Theorem elintab 4925

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 4924	. 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 1568   ∈ wcel 2143  {cab 2741  Vcvv 3455  ∩ cint 4913

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1573  df-ex 1810  df-nf 1814  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-int 4914

This theorem is referenced by:  elintrab  4926  intmin4  4943  intab  4944  dfom3  9617  dfom5  9620  tc2  9710  dfnn2  12247  brintclab  15040  efgi  19790  efgi2  19796  dfn0s2  28503  mclsax  36039  heibor1lem  38438  intabssd  44225  elmapintab  44302  cotrintab  44320  dffrege76  44645