MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  opelxp1 Structured version   Visualization version   GIF version

Theorem opelxp1 5564
Description: The first member of an ordered pair of classes in a Cartesian product belongs to first Cartesian product argument. (Contributed by NM, 28-May-2008.) (Revised by Mario Carneiro, 26-Apr-2015.)
Assertion
Ref Expression
opelxp1 (⟨𝐴, 𝐵⟩ ∈ (𝐶 × 𝐷) → 𝐴𝐶)

Proof of Theorem opelxp1
StepHypRef Expression
1 opelxp 5559 . 2 (⟨𝐴, 𝐵⟩ ∈ (𝐶 × 𝐷) ↔ (𝐴𝐶𝐵𝐷))
21simplbi 501 1 (⟨𝐴, 𝐵⟩ ∈ (𝐶 × 𝐷) → 𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2112  cop 4534   × cxp 5521
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-v 3446  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-opab 5096  df-xp 5529
This theorem is referenced by:  otelxp1  5566  dff3  6847  ressnop0  6896  swoord1  8307  swoord2  8308  isfin4p1  9730  canthp1lem2  10068  ciclcl  17067  txcmplem1  22249  txlm  22256  dvbsss  24508  nvvcop  28380  nvvop  28395  fldextfld1  31127  prsdm  31265  linedegen  33712  bj-opelresdm  34555  bj-idres  34570  opelopab3  35148
  Copyright terms: Public domain W3C validator