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

Theorem ineqri 3949
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 3939 . . 3 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
2 ineqri.1 . . 3 ((𝑥𝐴𝑥𝐵) ↔ 𝑥𝐶)
31, 2bitri 264 . 2 (𝑥 ∈ (𝐴𝐵) ↔ 𝑥𝐶)
43eqriv 2757 1 (𝐴𝐵) = 𝐶
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383   = wceq 1632  wcel 2139  cin 3714
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-v 3342  df-in 3722
This theorem is referenced by:  inidm  3965  inass  3966  dfin2  4003  indi  4016  inab  4038  in0  4111  pwin  5168  dfres3  5556  dmres  5577  inixp  33854
  Copyright terms: Public domain W3C validator