Theorem elintab 4922

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 4921	. 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 1538   ∈ wcel 2109  {cab 2707  Vcvv 3447  ∩ cint 4910

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-int 4911

This theorem is referenced by:  elintrab  4924  intmin4  4941  intab  4942  intidOLD  5418  dfom3  9600  dfom5  9603  tc2  9695  dfnn2  12199  brintclab  14967  efgi  19649  efgi2  19655  dfn0s2  28224  mclsax  35556  heibor1lem  37803  intabssd  43508  elmapintab  43585  cotrintab  43603  dffrege76  43928