Theorem elintab 4920

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 4919	. 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 209  ∀wal 1561   ∈ wcel 2145  {cab 2743  Vcvv 3457  ∩ cint 4908

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1566  df-ex 1803  df-nf 1807  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3080  df-int 4909

This theorem is referenced by:  elintrab  4921  intmin4  4938  intab  4939  dfom3  9604  dfom5  9607  tc2  9697  dfnn2  12237  brintclab  15028  efgi  19780  efgi2  19786  dfn0s2  28483  mclsax  35932  heibor1lem  38320  intabssd  44107  elmapintab  44184  cotrintab  44202  dffrege76  44527