Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  ineqri Structured version   Visualization version   GIF version

Theorem ineqri 4100
 Description: Inference from membership to intersection. (Contributed by NM, 21-Jun-1993.)
Hypothesis
Ref Expression
ineqri.1 ((𝑥𝐴𝑥𝐵) ↔ 𝑥𝐶)
Assertion
Ref Expression
ineqri (𝐴𝐵) = 𝐶
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶

Proof of Theorem ineqri
StepHypRef Expression
1 elin 4090 . . 3 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
2 ineqri.1 . . 3 ((𝑥𝐴𝑥𝐵) ↔ 𝑥𝐶)
31, 2bitri 276 . 2 (𝑥 ∈ (𝐴𝐵) ↔ 𝑥𝐶)
43eqriv 2792 1 (𝐴𝐵) = 𝐶
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 207   ∧ wa 396   = wceq 1522   ∈ wcel 2081   ∩ cin 3858 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-ext 2769 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-v 3439  df-in 3866 This theorem is referenced by:  inidm  4115  inass  4116  dfin2  4157  indi  4170  inab  4191  in0  4265  pwin  5342  dfres3  5739  dmres  5756  inixp  34535
 Copyright terms: Public domain W3C validator