Theorem opelxp1 5690

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 5684	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝐶 × 𝐷) ↔ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷))
2	1	simplbi 502	1 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝐶 × 𝐷) → 𝐴 ∈ 𝐶)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2145  ⟨cop 4590   × cxp 5646

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732  ax-sep 5249  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-opab 5168  df-xp 5654

This theorem is used by:  otelxp1  5693  dff3  7089  ressnop0  7146  swoord1  8729  swoord2  8730  isfin4p1  10350  canthp1lem2  10695  ciclcl  17924  txcmplem1  23907  txlm  23914  dvbsss  26169  nvvcop  31115  nvvop  31130  fldextfld1  34198  prsdm  34465  linedegen  36824  bj-opelresdm  37980  bj-idres  37995  opelopab3  38566  et-ltneverrefl  47797  fuco1  50345  fuco2  50347  fucoid2  50373  fucocolem2  50378  reldmlan2  50641  reldmran2  50642  lanrcl  50645  ranrcl  50646