Theorem opelxp1 5680

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) → 𝐴 ∈ 𝐶)

Proof of Theorem opelxp1

Step	Hyp	Ref	Expression
1		opelxp 5674	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝐶 × 𝐷) ↔ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷))
2	1	simplbi 497	1 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝐶 × 𝐷) → 𝐴 ∈ 𝐶)

Syntax hints:   → wi 4   ∈ wcel 2109  ⟨cop 4595   × cxp 5636

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-opab 5170  df-xp 5644

This theorem is referenced by:  otelxp1  5683  dff3  7072  ressnop0  7125  swoord1  8703  swoord2  8704  isfin4p1  10268  canthp1lem2  10606  ciclcl  17764  txcmplem1  23528  txlm  23535  dvbsss  25803  nvvcop  30523  nvvop  30538  fldextfld1  33643  prsdm  33904  linedegen  36131  bj-opelresdm  37133  bj-idres  37148  opelopab3  37712  et-ltneverrefl  46869  natglobalincr  46875  fuco1  49310  fuco2  49312  fucoid2  49338  fucocolem2  49343  reldmlan2  49606  reldmran2  49607  lanrcl  49610  ranrcl  49611