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Theorem opelxp1 5589
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 5584 . 2 (⟨𝐴, 𝐵⟩ ∈ (𝐶 × 𝐷) ↔ (𝐴𝐶𝐵𝐷))
21simplbi 498 1 (⟨𝐴, 𝐵⟩ ∈ (𝐶 × 𝐷) → 𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2105  cop 4563   × cxp 5546
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-opab 5120  df-xp 5554
This theorem is referenced by:  otelxp1  5591  dff3  6858  ressnop0  6907  swoord1  8309  swoord2  8310  isfin4p1  9725  canthp1lem2  10063  ciclcl  17060  txcmplem1  22177  txlm  22184  dvbsss  24427  nvvcop  28298  nvvop  28313  fldextfld1  30938  prsdm  31056  linedegen  33501  bj-opelresdm  34329  bj-idres  34344  opelopab3  34873
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