Theorem opelxp2 5674

Description: The second member of an ordered pair of classes in a Cartesian product belongs to second Cartesian product argument. (Contributed by Mario Carneiro, 26-Apr-2015.)

Assertion

Ref	Expression
opelxp2	⊢ (⟨𝐴, 𝐵⟩ ∈ (𝐶 × 𝐷) → 𝐵 ∈ 𝐷)

Proof of Theorem opelxp2

Step	Hyp	Ref	Expression
1		opelxp 5667	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝐶 × 𝐷) ↔ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷))
2	1	simprbi 496	1 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝐶 × 𝐷) → 𝐵 ∈ 𝐷)

Syntax hints:   → wi 4   ∈ wcel 2109  ⟨cop 4591   × cxp 5629

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 5246  ax-nul 5256  ax-pr 5382

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 3403  df-v 3446  df-dif 3914  df-un 3916  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-opab 5165  df-xp 5637

This theorem is referenced by:  dff4  7055  eceqoveq  8772  axdc4lem  10384  canthp1lem2  10582  cicrcl  17745  txcmplem1  23561  txlm  23568  brcgr  28880  nvex  30590  fldextfld2  33637  prsrn  33898  pprodss4v  35865  poimirlem27  37634  natglobalincr  46868  fuco1  49303  fuco2  49305  fucoid2  49331  fucocolem2  49336  reldmlan2  49599  reldmran2  49600  lanrcl  49603  ranrcl  49604