Theorem elintab 4901

Description: Membership in the intersection of a class abstraction. (Contributed by NM, 30-Aug-1993.)

Hypothesis

Ref	Expression
elintab.ex	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
elintab	⊢ (𝐴 ∈ ∩ {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝜑 → 𝐴 ∈ 𝑥))

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem elintab

Step	Hyp	Ref	Expression
1		elintab.ex	. 2 ⊢ 𝐴 ∈ V
2		elintabg 4900	. 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ ∩ {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝜑 → 𝐴 ∈ 𝑥)))
3	1, 2	ax-mp 5	1 ⊢ (𝐴 ∈ ∩ {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝜑 → 𝐴 ∈ 𝑥))

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1540   ∈ wcel 2114  {cab 2714  Vcvv 3429  ∩ cint 4889

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-int 4890

This theorem is referenced by:  elintrab  4902  intmin4  4919  intab  4920  dfom3  9568  dfom5  9571  tc2  9661  dfnn2  12187  brintclab  14963  efgi  19694  efgi2  19700  dfn0s2  28324  mclsax  35751  heibor1lem  38130  intabssd  43946  elmapintab  44023  cotrintab  44041  dffrege76  44366