MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  uniopel Structured version   Visualization version   GIF version

Theorem uniopel 5126
Description: Ordered pair membership is inherited by class union. (Contributed by NM, 13-May-2008.) (Revised by Mario Carneiro, 26-Apr-2015.)
Hypotheses
Ref Expression
opthw.1 𝐴 ∈ V
opthw.2 𝐵 ∈ V
Assertion
Ref Expression
uniopel (⟨𝐴, 𝐵⟩ ∈ 𝐶𝐴, 𝐵⟩ ∈ 𝐶)

Proof of Theorem uniopel
StepHypRef Expression
1 opthw.1 . . . 4 𝐴 ∈ V
2 opthw.2 . . . 4 𝐵 ∈ V
31, 2uniop 5125 . . 3 𝐴, 𝐵⟩ = {𝐴, 𝐵}
41, 2opi2 5086 . . 3 {𝐴, 𝐵} ∈ ⟨𝐴, 𝐵
53, 4eqeltri 2835 . 2 𝐴, 𝐵⟩ ∈ ⟨𝐴, 𝐵
6 elssuni 4619 . . 3 (⟨𝐴, 𝐵⟩ ∈ 𝐶 → ⟨𝐴, 𝐵⟩ ⊆ 𝐶)
76sseld 3743 . 2 (⟨𝐴, 𝐵⟩ ∈ 𝐶 → (𝐴, 𝐵⟩ ∈ ⟨𝐴, 𝐵⟩ → 𝐴, 𝐵⟩ ∈ 𝐶))
85, 7mpi 20 1 (⟨𝐴, 𝐵⟩ ∈ 𝐶𝐴, 𝐵⟩ ∈ 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2139  Vcvv 3340  {cpr 4323  cop 4327   cuni 4588
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-rex 3056  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589
This theorem is referenced by:  dmrnssfld  5539  unielrel  5821
  Copyright terms: Public domain W3C validator