Theorem opelxp2 5702

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 5695	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝐶 × 𝐷) ↔ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷))
2	1	simprbi 503	1 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝐶 × 𝐷) → 𝐵 ∈ 𝐷)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2145  ⟨cop 4593   × cxp 5657

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 2734  ax-sep 5255  ax-pr 5402

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 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-opab 5172  df-xp 5665

This theorem is used by:  dff4  7097  eceqoveq  8825  axdc4lem  10460  canthp1lem2  10663  cicrcl  17894  txcmplem1  23866  txlm  23873  brcgr  29341  nvex  31076  fldextfld2  34143  prsrn  34410  pprodss4v  36446  poimirlem27  38381  fuco1  50232  fuco2  50234  fucoid2  50260  fucocolem2  50265  reldmlan2  50528  reldmran2  50529  lanrcl  50532  ranrcl  50533