Theorem opelxp2 5291

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 5286	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝐶 × 𝐷) ↔ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷))
2	1	simprbi 484	1 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝐶 × 𝐷) → 𝐵 ∈ 𝐷)

Syntax hints:   → wi 4   ∈ wcel 2145  ⟨cop 4322   × cxp 5247

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-opab 4847  df-xp 5255

This theorem is referenced by:  dff4  6516  eceqoveq  8005  isfin4-3  9339  axdc4lem  9479  canthp1lem2  9677  cicrcl  16670  txcmplem1  21665  txlm  21672  brcgr  26001  nvex  27806  prsrn  30301  pprodss4v  32328  poimirlem27  33769