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Theorem gruop 9506
 Description: A Grothendieck universe contains ordered pairs of its elements. (Contributed by Mario Carneiro, 10-Jun-2013.)
Assertion
Ref Expression
gruop ((𝑈 ∈ Univ ∧ 𝐴𝑈𝐵𝑈) → ⟨𝐴, 𝐵⟩ ∈ 𝑈)

Proof of Theorem gruop
StepHypRef Expression
1 dfopg 4338 . . 3 ((𝐴𝑈𝐵𝑈) → ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}})
213adant1 1072 . 2 ((𝑈 ∈ Univ ∧ 𝐴𝑈𝐵𝑈) → ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}})
3 simp1 1054 . . 3 ((𝑈 ∈ Univ ∧ 𝐴𝑈𝐵𝑈) → 𝑈 ∈ Univ)
4 grusn 9505 . . . 4 ((𝑈 ∈ Univ ∧ 𝐴𝑈) → {𝐴} ∈ 𝑈)
543adant3 1074 . . 3 ((𝑈 ∈ Univ ∧ 𝐴𝑈𝐵𝑈) → {𝐴} ∈ 𝑈)
6 grupr 9498 . . 3 ((𝑈 ∈ Univ ∧ 𝐴𝑈𝐵𝑈) → {𝐴, 𝐵} ∈ 𝑈)
7 grupr 9498 . . 3 ((𝑈 ∈ Univ ∧ {𝐴} ∈ 𝑈 ∧ {𝐴, 𝐵} ∈ 𝑈) → {{𝐴}, {𝐴, 𝐵}} ∈ 𝑈)
83, 5, 6, 7syl3anc 1318 . 2 ((𝑈 ∈ Univ ∧ 𝐴𝑈𝐵𝑈) → {{𝐴}, {𝐴, 𝐵}} ∈ 𝑈)
92, 8eqeltrd 2688 1 ((𝑈 ∈ Univ ∧ 𝐴𝑈𝐵𝑈) → ⟨𝐴, 𝐵⟩ ∈ 𝑈)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  {csn 4125  {cpr 4127  ⟨cop 4131  Univcgru 9491 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-tr 4681  df-iota 5768  df-fv 5812  df-ov 6552  df-gru 9492 This theorem is referenced by:  gruf  9512
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