Theorem opelxp1 5676

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 5670	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝐶 × 𝐷) ↔ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷))
2	1	simplbi 496	1 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝐶 × 𝐷) → 𝐴 ∈ 𝐶)

Syntax hints:   → wi 4   ∈ wcel 2114  ⟨cop 4588   × cxp 5632

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5245  ax-pr 5381

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-opab 5163  df-xp 5640

This theorem is referenced by:  otelxp1  5679  dff3  7056  ressnop0  7110  swoord1  8680  swoord2  8681  isfin4p1  10239  canthp1lem2  10578  ciclcl  17740  txcmplem1  23602  txlm  23609  dvbsss  25876  nvvcop  30688  nvvop  30703  fldextfld1  33831  prsdm  34098  linedegen  36365  bj-opelresdm  37427  bj-idres  37442  opelopab3  37998  et-ltneverrefl  47258  natglobalincr  47264  fuco1  49709  fuco2  49711  fucoid2  49737  fucocolem2  49742  reldmlan2  50005  reldmran2  50006  lanrcl  50009  ranrcl  50010