  dfrcl2  39379  brtrclfv2  39432